Subgroups — C₂×C₆₇×#3277 #30740287

Order 17152.

2762 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#548×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3277	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×#464×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#449×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#451×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₁₆×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×Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂×#137×C₆₇	—	2	16	no	—
14	C₂×#137×C₆₇	C₂²	1	16	yes	complemented
15	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
22	#548×C₆₇	—	2	8	no	—
23	#548×C₆₇	C₂²	1	16	yes	complemented
24	C₂×#124×C₆₇	—	2	2	no	—
25	C₂×#124×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
27	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
28	C₂×#121×C₆₇	—	2	2	no	—
29	C₂×#121×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	—	2	2	no	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	#495×C₆₇	C₂²	1	8	yes	complemented
34	#468×C₆₇	C₂²	1	16	yes	complemented
35	#464×C₆₇	C₂²	1	16	yes	complemented
36	#451×C₆₇	C₂²	1	16	yes	complemented
37	#449×C₆₇	C₂²	1	16	yes	complemented
38	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
39	#436×C₆₇	C₂²	1	8	yes	complemented
40	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#103×C₆₇	C₂²	1	2	yes	—
43	C₂×#103×C₆₇	C₂²	1	2	yes	—
44	C₂×#102×C₆₇	C₂²	1	2	yes	—
45	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
46	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#41×C₆₇	—	2	4	no	—
48	C₂×#41×C₆₇	—	2	8	no	—
49	C₂×#41×C₆₇	—	4	4	no	—
50	C₂×#41×C₆₇	C₂³	1	8	yes	—
51	C₂²×Q₈×C₆₇	—	2	2	no	—
52	C₂²×Q₈×C₆₇	—	2	4	no	—
53	C₂²×Q₈×C₆₇	—	2	8	no	—
54	C₂²×Q₈×C₆₇	—	4	4	no	—
55	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
56	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
57	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×D₄×C₆₇	—	2	2	no	—
59	C₂²×D₄×C₆₇	—	2	2	no	—
60	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₇	—	2	2	no	—
63	C₂³×C₄×C₆₇	—	2	2	no	—
64	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
65	#137×C₆₇	—	2	64	no	—
66	C₂×Q₁₆×C₆₇	—	2	8	no	—
67	C₂×Q₁₆×C₆₇	—	4	2	no	—
68	C₂×Q₁₆×C₆₇	D₄	1	8	yes	complemented
69	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
70	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
71	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
72	C₂×SD₁₆×C₆₇	—	2	8	no	—
73	C₂×SD₁₆×C₆₇	—	2	8	no	—
74	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
76	C₂×D₈×C₆₇	—	4	2	no	—
77	C₂×D₈×C₆₇	C₂³	1	4	yes	—
78	C₂×M₁₆×C₆₇	—	2	8	no	—
79	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
80	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
83	#124×C₆₇	—	2	16	no	—
84	#124×C₆₇	—	2	4	no	—
85	#124×C₆₇	C₂³	1	8	yes	—
86	C₂²⋊Q₈×C₆₇	—	2	4	no	—
87	C₂²⋊Q₈×C₆₇	—	2	16	no	—
88	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
89	#121×C₆₇	—	2	16	no	—
90	#121×C₆₇	—	2	4	no	—
91	#121×C₆₇	C₂³	1	8	yes	—
92	C₄×Q₈×C₆₇	—	2	4	no	—
93	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
94	C₄×D₄×C₆₇	—	2	16	no	—
95	C₄×D₄×C₆₇	D₄	1	4	yes	—
96	C₄×D₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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	D₄	1	2	yes	—
106	C₂×C₄²×C₆₇	—	2	2	no	—
107	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	#107×C₆₇	C₂³	1	4	yes	—
109	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
110	#103×C₆₇	C₂³	1	8	yes	—
111	#103×C₆₇	C₂³	1	8	yes	—
112	#102×C₆₇	C₂³	1	8	yes	—
