Subgroups — C₂×C₆₇×#3355 #30740499

Order 17152.

1962 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#553×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3355	C₂	1	16	yes	maximal, direct product factor
3	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#482×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#480×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#478×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#469×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#428×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
13	#553×C₆₇	—	2	8	no	—
14	#553×C₆₇	C₂²	1	16	yes	complemented
15	C₂×#126×C₆₇	—	2	2	no	—
16	C₂×#125×C₆₇	C₂²	1	2	yes	—
17	C₂×#123×C₆₇	—	2	2	no	—
18	C₂×#121×C₆₇	—	2	2	no	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×#121×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₆₇	—	2	2	no	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#117×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	#499×C₆₇	C₂²	1	8	yes	complemented
27	#482×C₆₇	C₂²	1	16	yes	complemented
28	#480×C₆₇	C₂²	1	16	yes	complemented
29	#478×C₆₇	C₂²	1	16	yes	complemented
30	#469×C₆₇	C₂²	1	16	yes	complemented
31	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
32	#436×C₆₇	C₂²	1	8	yes	complemented
33	#428×C₆₇	C₂²	1	16	yes	complemented
34	C₂×#107×C₆₇	C₂²	1	2	yes	—
35	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#106×C₆₇	C₂²	1	2	yes	—
39	C₂×#106×C₆₇	C₂²	1	2	yes	—
40	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#102×C₆₇	C₂²	1	2	yes	—
42	C₂×#102×C₆₇	C₂²	1	2	yes	—
43	C₂×#102×C₆₇	C₂²	1	2	yes	—
44	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂²×D₄×C₆₇	—	2	2	no	—
48	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	—	2	2	no	—
51	C₂³×C₄×C₆₇	—	2	2	no	—
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂×D₈×C₆₇	—	2	4	no	—
54	C₂×D₈×C₆₇	C₂³	1	4	yes	—
55	C₂×M₁₆×C₆₇	—	2	8	no	—
56	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
57	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	#126×C₆₇	—	2	16	no	—
61	#125×C₆₇	—	2	4	no	—
62	#125×C₆₇	C₂³	1	8	yes	—
63	#123×C₆₇	—	2	16	no	—
64	#121×C₆₇	—	2	16	no	—
65	#121×C₆₇	—	2	4	no	—
66	#121×C₆₇	—	2	4	no	—
67	#121×C₆₇	C₂³	1	8	yes	—
68	#121×C₆₇	C₂³	1	8	yes	—
69	C₄×D₄×C₆₇	—	2	16	no	—
70	C₄×D₄×C₆₇	—	2	4	no	—
71	C₄×D₄×C₆₇	D₄	1	4	yes	—
72	C₄×D₄×C₆₇	C₂³	1	8	yes	—
73	C₄×D₄×C₆₇	C₂³	1	4	yes	—
74	#117×C₆₇	—	2	4	no	—
75	#117×C₆₇	C₂³	1	8	yes	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
90	C₂×C₄²×C₆₇	—	2	2	no	—
91	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
92	#107×C₆₇	C₂³	1	8	yes	—
93	#107×C₆₇	C₂³	1	4	yes	—
94	#107×C₆₇	C₂³	1	4	yes	—
95	#107×C₆₇	C₂³	1	4	yes	—
96	#106×C₆₇	C₂³	1	8	yes	—
97	#106×C₆₇	C₂³	1	8	yes	—
98	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
99	#102×C₆₇	C₂³	1	8	yes	—
100	#102×C₆₇	C₂³	1	8	yes	—
101	#102×C₆₇	C₂³	1	8	yes	—
102	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
103	C₄×C₈×C₆₇	C₂³	1	4	yes	—
104	C₂⁴×C₆₇	—	2	2	no	—
105	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
106	C₂×D₄×C₆₇	—	2	8	no	—
107	C₂×D₄×C₆₇	—	2	8	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	4	no	—
110	C₂×D₄×C₆₇	—	2	4	no	—
111	C₂×D₄×C₆₇	—	4	2	no	—
112	C₂×D₄×C₆₇	—	4	8	no	—
113	C₂×D₄×C₆₇	—	8	4	no	—
114	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
115	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	8	no	—
122	C₂²×C₄×C₆₇	—	2	8	no	—
123	C₂²×C₄×C₆₇	—	8	4	no	—
124	C₂²×C₄×C₆₇	—	8	4	no	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	D₈×C₆₇	—	2	8	no	—
132	M₁₆×C₆₇	—	2	16	no	—
133	C₂×C₈×C₆₇	—	2	4	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	#41	1	4	yes	—
136	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
137	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
138	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	8	no	—
142	C₄⋊C₄×C₆₇	—	2	8	no	—
143	C₄⋊C₄×C₆₇	—	2	8	no	—
144	C₄⋊C₄×C₆₇	—	2	2	no	—
145	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
146	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
147	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
