Subgroups — C₂×C₆₇×#3273 #30740283

Order 17152.

3042 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#546×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3273	C₂	1	16	yes	maximal, direct product factor
4	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#468×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#463×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#452×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#447×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₆₇	C₂²	1	2	yes	—
12	C₂³×D₄×C₆₇	C₂²	1	2	yes	complemented
13	C₂×AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
14	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	16	yes	complemented
15	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
17	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
18	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
19	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
22	#546×C₆₇	—	2	8	no	—
23	#546×C₆₇	C₂²	1	16	yes	complemented
24	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
25	C₂×#123×C₆₇	—	2	2	no	—
26	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
27	C₂×#121×C₆₇	—	2	2	no	—
28	C₂×#121×C₆₇	C₂²	1	2	yes	—
29	C₂×#120×C₆₇	—	2	2	no	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
32	C₂²×C₂²⋊C₄×C₆₇	—	2	2	no	—
33	#495×C₆₇	C₂²	1	8	yes	complemented
34	#468×C₆₇	C₂²	1	16	yes	complemented
35	#463×C₆₇	C₂²	1	16	yes	complemented
36	#452×C₆₇	C₂²	1	16	yes	complemented
37	#447×C₆₇	C₂²	1	16	yes	complemented
38	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
39	#436×C₆₇	C₂²	1	8	yes	complemented
40	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
43	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
44	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
45	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
46	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
47	C₂⁵×C₆₇	—	2	2	no	—
48	C₂×#41×C₆₇	—	2	4	no	—
49	C₂×#41×C₆₇	—	2	8	no	—
50	C₂×#41×C₆₇	—	4	4	no	—
51	C₂×#41×C₆₇	C₂³	1	8	yes	—
52	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×D₄×C₆₇	—	2	8	no	—
54	C₂²×D₄×C₆₇	—	2	2	no	—
55	C₂²×D₄×C₆₇	—	2	2	no	—
56	C₂²×D₄×C₆₇	—	2	2	no	—
57	C₂²×D₄×C₆₇	—	2	4	no	—
58	C₂²×D₄×C₆₇	—	4	4	no	—
59	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
61	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
62	C₂²×D₄×C₆₇	C₂³	1	8	yes	complemented
63	C₂³×C₄×C₆₇	—	2	2	no	—
64	C₂³×C₄×C₆₇	—	2	2	no	—
65	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
66	AGL(1,ℤ/8)×C₆₇	—	2	64	no	—
67	C₂×Q₁₆×C₆₇	—	4	2	no	—
68	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	complemented
69	C₂×SD₁₆×C₆₇	—	2	8	no	—
70	C₂×SD₁₆×C₆₇	—	2	8	no	—
71	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	complemented
72	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	complemented
73	C₂×D₈×C₆₇	—	2	8	no	—
74	C₂×D₈×C₆₇	—	4	2	no	—
75	C₂×D₈×C₆₇	D₄	1	8	yes	complemented
76	C₂×D₈×C₆₇	C₂³	1	8	yes	—
77	C₂×D₈×C₆₇	C₂³	1	4	yes	—
78	C₂×D₈×C₆₇	C₂³	1	4	yes	—