113	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
114	C₄×C₈×C₆₇	C₂³	1	4	yes	—
115	C₂⁴×C₆₇	—	2	2	no	—
116	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
117	#41×C₆₇	—	2	16	no	—
118	#41×C₆₇	—	4	8	no	—
119	#41×C₆₇	—	4	16	no	—
120	C₂×Q₈×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	—	2	8	no	—
122	C₂×Q₈×C₆₇	—	2	2	no	—
123	C₂×Q₈×C₆₇	—	2	4	no	—
124	C₂×Q₈×C₆₇	—	2	16	no	—
125	C₂×Q₈×C₆₇	—	2	4	no	—
126	C₂×Q₈×C₆₇	—	2	4	no	—
127	C₂×Q₈×C₆₇	—	4	8	no	—
128	C₂×Q₈×C₆₇	—	4	8	no	—
129	C₂×Q₈×C₆₇	—	4	16	no	—
130	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
131	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
132	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
133	C₂×D₄×C₆₇	—	2	8	no	—
134	C₂×D₄×C₆₇	—	2	8	no	—
135	C₂×D₄×C₆₇	—	2	4	no	—
136	C₂×D₄×C₆₇	—	2	2	no	—
137	C₂×D₄×C₆₇	—	2	4	no	—
138	C₂×D₄×C₆₇	—	4	2	no	—
139	C₂×D₄×C₆₇	—	4	8	no	—
140	C₂×D₄×C₆₇	—	4	8	no	—
141	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
142	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	4	no	—
146	C₂²×C₄×C₆₇	—	2	8	no	—
147	C₂²×C₄×C₆₇	—	2	2	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	8	no	—
150	C₂²×C₄×C₆₇	—	4	8	no	—
151	C₂²×C₄×C₆₇	—	4	8	no	—
152	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₆₇	—	2	16	no	—
159	Q₁₆×C₆₇	—	4	4	no	—
160	Q₁₆×C₆₇	C₂×D₄	1	16	yes	complemented
161	SD₁₆×C₆₇	—	2	16	no	—
162	SD₁₆×C₆₇	—	2	16	no	—
163	D₈×C₆₇	—	4	4	no	—
164	M₁₆×C₆₇	—	2	16	no	—
165	C₂×C₈×C₆₇	—	2	4	no	—
166	C₂×C₈×C₆₇	—	2	2	no	—
167	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
168	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
169	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
170	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
171	C₄⋊C₄×C₆₇	—	2	8	no	—
172	C₄⋊C₄×C₆₇	—	2	8	no	—
173	C₄⋊C₄×C₆₇	—	2	2	no	—
174	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
175	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
176	C₂²⋊C₄×C₆₇	—	2	8	no	—
177	C₂²⋊C₄×C₆₇	—	2	8	no	—
178	C₂²⋊C₄×C₆₇	—	2	8	no	—
179	C₂²⋊C₄×C₆₇	—	2	8	no	—
180	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
181	C₄²×C₆₇	—	2	8	no	—
182	C₄²×C₆₇	C₂×D₄	1	2	yes	—
183	C₄²×C₆₇	C₂⁴	1	2	yes	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	2	4	no	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	2	4	no	—
188	C₂³×C₆₇	—	4	4	no	—
189	C₂³×C₆₇	—	4	4	no	—
190	C₂³×C₆₇	—	4	4	no	—
191	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
192	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	Q₈×C₆₇	—	2	8	no	—
194	Q₈×C₆₇	—	2	4	no	—
195	Q₈×C₆₇	—	2	4	no	—
196	Q₈×C₆₇	—	2	4	no	—
197	Q₈×C₆₇	—	4	16	no	—
198	Q₈×C₆₇	—	4	4	no	—
199	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
200	D₄×C₆₇	—	2	4	no	—
201	D₄×C₆₇	—	2	4	no	—
202	D₄×C₆₇	—	4	4	no	—
203	D₄×C₆₇	—	4	16	no	—
204	D₄×C₆₇	—	4	16	no	—
205	D₄×C₆₇	—	4	4	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	8	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	8	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	4	no	—
216	C₂×C₄×C₆₇	—	4	16	no	—
217	C₂×C₄×C₆₇	—	4	16	no	—
218	C₂×C₄×C₆₇	—	4	4	no	—
219	C₂×C₄×C₆₇	—	4	4	no	—
220	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
226	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
227	C₈×C₆₇	—	2	4	no	—
228	C₈×C₆₇	—	2	2	no	—
229	C₈×C₆₇	C₂²×D₄	1	4	yes	—
230	C₂²×C₆₇	—	2	4	no	—
231	C₂²×C₆₇	—	2	4	no	—
232	C₂²×C₆₇	—	2	4	no	—
233	C₂²×C₆₇	—	4	8	no	—
234	C₂²×C₆₇	—	4	4	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	—	4	4	no	—
237	C₂²×C₆₇	—	8	8	no	—
238	C₂²×C₆₇	D₄²	1	2	yes	—
239	C₂²×C₆₇	#575	1	2	yes	—
