Subgroups — C₂×C₆₇×#3278 #30740288

Order 17152.

2794 conjugacy classes of subgroups.

The subgroup lattice has 604 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×#550×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3278	C₂	1	16	yes	maximal, direct product factor
5	C₂×#492×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#468×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#462×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#452×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#448×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	2	yes	—
14	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#137×C₆₇	—	2	16	no	—
16	C₂×#137×C₆₇	C₂²	1	16	yes	complemented
17	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
22	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
23	#550×C₆₇	—	2	4	no	—
24	#550×C₆₇	C₂²	1	8	yes	—
25	D₄×Q₈×C₆₇	—	2	4	no	—
26	D₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
27	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
28	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#127×C₆₇	—	2	1	no	—
30	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
32	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
33	C₂×#121×C₆₇	—	2	2	no	—
34	C₂×#121×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₆₇	—	2	1	no	—
38	C₂×C₄×D₄×C₆₇	—	2	1	no	—
39	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
40	#492×C₆₇	C₂²	1	8	yes	complemented
41	#468×C₆₇	C₂²	1	16	yes	complemented
42	#462×C₆₇	C₂²	1	8	yes	complemented
43	#460×C₆₇	C₂²	1	8	yes	complemented
44	#452×C₆₇	C₂²	1	16	yes	complemented
45	#448×C₆₇	C₂²	1	16	yes	complemented
46	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
47	#436×C₆₇	C₂²	1	8	yes	complemented
48	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×#103×C₆₇	C₂²	1	2	yes	—
51	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
52	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×#102×C₆₇	C₂²	1	2	yes	—
54	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
55	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
56	C₂×#41×C₆₇	—	2	4	no	—
57	C₂×#41×C₆₇	—	2	8	no	—
58	C₂×#41×C₆₇	—	4	4	no	—
59	C₂×#41×C₆₇	C₂³	1	8	yes	—
60	C₂²×Q₈×C₆₇	—	2	2	no	—
61	C₂²×Q₈×C₆₇	—	2	4	no	—
62	C₂²×Q₈×C₆₇	—	2	8	no	—
63	C₂²×Q₈×C₆₇	—	4	4	no	—
64	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
65	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
66	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₆₇	—	2	2	no	—
69	C₂²×D₄×C₆₇	—	2	2	no	—
70	C₂²×D₄×C₆₇	—	2	1	no	—
71	C₂²×D₄×C₆₇	—	2	1	no	—
72	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
73	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
74	C₂³×C₄×C₆₇	—	2	2	no	—
75	C₂³×C₄×C₆₇	—	2	2	no	—
76	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
77	#137×C₆₇	—	2	64	no	—
78	C₂×Q₁₆×C₆₇	—	2	8	no	—
79	C₂×Q₁₆×C₆₇	D₄	1	8	yes	complemented
80	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
81	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
82	C₂×SD₁₆×C₆₇	—	2	8	no	—
83	C₂×SD₁₆×C₆₇	—	2	8	no	—
84	C₂×SD₁₆×C₆₇	—	4	4	no	—
85	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
86	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
87	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
88	C₂×M₁₆×C₆₇	—	2	8	no	—
89	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
90	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
91	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊Q₈×C₆₇	—	2	8	no	—
94	C₄⋊Q₈×C₆₇	—	2	2	no	—
95	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