79	C₂×M₁₆×C₆₇	—	2	8	no	—
80	C₂×M₁₆×C₆₇	C₂³	1	8	yes	complemented
81	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
82	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	#124×C₆₇	—	2	4	no	—
85	#124×C₆₇	C₂³	1	8	yes	complemented
86	#123×C₆₇	—	2	16	no	—
87	C₂²⋊Q₈×C₆₇	—	2	4	no	—
88	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
89	#121×C₆₇	—	2	4	no	—
90	#121×C₆₇	—	2	16	no	—
91	#121×C₆₇	C₂³	1	8	yes	—
92	#120×C₆₇	—	2	16	no	—
93	C₄×D₄×C₆₇	—	2	4	no	—
94	C₄×D₄×C₆₇	D₄	1	4	yes	—
95	C₄×D₄×C₆₇	C₂³	1	4	yes	—
96	C₄×D₄×C₆₇	C₂³	1	8	yes	complemented
97	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
101	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
102	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
103	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
104	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
106	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
107	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	#107×C₆₇	C₂³	1	4	yes	complemented
109	C₄⋊C₈×C₆₇	C₂³	1	4	yes	complemented
110	#103×C₆₇	C₂³	1	8	yes	complemented
111	#102×C₆₇	C₂³	1	8	yes	complemented
112	#102×C₆₇	C₂³	1	8	yes	complemented
113	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	complemented
114	C₄×C₈×C₆₇	C₂³	1	4	yes	complemented
115	C₂⁴×C₆₇	—	2	2	no	—
116	C₂⁴×C₆₇	—	2	2	no	—
117	C₂⁴×C₆₇	—	2	8	no	—
118	C₂⁴×C₆₇	—	4	8	no	—
119	C₂⁴×C₆₇	—	4	8	no	—
120	C₂⁴×C₆₇	—	8	4	no	—
121	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
122	#41×C₆₇	—	2	16	no	—
123	#41×C₆₇	—	4	8	no	—
124	#41×C₆₇	—	4	16	no	—
125	C₂×Q₈×C₆₇	—	2	2	no	—
126	C₂×Q₈×C₆₇	—	2	4	no	—
127	C₂×Q₈×C₆₇	—	2	4	no	—
128	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
129	C₂×D₄×C₆₇	—	2	16	no	—
130	C₂×D₄×C₆₇	—	2	8	no	—
131	C₂×D₄×C₆₇	—	2	8	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	—	2	4	no	—
134	C₂×D₄×C₆₇	—	2	8	no	—
135	C₂×D₄×C₆₇	—	2	2	no	—
136	C₂×D₄×C₆₇	—	2	4	no	—
137	C₂×D₄×C₆₇	—	2	4	no	—
138	C₂×D₄×C₆₇	—	4	8	no	—
139	C₂×D₄×C₆₇	—	4	16	no	—
140	C₂×D₄×C₆₇	—	4	2	no	—
141	C₂×D₄×C₆₇	—	4	8	no	—
142	C₂×D₄×C₆₇	—	4	8	no	—
143	C₂×D₄×C₆₇	—	8	4	no	—
144	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
145	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
146	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
147	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	2	no	—
152	C₂²×C₄×C₆₇	—	2	8	no	—
153	C₂²×C₄×C₆₇	—	2	8	no	—
154	C₂²×C₄×C₆₇	—	4	8	no	—
155	C₂²×C₄×C₆₇	—	8	4	no	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
162	Q₁₆×C₆₇	—	4	4	no	—
163	SD₁₆×C₆₇	—	2	16	no	—
164	SD₁₆×C₆₇	—	2	16	no	—
165	D₈×C₆₇	—	2	16	no	—
166	D₈×C₆₇	—	4	4	no	—
167	D₈×C₆₇	C₂×D₄	1	16	yes	complemented
168	M₁₆×C₆₇	—	2	16	no	—
169	C₂×C₈×C₆₇	—	2	4	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
172	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
173	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
174	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
175	C₄⋊C₄×C₆₇	—	2	8	no	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