240	C₂²×C₆₇	#578	1	2	yes	—
241	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
242	C₄×C₆₇	—	2	4	no	—
243	C₄×C₆₇	—	2	8	no	—
244	C₄×C₆₇	—	2	2	no	—
245	C₄×C₆₇	—	2	2	no	—
246	C₄×C₆₇	—	4	4	no	—
247	C₄×C₆₇	—	4	4	no	—
248	C₄×C₆₇	D₄²	1	2	yes	—
249	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
250	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
251	C₂×#3277	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂×C₆₇	—	4	4	no	—
253	C₂×C₆₇	—	8	4	no	—
254	C₂×C₆₇	#3277	1	4	yes	direct product factor
255	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
256	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
257	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
258	C₂²×#137	C₂×C₆₇	1	2	yes	complemented
259	C₂×#548	C₂×C₆₇	1	2	yes	complemented
260	#3277	C₂×C₆₇	1	16	yes	direct product factor
261	C₂×#495	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₂×#468	C₂×C₆₇	1	2	yes	complemented
263	C₂×#464	C₂×C₆₇	1	2	yes	complemented
264	C₂×#449	C₂×C₆₇	1	2	yes	complemented
265	C₂×#451	C₂×C₆₇	1	2	yes	complemented
266	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
268	C₆₇	C₂×#3277	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²×#41	C₂²×C₆₇	1	2	yes	—
270	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
271	C₂×#137	—	2	16	no	—
272	C₂×#137	C₂²×C₆₇	1	16	yes	complemented
273	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
274	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
275	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
276	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
277	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
278	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
279	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
280	#548	—	2	8	no	—
281	#548	C₂²×C₆₇	1	16	yes	complemented
282	C₂×#124	—	2	2	no	—
283	C₂×#124	C₂²×C₆₇	1	2	yes	—
284	C₂×C₂²⋊Q₈	—	2	2	no	—
285	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
286	C₂×#121	—	2	2	no	—
287	C₂×#121	C₂²×C₆₇	1	2	yes	—
288	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
289	C₂×C₄×D₄	—	2	2	no	—
290	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
291	#495	C₂²×C₆₇	1	8	yes	complemented
292	#468	C₂²×C₆₇	1	16	yes	complemented
293	#464	C₂²×C₆₇	1	16	yes	complemented
294	#451	C₂²×C₆₇	1	16	yes	complemented
295	#449	C₂²×C₆₇	1	16	yes	complemented
296	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
297	#436	C₂²×C₆₇	1	8	yes	complemented
298	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×#103	C₂²×C₆₇	1	2	yes	—
301	C₂×#103	C₂²×C₆₇	1	2	yes	—
302	C₂×#102	C₂²×C₆₇	1	2	yes	—
303	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
304	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
305	C₂×#41	—	2	4	no	—
306	C₂×#41	—	2	8	no	—
307	C₂×#41	—	4	4	no	—
308	C₂×#41	C₂³×C₆₇	1	8	yes	—
309	C₂²×Q₈	—	2	2	no	—
310	C₂²×Q₈	—	2	4	no	—
311	C₂²×Q₈	—	2	8	no	—
312	C₂²×Q₈	—	4	4	no	—
313	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
314	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
315	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
316	C₂²×D₄	—	2	2	no	—
317	C₂²×D₄	—	2	2	no	—
318	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
319	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
320	C₂³×C₄	—	2	2	no	—
321	C₂³×C₄	—	2	2	no	—
322	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
323	#137	—	2	64	no	—
324	C₂×Q₁₆	—	2	8	no	—
325	C₂×Q₁₆	—	4	2	no	—
326	C₂×Q₁₆	D₄×C₆₇	1	8	yes	complemented
327	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