96	#127×C₆₇	—	2	8	no	—
97	#127×C₆₇	—	2	2	no	—
98	#127×C₆₇	C₂³	1	4	yes	—
99	C₂²⋊Q₈×C₆₇	—	2	4	no	—
100	C₂²⋊Q₈×C₆₇	—	2	16	no	—
101	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
102	#121×C₆₇	—	2	4	no	—
103	#121×C₆₇	—	2	16	no	—
104	#121×C₆₇	C₂³	1	8	yes	—
105	C₄×Q₈×C₆₇	—	2	2	no	—
106	C₄×Q₈×C₆₇	—	2	2	no	—
107	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
108	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
109	C₄×D₄×C₆₇	—	2	8	no	—
110	C₄×D₄×C₆₇	—	2	8	no	—
111	C₄×D₄×C₆₇	D₄	1	4	yes	—
112	C₄×D₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
114	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
117	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
124	C₂×C₄²×C₆₇	—	2	1	no	—
125	C₂×C₄²×C₆₇	—	2	1	no	—
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	#107×C₆₇	C₂³	1	4	yes	—
128	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
129	#103×C₆₇	C₂³	1	4	yes	—
130	#103×C₆₇	C₂³	1	8	yes	—
131	#103×C₆₇	C₂³	1	4	yes	—
132	#102×C₆₇	C₂³	1	8	yes	—
133	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
134	C₄×C₈×C₆₇	C₂³	1	4	yes	—
135	C₂⁴×C₆₇	—	2	2	no	—
136	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
137	#41×C₆₇	—	2	16	no	—
138	#41×C₆₇	—	4	8	no	—
139	#41×C₆₇	—	4	16	no	—
140	C₂×Q₈×C₆₇	—	2	1	no	—
141	C₂×Q₈×C₆₇	—	2	1	no	—
142	C₂×Q₈×C₆₇	—	2	8	no	—
143	C₂×Q₈×C₆₇	—	2	2	no	—
144	C₂×Q₈×C₆₇	—	2	2	no	—
145	C₂×Q₈×C₆₇	—	2	2	no	—
146	C₂×Q₈×C₆₇	—	2	16	no	—
147	C₂×Q₈×C₆₇	—	2	4	no	—
148	C₂×Q₈×C₆₇	—	2	4	no	—
149	C₂×Q₈×C₆₇	—	4	8	no	—
150	C₂×Q₈×C₆₇	—	4	8	no	—
151	C₂×Q₈×C₆₇	—	4	16	no	—
152	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
153	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
154	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
155	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
156	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
157	C₂×D₄×C₆₇	—	2	8	no	—
158	C₂×D₄×C₆₇	—	2	8	no	—
159	C₂×D₄×C₆₇	—	2	4	no	—
160	C₂×D₄×C₆₇	—	2	4	no	—
161	C₂×D₄×C₆₇	—	2	4	no	—
162	C₂×D₄×C₆₇	—	2	2	no	—
163	C₂×D₄×C₆₇	—	2	4	no	—
164	C₂×D₄×C₆₇	—	2	4	no	—
165	C₂×D₄×C₆₇	—	4	8	no	—
166	C₂×D₄×C₆₇	—	4	4	no	—
167	C₂×D₄×C₆₇	—	8	2	no	—
168	C₂×D₄×C₆₇	—	8	4	no	—
169	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
170	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₆₇	—	2	1	no	—
172	C₂²×C₄×C₆₇	—	2	1	no	—
173	C₂²×C₄×C₆₇	—	2	4	no	—
174	C₂²×C₄×C₆₇	—	2	4	no	—
175	C₂²×C₄×C₆₇	—	2	8	no	—
176	C₂²×C₄×C₆₇	—	2	1	no	—
177	C₂²×C₄×C₆₇	—	2	1	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	—	2	8	no	—
180	C₂²×C₄×C₆₇	—	4	8	no	—
181	C₂²×C₄×C₆₇	—	4	8	no	—
182	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
188	Q₁₆×C₆₇	—	2	16	no	—
189	Q₁₆×C₆₇	C₂×D₄	1	16	yes	complemented
190	SD₁₆×C₆₇	—	2	16	no	—
191	SD₁₆×C₆₇	—	2	16	no	—
192	SD₁₆×C₆₇	—	4	8	no	—
193	M₁₆×C₆₇	—	2	16	no	—
194	C₂×C₈×C₆₇	—	2	4	no	—
195	C₂×C₈×C₆₇	—	2	2	no	—
196	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
197	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
198	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
199	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
200	C₄⋊C₄×C₆₇	—	2	4	no	—
201	C₄⋊C₄×C₆₇	—	2	4	no	—
202	C₄⋊C₄×C₆₇	—	2	4	no	—
203	C₄⋊C₄×C₆₇	—	2	4	no	—
204	C₄⋊C₄×C₆₇	—	2	8	no	—
205	C₄⋊C₄×C₆₇	—	2	1	no	—
206	C₄⋊C₄×C₆₇	—	2	1	no	—
207	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
208	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
209	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
210	C₂²⋊C₄×C₆₇	—	2	8	no	—
211	C₂²⋊C₄×C₆₇	—	2	8	no	—