179	C₂²⋊C₄×C₆₇	—	2	8	no	—
180	C₂²⋊C₄×C₆₇	—	2	8	no	—
181	C₂²⋊C₄×C₆₇	—	2	8	no	—
182	C₂²⋊C₄×C₆₇	—	2	8	no	—
183	C₂²⋊C₄×C₆₇	—	2	8	no	—
184	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
185	C₄²×C₆₇	C₂×D₄	1	2	yes	—
186	C₄²×C₆₇	C₂⁴	1	2	yes	—
187	C₂³×C₆₇	—	2	4	no	—
188	C₂³×C₆₇	—	2	8	no	—
189	C₂³×C₆₇	—	2	4	no	—
190	C₂³×C₆₇	—	2	4	no	—
191	C₂³×C₆₇	—	2	4	no	—
192	C₂³×C₆₇	—	2	4	no	—
193	C₂³×C₆₇	—	4	16	no	—
194	C₂³×C₆₇	—	4	16	no	—
195	C₂³×C₆₇	—	4	4	no	—
196	C₂³×C₆₇	—	4	4	no	—
197	C₂³×C₆₇	—	4	4	no	—
198	C₂³×C₆₇	—	4	4	no	—
199	C₂³×C₆₇	—	4	4	no	—
200	C₂³×C₆₇	—	8	8	no	—
201	C₂³×C₆₇	—	8	32	no	—
202	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
203	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	Q₈×C₆₇	—	2	4	no	—
205	Q₈×C₆₇	—	2	4	no	—
206	Q₈×C₆₇	—	4	4	no	—
207	D₄×C₆₇	—	2	8	no	—
208	D₄×C₆₇	—	2	4	no	—
209	D₄×C₆₇	—	2	4	no	—
210	D₄×C₆₇	—	2	4	no	—
211	D₄×C₆₇	—	4	4	no	—
212	D₄×C₆₇	—	4	16	no	—
213	D₄×C₆₇	—	4	16	no	—
214	D₄×C₆₇	—	4	4	no	—
215	D₄×C₆₇	—	8	8	no	—
216	D₄×C₆₇	C₂²×D₄	1	8	yes	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	8	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	4	16	no	—
226	C₂×C₄×C₆₇	—	4	4	no	—
227	C₂×C₄×C₆₇	—	4	4	no	—
228	C₂×C₄×C₆₇	—	8	8	no	—
229	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
232	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
235	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
236	C₈×C₆₇	—	2	4	no	—
237	C₈×C₆₇	—	2	2	no	—
238	C₈×C₆₇	C₂²×D₄	1	4	yes	—
239	C₂²×C₆₇	—	2	4	no	—
240	C₂²×C₆₇	—	2	8	no	—
241	C₂²×C₆₇	—	2	4	no	—
242	C₂²×C₆₇	—	2	4	no	—
243	C₂²×C₆₇	—	4	8	no	—
244	C₂²×C₆₇	—	4	16	no	—
245	C₂²×C₆₇	—	4	4	no	—
246	C₂²×C₆₇	—	4	4	no	—
247	C₂²×C₆₇	—	4	4	no	—
248	C₂²×C₆₇	—	4	4	no	—
249	C₂²×C₆₇	—	4	4	no	—
250	C₂²×C₆₇	—	8	32	no	—
251	C₂²×C₆₇	—	8	8	no	—
252	C₂²×C₆₇	D₄²	1	2	yes	—
253	C₂²×C₆₇	#575	1	2	yes	—
254	C₂²×C₆₇	#576	1	2	yes	—
255	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
256	C₄×C₆₇	—	2	4	no	—
257	C₄×C₆₇	—	2	2	no	—
258	C₄×C₆₇	—	2	2	no	—
259	C₄×C₆₇	—	4	4	no	—
260	C₄×C₆₇	—	4	4	no	—
261	C₄×C₆₇	D₄²	1	2	yes	—
262	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
263	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
264	C₂×#3273	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
265	C₂×C₆₇	—	4	4	no	—
266	C₂×C₆₇	—	4	8	no	—
267	C₂×C₆₇	—	8	4	no	—
268	C₂×C₆₇	#3273	1	4	yes	direct product factor
269	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
270	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
271	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
272	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	2	yes	complemented
273	C₂×#546	C₂×C₆₇	1	2	yes	complemented
274	#3273	C₂×C₆₇	1	16	yes	direct product factor
275	C₂×#495	C₂×C₆₇	1	1	yes	characteristic, complemented
276	C₂×#468	C₂×C₆₇	1	2	yes	complemented
277	C₂×#463	C₂×C₆₇	1	2	yes	complemented
278	C₂×#452	C₂×C₆₇	1	2	yes	complemented
279	C₂×#447	C₂×C₆₇	1	2	yes	complemented
280	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