328	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
329	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
330	C₂×SD₁₆	—	2	8	no	—
331	C₂×SD₁₆	—	2	8	no	—
332	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
333	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
334	C₂×D₈	—	4	2	no	—
335	C₂×D₈	C₂³×C₆₇	1	4	yes	—
336	C₂×M₁₆	—	2	8	no	—
337	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
338	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
339	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
340	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
341	#124	—	2	16	no	—
342	#124	—	2	4	no	—
343	#124	C₂³×C₆₇	1	8	yes	—
344	C₂²⋊Q₈	—	2	4	no	—
345	C₂²⋊Q₈	—	2	16	no	—
346	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
347	#121	—	2	16	no	—
348	#121	—	2	4	no	—
349	#121	C₂³×C₆₇	1	8	yes	—
350	C₄×Q₈	—	2	4	no	—
351	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
352	C₄×D₄	—	2	16	no	—
353	C₄×D₄	D₄×C₆₇	1	4	yes	—
354	C₄×D₄	C₂³×C₆₇	1	4	yes	—
355	C₂×C₄⋊C₄	—	2	2	no	—
356	C₂×C₄⋊C₄	—	2	2	no	—
357	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
364	C₂×C₄²	—	2	2	no	—
365	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
366	#107	C₂³×C₆₇	1	4	yes	—
367	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
368	#103	C₂³×C₆₇	1	8	yes	—
369	#103	C₂³×C₆₇	1	8	yes	—
370	#102	C₂³×C₆₇	1	8	yes	—
371	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
372	C₄×C₈	C₂³×C₆₇	1	4	yes	—
373	C₂⁴	—	2	2	no	—
374	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
375	#41	—	2	16	no	—
376	#41	—	4	8	no	—
377	#41	—	4	16	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	8	no	—
380	C₂×Q₈	—	2	2	no	—
381	C₂×Q₈	—	2	4	no	—
382	C₂×Q₈	—	2	16	no	—
383	C₂×Q₈	—	2	4	no	—
384	C₂×Q₈	—	2	4	no	—
385	C₂×Q₈	—	4	8	no	—
386	C₂×Q₈	—	4	8	no	—
387	C₂×Q₈	—	4	16	no	—
388	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
389	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
390	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	—	2	8	no	—
393	C₂×D₄	—	2	4	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	4	no	—
396	C₂×D₄	—	4	2	no	—
397	C₂×D₄	—	4	8	no	—
398	C₂×D₄	—	4	8	no	—
399	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
400	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	—	2	8	no	—
408	C₂²×C₄	—	4	8	no	—
409	C₂²×C₄	—	4	8	no	—
410	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
411	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
412	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
413	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
414	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
415	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
416	Q₁₆	—	2	16	no	—
417	Q₁₆	—	4	4	no	—
418	Q₁₆	C₂×D₄×C₆₇	1	16	yes	complemented
419	SD₁₆	—	2	16	no	—
420	SD₁₆	—	2	16	no	—
421	D₈	—	4	4	no	—
422	M₁₆	—	2	16	no	—
423	C₂×C₈	—	2	4	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
426	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
427	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
428	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
429	C₄⋊C₄	—	2	8	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
433	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
434	C₂²⋊C₄	—	2	8	no	—
435	C₂²⋊C₄	—	2	8	no	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
439	C₄²	—	2	8	no	—
440	C₄²	C₂×D₄×C₆₇	1	2	yes	—
441	C₄²	C₂⁴×C₆₇	1	2	yes	—
442	C₂³	—	2	4	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	4	4	no	—
447	C₂³	—	4	4	no	—
448	C₂³	—	4	4	no	—