212	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
213	C₄²×C₆₇	—	2	4	no	—
214	C₄²×C₆₇	—	2	4	no	—
215	C₄²×C₆₇	C₂×D₄	1	2	yes	—
216	C₄²×C₆₇	C₂⁴	1	2	yes	—
217	C₂³×C₆₇	—	2	4	no	—
218	C₂³×C₆₇	—	2	4	no	—
219	C₂³×C₆₇	—	2	4	no	—
220	C₂³×C₆₇	—	2	4	no	—
221	C₂³×C₆₇	—	4	4	no	—
222	C₂³×C₆₇	—	4	4	no	—
223	C₂³×C₆₇	—	4	4	no	—
224	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
225	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
226	Q₈×C₆₇	—	2	4	no	—
227	Q₈×C₆₇	—	2	4	no	—
228	Q₈×C₆₇	—	2	2	no	—
229	Q₈×C₆₇	—	2	2	no	—
230	Q₈×C₆₇	—	2	4	no	—
231	Q₈×C₆₇	—	2	4	no	—
232	Q₈×C₆₇	—	4	16	no	—
233	Q₈×C₆₇	—	4	4	no	—
234	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
235	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
236	D₄×C₆₇	—	2	8	no	—
237	D₄×C₆₇	—	2	4	no	—
238	D₄×C₆₇	—	2	4	no	—
239	D₄×C₆₇	—	4	16	no	—
240	D₄×C₆₇	—	4	4	no	—
241	D₄×C₆₇	—	4	8	no	—
242	D₄×C₆₇	—	8	4	no	—
243	D₄×C₆₇	—	8	8	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	8	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	4	no	—
256	C₂×C₄×C₆₇	—	4	16	no	—
257	C₂×C₄×C₆₇	—	4	16	no	—
258	C₂×C₄×C₆₇	—	4	4	no	—
259	C₂×C₄×C₆₇	—	4	2	no	—
260	C₂×C₄×C₆₇	—	4	2	no	—
261	C₂×C₄×C₆₇	#137	1	4	yes	complemented
262	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
268	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
269	C₈×C₆₇	—	2	4	no	—
270	C₈×C₆₇	—	2	2	no	—
271	C₈×C₆₇	C₂²×D₄	1	4	yes	—
272	C₂²×C₆₇	—	2	4	no	—
273	C₂²×C₆₇	—	2	4	no	—
274	C₂²×C₆₇	—	2	4	no	—
275	C₂²×C₆₇	—	4	8	no	—
276	C₂²×C₆₇	—	4	4	no	—
277	C₂²×C₆₇	—	4	4	no	—
278	C₂²×C₆₇	—	4	4	no	—
279	C₂²×C₆₇	—	8	8	no	—
280	C₂²×C₆₇	D₄²	1	2	yes	—
281	C₂²×C₆₇	C₂×#137	1	2	yes	—
282	C₂²×C₆₇	#577	1	2	yes	—
283	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
284	C₄×C₆₇	—	2	4	no	—
285	C₄×C₆₇	—	2	4	no	—
286	C₄×C₆₇	—	2	4	no	—
287	C₄×C₆₇	—	2	2	no	—
288	C₄×C₆₇	—	4	4	no	—
289	C₄×C₆₇	—	4	2	no	—
290	C₄×C₆₇	—	4	2	no	—
291	C₄×C₆₇	D₄²	1	2	yes	—
292	C₄×C₆₇	C₂×#137	1	4	yes	complemented
293	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
294	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
295	C₂×#3278	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
296	C₂×C₆₇	—	4	4	no	—
297	C₂×C₆₇	—	8	4	no	—
298	C₂×C₆₇	#3278	1	4	yes	direct product factor
299	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
300	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
301	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
302	C₂²×#137	C₂×C₆₇	1	2	yes	complemented
303	C₂×#550	C₂×C₆₇	1	1	yes	characteristic
304	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#3278	C₂×C₆₇	1	16	yes	direct product factor
306	C₂×#492	C₂×C₆₇	1	1	yes	characteristic, complemented
307	C₂×#468	C₂×C₆₇	1	2	yes	complemented
308	C₂×#462	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₂×#460	C₂×C₆₇	1	1	yes	characteristic, complemented
310	C₂×#452	C₂×C₆₇	1	2	yes	complemented
311	C₂×#448	C₂×C₆₇	1	2	yes	complemented
312	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
313	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
314	C₆₇	C₂×#3278	1	1	yes	characteristic, direct product factor, Sylow
315	C₂²×#41	C₂²×C₆₇	1	2	yes	—
316	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
317	C₂×#137	—	2	16	no	—
318	C₂×#137	C₂²×C₆₇	1	16	yes	complemented
319	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
320	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
321	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
322	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