281	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
282	C₆₇	C₂×#3273	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²×#41	C₂²×C₆₇	1	2	yes	—
284	C₂³×D₄	C₂²×C₆₇	1	2	yes	complemented
285	C₂×AGL(1,ℤ/8)	—	2	16	no	—
286	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	16	yes	complemented
287	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
288	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
289	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
290	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
291	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
292	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
293	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
294	#546	—	2	8	no	—
295	#546	C₂²×C₆₇	1	16	yes	complemented
296	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
297	C₂×#123	—	2	2	no	—
298	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
299	C₂×#121	—	2	2	no	—
300	C₂×#121	C₂²×C₆₇	1	2	yes	—
301	C₂×#120	—	2	2	no	—
302	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
304	C₂²×C₂²⋊C₄	—	2	2	no	—
305	#495	C₂²×C₆₇	1	8	yes	complemented
306	#468	C₂²×C₆₇	1	16	yes	complemented
307	#463	C₂²×C₆₇	1	16	yes	complemented
308	#452	C₂²×C₆₇	1	16	yes	complemented
309	#447	C₂²×C₆₇	1	16	yes	complemented
310	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
311	#436	C₂²×C₆₇	1	8	yes	complemented
312	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
313	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
314	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
315	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
316	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
317	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
318	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
319	C₂⁵	—	2	2	no	—
320	C₂×#41	—	2	4	no	—
321	C₂×#41	—	2	8	no	—
322	C₂×#41	—	4	4	no	—
323	C₂×#41	C₂³×C₆₇	1	8	yes	—
324	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
325	C₂²×D₄	—	2	8	no	—
326	C₂²×D₄	—	2	2	no	—
327	C₂²×D₄	—	2	2	no	—
328	C₂²×D₄	—	2	2	no	—
329	C₂²×D₄	—	2	4	no	—
330	C₂²×D₄	—	4	4	no	—
331	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
332	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
333	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
334	C₂²×D₄	C₂³×C₆₇	1	8	yes	complemented
335	C₂³×C₄	—	2	2	no	—
336	C₂³×C₄	—	2	2	no	—
337	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
338	AGL(1,ℤ/8)	—	2	64	no	—
339	C₂×Q₁₆	—	4	2	no	—
340	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	complemented
341	C₂×SD₁₆	—	2	8	no	—
342	C₂×SD₁₆	—	2	8	no	—
343	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	complemented
344	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	complemented
345	C₂×D₈	—	2	8	no	—
346	C₂×D₈	—	4	2	no	—
347	C₂×D₈	D₄×C₆₇	1	8	yes	complemented
348	C₂×D₈	C₂³×C₆₇	1	8	yes	—
349	C₂×D₈	C₂³×C₆₇	1	4	yes	—
350	C₂×D₈	C₂³×C₆₇	1	4	yes	—
351	C₂×M₁₆	—	2	8	no	—
352	C₂×M₁₆	C₂³×C₆₇	1	8	yes	complemented
353	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