449	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
450	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
451	Q₈	—	2	8	no	—
452	Q₈	—	2	4	no	—
453	Q₈	—	2	4	no	—
454	Q₈	—	2	4	no	—
455	Q₈	—	4	16	no	—
456	Q₈	—	4	4	no	—
457	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
458	D₄	—	2	4	no	—
459	D₄	—	2	4	no	—
460	D₄	—	4	4	no	—
461	D₄	—	4	16	no	—
462	D₄	—	4	16	no	—
463	D₄	—	4	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	8	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	4	16	no	—
475	C₂×C₄	—	4	16	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
479	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
480	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
481	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
483	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
484	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
485	C₈	—	2	4	no	—
486	C₈	—	2	2	no	—
487	C₈	C₂²×D₄×C₆₇	1	4	yes	—
488	C₂²	—	2	4	no	—
489	C₂²	—	2	4	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	4	8	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	8	8	no	—
496	C₂²	D₄²×C₆₇	1	2	yes	—
497	C₂²	#575×C₆₇	1	2	yes	—
498	C₂²	#578×C₆₇	1	2	yes	—
499	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	4	no	—
503	C₄	—	2	8	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	4	no	—
506	C₄	D₄²×C₆₇	1	2	yes	—
507	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
508	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
509	C₂	—	4	4	no	—
510	C₂	—	8	4	no	—
511	C₂	C₆₇×#3277	1	4	yes	direct product factor
512	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
513	C₂	C₂×#575×C₆₇	1	1	yes	characteristic
514	C₂	C₂×#578×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×#137×C₆₇	C₂	semidirect
2. C₂×#548×C₆₇	C₂	semidirect
3. C₆₇×#3277	C₂	direct (N × H)
4. C₂×#495×C₆₇	C₂	semidirect
5. C₂×#468×C₆₇	C₂	semidirect
6. C₂×#464×C₆₇	C₂	semidirect
7. C₂×#449×C₆₇	C₂	semidirect
8. C₂×#451×C₆₇	C₂	semidirect
9. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
10. C₂×#436×C₆₇	C₂	semidirect
14. C₂×#137×C₆₇	C₂²	semidirect
23. #548×C₆₇	C₂²	semidirect
33. #495×C₆₇	C₂²	semidirect
34. #468×C₆₇	C₂²	semidirect
35. #464×C₆₇	C₂²	semidirect
36. #451×C₆₇	C₂²	semidirect
37. #449×C₆₇	C₂²	semidirect
38. C₄×Q₁₆×C₆₇	C₂²	semidirect
39. #436×C₆₇	C₂²	semidirect
68. C₂×Q₁₆×C₆₇	D₄	semidirect
160. Q₁₆×C₆₇	C₂×D₄	semidirect
251. C₂×#3277	C₆₇	direct (N × H)
254. C₂×C₆₇	#3277	direct (N × H)
258. C₂²×#137	C₂×C₆₇	semidirect
259. C₂×#548	C₂×C₆₇	semidirect
260. #3277	C₂×C₆₇	direct (N × H)
261. C₂×#495	C₂×C₆₇	semidirect
262. C₂×#468	C₂×C₆₇	semidirect
263. C₂×#464	C₂×C₆₇	semidirect
264. C₂×#449	C₂×C₆₇	semidirect
265. C₂×#451	C₂×C₆₇	semidirect
266. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
267. C₂×#436	C₂×C₆₇	semidirect
268. C₆₇	C₂×#3277	direct (N × H)
272. C₂×#137	C₂²×C₆₇	semidirect
281. #548	C₂²×C₆₇	semidirect
291. #495	C₂²×C₆₇	semidirect
292. #468	C₂²×C₆₇	semidirect
293. #464	C₂²×C₆₇	semidirect
294. #451	C₂²×C₆₇	semidirect
295. #449	C₂²×C₆₇	semidirect
296. C₄×Q₁₆	C₂²×C₆₇	semidirect
297. #436	C₂²×C₆₇	semidirect
326. C₂×Q₁₆	D₄×C₆₇	semidirect
418. Q₁₆	C₂×D₄×C₆₇	semidirect
511. C₂	C₆₇×#3277	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×#137×C₆₇	2
C₂×#548×C₆₇	2
C₆₇×#3277	2
C₂×#495×C₆₇	2
C₂×#468×C₆₇	2
C₂×#464×C₆₇	2
C₂×#449×C₆₇	2
C₂×#451×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×#436×C₆₇	2
C₂×#3277	67

Subgroups — C2×C67×#3277 #30740287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3277 #30740287