323	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
324	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
325	#550	—	2	4	no	—
326	#550	C₂²×C₆₇	1	8	yes	—
327	D₄×Q₈	—	2	4	no	—
328	D₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
329	C₂×C₄⋊Q₈	—	2	1	no	—
330	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×#127	—	2	1	no	—
332	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₂²⋊Q₈	—	2	2	no	—
334	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
335	C₂×#121	—	2	2	no	—
336	C₂×#121	C₂²×C₆₇	1	2	yes	—
337	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
338	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×C₄×D₄	—	2	1	no	—
340	C₂×C₄×D₄	—	2	1	no	—
341	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
342	#492	C₂²×C₆₇	1	8	yes	complemented
343	#468	C₂²×C₆₇	1	16	yes	complemented
344	#462	C₂²×C₆₇	1	8	yes	complemented
345	#460	C₂²×C₆₇	1	8	yes	complemented
346	#452	C₂²×C₆₇	1	16	yes	complemented
347	#448	C₂²×C₆₇	1	16	yes	complemented
348	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
349	#436	C₂²×C₆₇	1	8	yes	complemented
350	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
351	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
352	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
353	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
354	C₂×#103	C₂²×C₆₇	1	2	yes	—
355	C₂×#102	C₂²×C₆₇	1	2	yes	—
356	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
357	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
358	C₂×#41	—	2	4	no	—
359	C₂×#41	—	2	8	no	—
360	C₂×#41	—	4	4	no	—
361	C₂×#41	C₂³×C₆₇	1	8	yes	—
362	C₂²×Q₈	—	2	2	no	—
363	C₂²×Q₈	—	2	4	no	—
364	C₂²×Q₈	—	2	8	no	—
365	C₂²×Q₈	—	4	4	no	—
366	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
367	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
368	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
369	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
370	C₂²×D₄	—	2	2	no	—
371	C₂²×D₄	—	2	2	no	—
372	C₂²×D₄	—	2	1	no	—
373	C₂²×D₄	—	2	1	no	—
374	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
375	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
376	C₂³×C₄	—	2	2	no	—
377	C₂³×C₄	—	2	2	no	—
378	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
379	#137	—	2	64	no	—
380	C₂×Q₁₆	—	2	8	no	—
381	C₂×Q₁₆	D₄×C₆₇	1	8	yes	complemented
382	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
383	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
384	C₂×SD₁₆	—	2	8	no	—
385	C₂×SD₁₆	—	2	8	no	—
386	C₂×SD₁₆	—	4	4	no	—
387	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
388	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
389	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
390	C₂×M₁₆	—	2	8	no	—
391	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
392	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
393	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
394	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
395	C₄⋊Q₈	—	2	8	no	—
396	C₄⋊Q₈	—	2	2	no	—
397	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
398	#127	—	2	8	no	—
399	#127	—	2	2	no	—
400	#127	C₂³×C₆₇	1	4	yes	—
401	C₂²⋊Q₈	—	2	4	no	—
402	C₂²⋊Q₈	—	2	16	no	—
403	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
404	#121	—	2	4	no	—
405	#121	—	2	16	no	—
406	#121	C₂³×C₆₇	1	8	yes	—
407	C₄×Q₈	—	2	2	no	—
408	C₄×Q₈	—	2	2	no	—
409	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
410	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
411	C₄×D₄	—	2	8	no	—
412	C₄×D₄	—	2	8	no	—
413	C₄×D₄	D₄×C₆₇	1	4	yes	—