354	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
355	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
356	#124	—	2	4	no	—
357	#124	C₂³×C₆₇	1	8	yes	complemented
358	#123	—	2	16	no	—
359	C₂²⋊Q₈	—	2	4	no	—
360	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
361	#121	—	2	4	no	—
362	#121	—	2	16	no	—
363	#121	C₂³×C₆₇	1	8	yes	—
364	#120	—	2	16	no	—
365	C₄×D₄	—	2	4	no	—
366	C₄×D₄	D₄×C₆₇	1	4	yes	—
367	C₄×D₄	C₂³×C₆₇	1	4	yes	—
368	C₄×D₄	C₂³×C₆₇	1	8	yes	complemented
369	C₂×C₄⋊C₄	—	2	2	no	—
370	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	2	no	—
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₂²⋊C₄	—	2	2	no	—
377	C₂×C₂²⋊C₄	—	2	16	no	—
378	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
379	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
380	#107	C₂³×C₆₇	1	4	yes	complemented
381	C₄⋊C₈	C₂³×C₆₇	1	4	yes	complemented
382	#103	C₂³×C₆₇	1	8	yes	complemented
383	#102	C₂³×C₆₇	1	8	yes	complemented
384	#102	C₂³×C₆₇	1	8	yes	complemented
385	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	complemented
386	C₄×C₈	C₂³×C₆₇	1	4	yes	complemented
387	C₂⁴	—	2	2	no	—
388	C₂⁴	—	2	2	no	—
389	C₂⁴	—	2	8	no	—
390	C₂⁴	—	4	8	no	—
391	C₂⁴	—	4	8	no	—
392	C₂⁴	—	8	4	no	—
393	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
394	#41	—	2	16	no	—
395	#41	—	4	8	no	—
396	#41	—	4	16	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
401	C₂×D₄	—	2	16	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	8	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	4	8	no	—
411	C₂×D₄	—	4	16	no	—
412	C₂×D₄	—	4	2	no	—
413	C₂×D₄	—	4	8	no	—
414	C₂×D₄	—	4	8	no	—
415	C₂×D₄	—	8	4	no	—
416	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
417	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
418	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
419	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	2	8	no	—
426	C₂²×C₄	—	4	8	no	—
427	C₂²×C₄	—	8	4	no	—
428	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
429	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
431	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
432	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
433	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
434	Q₁₆	—	4	4	no	—
435	SD₁₆	—	2	16	no	—
436	SD₁₆	—	2	16	no	—
437	D₈	—	2	16	no	—
438	D₈	—	4	4	no	—
439	D₈	C₂×D₄×C₆₇	1	16	yes	complemented
440	M₁₆	—	2	16	no	—
441	C₂×C₈	—	2	4	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
444	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
445	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
446	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
447	C₄⋊C₄	—	2	8	no	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
450	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	2	8	no	—
455	C₂²⋊C₄	—	2	8	no	—
456	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
457	C₄²	C₂×D₄×C₆₇	1	2	yes	—
458	C₄²	C₂⁴×C₆₇	1	2	yes	—
459	C₂³	—	2	4	no	—
460	C₂³	—	2	8	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	4	no	—
463	C₂³	—	2	4	no	—
464	C₂³	—	2	4	no	—
465	C₂³	—	4	16	no	—
466	C₂³	—	4	16	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	4	4	no	—
469	C₂³	—	4	4	no	—
470	C₂³	—	4	4	no	—
471	C₂³	—	4	4	no	—
472	C₂³	—	8	8	no	—
473	C₂³	—	8	32	no	—
474	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
475	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
476	Q₈	—	2	4	no	—
477	Q₈	—	2	4	no	—
478	Q₈	—	4	4	no	—
479	D₄	—	2	8	no	—
480	D₄	—	2	4	no	—
481	D₄	—	2	4	no	—
482	D₄	—	2	4	no	—
483	D₄	—	4	4	no	—
484	D₄	—	4	16	no	—
485	D₄	—	4	16	no	—
486	D₄	—	4	4	no	—
487	D₄	—	8	8	no	—
488	D₄	C₂²×D₄×C₆₇	1	8	yes	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	8	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	4	16	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	4	4	no	—
500	C₂×C₄	—	8	8	no	—
501	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
503	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
504	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
506	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
507	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
508	C₈	—	2	4	no	—
509	C₈	—	2	2	no	—
510	C₈	C₂²×D₄×C₆₇	1	4	yes	—
511	C₂²	—	2	4	no	—
512	C₂²	—	2	4	no	—
513	C₂²	—	2	4	no	—
514	C₂²	—	2	8	no	—
515	C₂²	—	4	8	no	—
516	C₂²	—	4	16	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	4	4	no	—
520	C₂²	—	4	4	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	8	32	no	—
523	C₂²	—	8	8	no	—
524	C₂²	D₄²×C₆₇	1	2	yes	—
525	C₂²	#575×C₆₇	1	2	yes	—
526	C₂²	#576×C₆₇	1	2	yes	—
527	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
528	C₄	—	2	2	no	—
529	C₄	—	2	2	no	—
530	C₄	—	2	4	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	4	no	—
533	C₄	D₄²×C₆₇	1	2	yes	—
534	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
535	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
536	C₂	—	4	4	no	—
537	C₂	—	4	8	no	—
538	C₂	—	8	4	no	—
539	C₂	C₆₇×#3273	1	4	yes	direct product factor
540	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
541	C₂	C₂×#575×C₆₇	1	1	yes	characteristic
542	C₂	C₂×#576×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
2. C₂×#546×C₆₇	C₂	semidirect
3. C₆₇×#3273	C₂	direct (N × H)
4. C₂×#495×C₆₇	C₂	semidirect
5. C₂×#468×C₆₇	C₂	semidirect
6. C₂×#463×C₆₇	C₂	semidirect
7. C₂×#452×C₆₇	C₂	semidirect
8. C₂×#447×C₆₇	C₂	semidirect
9. C₂×C₄×D₈×C₆₇	C₂	semidirect
10. C₂×#436×C₆₇	C₂	semidirect
12. C₂³×D₄×C₆₇	C₂²	semidirect
14. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
15. C₂²×Q₁₆×C₆₇	C₂²	semidirect
16. C₂²×SD₁₆×C₆₇	C₂²	semidirect
17. C₂²×SD₁₆×C₆₇	C₂²	semidirect
21. C₂²×M₁₆×C₆₇	C₂²	semidirect
23. #546×C₆₇	C₂²	semidirect
24. C₂×#124×C₆₇	C₂²	semidirect
26. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
31. C₂×C₄×D₄×C₆₇	C₂²	semidirect
33. #495×C₆₇	C₂²	semidirect
34. #468×C₆₇	C₂²	semidirect
35. #463×C₆₇	C₂²	semidirect
36. #452×C₆₇	C₂²	semidirect
37. #447×C₆₇	C₂²	semidirect
38. C₄×D₈×C₆₇	C₂²	semidirect
39. #436×C₆₇	C₂²	semidirect