414	C₄×D₄	C₂³×C₆₇	1	4	yes	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	—	2	1	no	—
419	C₂×C₄⋊C₄	—	2	2	no	—
420	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
423	C₂×C₂²⋊C₄	—	2	2	no	—
424	C₂×C₂²⋊C₄	—	2	2	no	—
425	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
426	C₂×C₄²	—	2	1	no	—
427	C₂×C₄²	—	2	1	no	—
428	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
429	#107	C₂³×C₆₇	1	4	yes	—
430	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
431	#103	C₂³×C₆₇	1	4	yes	—
432	#103	C₂³×C₆₇	1	4	yes	—
433	#103	C₂³×C₆₇	1	8	yes	—
434	#102	C₂³×C₆₇	1	8	yes	—
435	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
436	C₄×C₈	C₂³×C₆₇	1	4	yes	—
437	C₂⁴	—	2	2	no	—
438	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
439	#41	—	2	16	no	—
440	#41	—	4	8	no	—
441	#41	—	4	16	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	8	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	16	no	—
449	C₂×Q₈	—	2	4	no	—
450	C₂×Q₈	—	2	4	no	—
451	C₂×Q₈	—	4	8	no	—
452	C₂×Q₈	—	4	8	no	—
453	C₂×Q₈	—	4	16	no	—
454	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
455	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
456	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
457	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
458	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
459	C₂×D₄	—	2	8	no	—
460	C₂×D₄	—	2	8	no	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	4	8	no	—
468	C₂×D₄	—	4	4	no	—
469	C₂×D₄	—	8	2	no	—
470	C₂×D₄	—	8	4	no	—
471	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
472	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	8	no	—
481	C₂²×C₄	—	2	8	no	—
482	C₂²×C₄	—	4	8	no	—
483	C₂²×C₄	—	4	8	no	—
484	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
486	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
487	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
488	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
489	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
490	Q₁₆	—	2	16	no	—
491	Q₁₆	C₂×D₄×C₆₇	1	16	yes	complemented
492	SD₁₆	—	2	16	no	—
493	SD₁₆	—	2	16	no	—
494	SD₁₆	—	4	8	no	—
495	M₁₆	—	2	16	no	—
496	C₂×C₈	—	2	4	no	—
497	C₂×C₈	—	2	2	no	—
498	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
499	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
500	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
501	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	4	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	8	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
510	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
511	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
512	C₂²⋊C₄	—	2	8	no	—
513	C₂²⋊C₄	—	2	8	no	—
514	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
515	C₄²	—	2	4	no	—
516	C₄²	—	2	4	no	—
517	C₄²	C₂×D₄×C₆₇	1	2	yes	—
518	C₄²	C₂⁴×C₆₇	1	2	yes	—
519	C₂³	—	2	4	no	—
520	C₂³	—	2	4	no	—
521	C₂³	—	2	4	no	—
522	C₂³	—	2	4	no	—
523	C₂³	—	4	4	no	—
524	C₂³	—	4	4	no	—
525	C₂³	—	4	4	no	—
526	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
527	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
528	Q₈	—	2	4	no	—
529	Q₈	—	2	4	no	—
530	Q₈	—	2	2	no	—
531	Q₈	—	2	2	no	—
532	Q₈	—	2	4	no	—
533	Q₈	—	2	4	no	—
534	Q₈	—	4	16	no	—
535	Q₈	—	4	4	no	—
536	Q₈	C₂²×D₄×C₆₇	1	4	yes	—
537	Q₈	C₂²×D₄×C₆₇	1	4	yes	—
538	D₄	—	2	8	no	—
539	D₄	—	2	4	no	—
540	D₄	—	2	4	no	—
541	D₄	—	4	16	no	—
542	D₄	—	4	4	no	—
543	D₄	—	4	8	no	—
544	D₄	—	8	4	no	—