40. C₂×#107×C₆₇	C₂²	semidirect
41. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
42. C₂×#103×C₆₇	C₂²	semidirect
43. C₂×#102×C₆₇	C₂²	semidirect
44. C₂×#102×C₆₇	C₂²	semidirect
45. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
46. C₂×C₄×C₈×C₆₇	C₂²	semidirect
62. C₂²×D₄×C₆₇	C₂³	semidirect
68. C₂×Q₁₆×C₆₇	C₂³	semidirect
71. C₂×SD₁₆×C₆₇	C₂³	semidirect
72. C₂×SD₁₆×C₆₇	C₂³	semidirect
75. C₂×D₈×C₆₇	D₄	semidirect
80. C₂×M₁₆×C₆₇	C₂³	semidirect
85. #124×C₆₇	C₂³	semidirect
88. C₂²⋊Q₈×C₆₇	C₂³	semidirect
96. C₄×D₄×C₆₇	C₂³	semidirect
108. #107×C₆₇	C₂³	semidirect
109. C₄⋊C₈×C₆₇	C₂³	semidirect
110. #103×C₆₇	C₂³	semidirect
111. #102×C₆₇	C₂³	semidirect
112. #102×C₆₇	C₂³	semidirect
113. C₂²⋊C₈×C₆₇	C₂³	semidirect
114. C₄×C₈×C₆₇	C₂³	semidirect
167. D₈×C₆₇	C₂×D₄	semidirect
264. C₂×#3273	C₆₇	direct (N × H)
268. C₂×C₆₇	#3273	direct (N × H)
272. C₂²×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
273. C₂×#546	C₂×C₆₇	semidirect
274. #3273	C₂×C₆₇	direct (N × H)
275. C₂×#495	C₂×C₆₇	semidirect
276. C₂×#468	C₂×C₆₇	semidirect
277. C₂×#463	C₂×C₆₇	semidirect
278. C₂×#452	C₂×C₆₇	semidirect
279. C₂×#447	C₂×C₆₇	semidirect
280. C₂×C₄×D₈	C₂×C₆₇	semidirect
281. C₂×#436	C₂×C₆₇	semidirect
282. C₆₇	C₂×#3273	direct (N × H)
284. C₂³×D₄	C₂²×C₆₇	semidirect
286. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
287. C₂²×Q₁₆	C₂²×C₆₇	semidirect
288. C₂²×SD₁₆	C₂²×C₆₇	semidirect
289. C₂²×SD₁₆	C₂²×C₆₇	semidirect
293. C₂²×M₁₆	C₂²×C₆₇	semidirect
295. #546	C₂²×C₆₇	semidirect
296. C₂×#124	C₂²×C₆₇	semidirect
298. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
303. C₂×C₄×D₄	C₂²×C₆₇	semidirect
305. #495	C₂²×C₆₇	semidirect
306. #468	C₂²×C₆₇	semidirect
307. #463	C₂²×C₆₇	semidirect
308. #452	C₂²×C₆₇	semidirect
309. #447	C₂²×C₆₇	semidirect
310. C₄×D₈	C₂²×C₆₇	semidirect
311. #436	C₂²×C₆₇	semidirect
312. C₂×#107	C₂²×C₆₇	semidirect
313. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
314. C₂×#103	C₂²×C₆₇	semidirect
315. C₂×#102	C₂²×C₆₇	semidirect
316. C₂×#102	C₂²×C₆₇	semidirect
317. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
318. C₂×C₄×C₈	C₂²×C₆₇	semidirect
334. C₂²×D₄	C₂³×C₆₇	semidirect
340. C₂×Q₁₆	C₂³×C₆₇	semidirect
343. C₂×SD₁₆	C₂³×C₆₇	semidirect
344. C₂×SD₁₆	C₂³×C₆₇	semidirect
347. C₂×D₈	D₄×C₆₇	semidirect
352. C₂×M₁₆	C₂³×C₆₇	semidirect
357. #124	C₂³×C₆₇	semidirect
360. C₂²⋊Q₈	C₂³×C₆₇	semidirect
368. C₄×D₄	C₂³×C₆₇	semidirect
380. #107	C₂³×C₆₇	semidirect
381. C₄⋊C₈	C₂³×C₆₇	semidirect
382. #103	C₂³×C₆₇	semidirect
383. #102	C₂³×C₆₇	semidirect
384. #102	C₂³×C₆₇	semidirect
385. C₂²⋊C₈	C₂³×C₆₇	semidirect
386. C₄×C₈	C₂³×C₆₇	semidirect
439. D₈	C₂×D₄×C₆₇	semidirect
539. C₂	C₆₇×#3273	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×AGL(1,ℤ/8)×C₆₇	2
C₂×#546×C₆₇	2
C₆₇×#3273	2
C₂×#495×C₆₇	2
C₂×#468×C₆₇	2
C₂×#463×C₆₇	2
C₂×#452×C₆₇	2
C₂×#447×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂×#436×C₆₇	2
C₂×#3273	67

Subgroups — C2×C67×#3273 #30740283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3273 #30740283