545	D₄	—	8	8	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	8	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	4	16	no	—
559	C₂×C₄	—	4	16	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	2	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	#137×C₆₇	1	4	yes	complemented
564	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
566	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
569	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
570	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
571	C₈	—	2	4	no	—
572	C₈	—	2	2	no	—
573	C₈	C₂²×D₄×C₆₇	1	4	yes	—
574	C₂²	—	2	4	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	4	8	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	8	8	no	—
582	C₂²	D₄²×C₆₇	1	2	yes	—
583	C₂²	C₂×#137×C₆₇	1	2	yes	—
584	C₂²	#577×C₆₇	1	2	yes	—
585	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
586	C₄	—	2	2	no	—
587	C₄	—	2	4	no	—
588	C₄	—	2	4	no	—
589	C₄	—	2	4	no	—
590	C₄	—	4	2	no	—
591	C₄	—	4	2	no	—
592	C₄	—	4	4	no	—
593	C₄	D₄²×C₆₇	1	2	yes	—
594	C₄	C₂×#137×C₆₇	1	4	yes	complemented
595	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
596	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
597	C₂	—	4	4	no	—
598	C₂	—	8	4	no	—
599	C₂	C₆₇×#3278	1	4	yes	direct product factor
600	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
601	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
602	C₂	C₂×#577×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#550×C₆₇	C₂
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
3. C₂×D₄×Q₈×C₆₇	C₂	semidirect
4. C₆₇×#3278	C₂	direct (N × H)
5. C₂×#492×C₆₇	C₂	semidirect
6. C₂×#468×C₆₇	C₂	semidirect
7. C₂×#462×C₆₇	C₂	semidirect
8. C₂×#460×C₆₇	C₂	semidirect
9. C₂×#452×C₆₇	C₂	semidirect
10. C₂×#448×C₆₇	C₂	semidirect
11. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
12. C₂×#436×C₆₇	C₂	semidirect
16. C₂×#137×C₆₇	C₂²	semidirect
26. D₄×Q₈×C₆₇	C₂²	semidirect
40. #492×C₆₇	C₂²	semidirect
41. #468×C₆₇	C₂²	semidirect
42. #462×C₆₇	C₂²	semidirect
43. #460×C₆₇	C₂²	semidirect
44. #452×C₆₇	C₂²	semidirect
45. #448×C₆₇	C₂²	semidirect
46. C₄×Q₁₆×C₆₇	C₂²	semidirect
47. #436×C₆₇	C₂²	semidirect
79. C₂×Q₁₆×C₆₇	D₄	semidirect
189. Q₁₆×C₆₇	C₂×D₄	semidirect
261. C₂×C₄×C₆₇	#137	semidirect
292. C₄×C₆₇	C₂×#137	semidirect
295. C₂×#3278	C₆₇	direct (N × H)
298. C₂×C₆₇	#3278	direct (N × H)
302. C₂²×#137	C₂×C₆₇	semidirect
304. C₂×D₄×Q₈	C₂×C₆₇	semidirect
305. #3278	C₂×C₆₇	direct (N × H)
306. C₂×#492	C₂×C₆₇	semidirect
307. C₂×#468	C₂×C₆₇	semidirect
308. C₂×#462	C₂×C₆₇	semidirect
309. C₂×#460	C₂×C₆₇	semidirect
310. C₂×#452	C₂×C₆₇	semidirect
311. C₂×#448	C₂×C₆₇	semidirect
312. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
313. C₂×#436	C₂×C₆₇	semidirect
314. C₆₇	C₂×#3278	direct (N × H)
318. C₂×#137	C₂²×C₆₇	semidirect
328. D₄×Q₈	C₂²×C₆₇	semidirect
342. #492	C₂²×C₆₇	semidirect
343. #468	C₂²×C₆₇	semidirect
344. #462	C₂²×C₆₇	semidirect
345. #460	C₂²×C₆₇	semidirect
346. #452	C₂²×C₆₇	semidirect
347. #448	C₂²×C₆₇	semidirect
348. C₄×Q₁₆	C₂²×C₆₇	semidirect
349. #436	C₂²×C₆₇	semidirect
381. C₂×Q₁₆	D₄×C₆₇	semidirect
491. Q₁₆	C₂×D₄×C₆₇	semidirect
563. C₂×C₄	#137×C₆₇	semidirect
594. C₄	C₂×#137×C₆₇	semidirect
599. C₂	C₆₇×#3278	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×#550×C₆₇	2
C₂×D₄×Q₈×C₆₇	2
C₆₇×#3278	2
C₂×#492×C₆₇	2
C₂×#468×C₆₇	2
C₂×#462×C₆₇	2
C₂×#460×C₆₇	2
C₂×#452×C₆₇	2
C₂×#448×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×#436×C₆₇	2
C₂×#3278	67

Subgroups — C2×C67×#3278 #30740288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3278 #30740288