148	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
149	C₂²⋊C₄×C₆₇	—	2	8	no	—
150	C₂²⋊C₄×C₆₇	—	2	8	no	—
151	C₂²⋊C₄×C₆₇	—	2	8	no	—
152	C₂²⋊C₄×C₆₇	—	2	8	no	—
153	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
154	C₄²×C₆₇	—	2	8	no	—
155	C₄²×C₆₇	C₂×D₄	1	2	yes	—
156	C₄²×C₆₇	C₂⁴	1	2	yes	—
157	C₂³×C₆₇	—	2	4	no	—
158	C₂³×C₆₇	—	2	4	no	—
159	C₂³×C₆₇	—	2	4	no	—
160	C₂³×C₆₇	—	2	4	no	—
161	C₂³×C₆₇	—	4	4	no	—
162	C₂³×C₆₇	—	4	4	no	—
163	C₂³×C₆₇	—	4	4	no	—
164	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
165	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	D₄×C₆₇	—	2	4	no	—
167	D₄×C₆₇	—	2	4	no	—
168	D₄×C₆₇	—	4	4	no	—
169	D₄×C₆₇	—	4	16	no	—
170	D₄×C₆₇	—	4	4	no	—
171	D₄×C₆₇	—	8	8	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	8	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	4	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	4	4	no	—
182	C₂×C₄×C₆₇	—	4	4	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	8	8	no	—
185	C₂×C₄×C₆₇	—	8	8	no	—
186	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
187	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
188	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
191	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
192	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
193	C₈×C₆₇	—	2	4	no	—
194	C₈×C₆₇	—	2	2	no	—
195	C₈×C₆₇	C₂×#41	1	4	yes	—
196	C₂²×C₆₇	—	2	4	no	—
197	C₂²×C₆₇	—	2	4	no	—
198	C₂²×C₆₇	—	2	4	no	—
199	C₂²×C₆₇	—	4	8	no	—
200	C₂²×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	—	8	8	no	—
204	C₂²×C₆₇	#547	1	2	yes	—
205	C₂²×C₆₇	#575	1	2	yes	—
206	C₂²×C₆₇	#576	1	2	yes	—
207	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
208	C₄×C₆₇	—	2	4	no	—
209	C₄×C₆₇	—	2	2	no	—
210	C₄×C₆₇	—	2	2	no	—
211	C₄×C₆₇	—	4	4	no	—
212	C₄×C₆₇	—	4	4	no	—
213	C₄×C₆₇	—	4	4	no	—
214	C₄×C₆₇	#547	1	2	yes	—
215	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
216	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
217	C₂×#3355	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂×C₆₇	—	4	4	no	—
219	C₂×C₆₇	—	8	4	no	—
220	C₂×C₆₇	#3355	1	4	yes	direct product factor
221	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
222	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
223	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
224	C₂×#553	C₂×C₆₇	1	2	yes	complemented
225	#3355	C₂×C₆₇	1	16	yes	direct product factor
226	C₂×#499	C₂×C₆₇	1	1	yes	characteristic, complemented
227	C₂×#482	C₂×C₆₇	1	2	yes	complemented
228	C₂×#480	C₂×C₆₇	1	2	yes	complemented
229	C₂×#478	C₂×C₆₇	1	2	yes	complemented
230	C₂×#469	C₂×C₆₇	1	2	yes	complemented
231	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
232	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₂×#428	C₂×C₆₇	1	2	yes	complemented
234	C₆₇	C₂×#3355	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
237	#553	—	2	8	no	—
238	#553	C₂²×C₆₇	1	16	yes	complemented
239	C₂×#126	—	2	2	no	—
240	C₂×#125	C₂²×C₆₇	1	2	yes	—
241	C₂×#123	—	2	2	no	—
242	C₂×#121	—	2	2	no	—
243	C₂×#121	C₂²×C₆₇	1	2	yes	—
244	C₂×#121	C₂²×C₆₇	1	2	yes	—
245	C₂×C₄×D₄	—	2	2	no	—
246	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
247	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
248	C₂×#117	C₂²×C₆₇	1	2	yes	—
249	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
250	#499	C₂²×C₆₇	1	8	yes	complemented
251	#482	C₂²×C₆₇	1	16	yes	complemented
252	#480	C₂²×C₆₇	1	16	yes	complemented
253	#478	C₂²×C₆₇	1	16	yes	complemented
254	#469	C₂²×C₆₇	1	16	yes	complemented
255	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
256	#436	C₂²×C₆₇	1	8	yes	complemented
257	#428	C₂²×C₆₇	1	16	yes	complemented
258	C₂×#107	C₂²×C₆₇	1	2	yes	—
259	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×#106	C₂²×C₆₇	1	2	yes	—
263	C₂×#106	C₂²×C₆₇	1	2	yes	—
264	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
265	C₂×#102	C₂²×C₆₇	1	2	yes	—
266	C₂×#102	C₂²×C₆₇	1	2	yes	—
267	C₂×#102	C₂²×C₆₇	1	2	yes	—
268	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
269	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
270	C₂²×D₄	—	2	2	no	—
271	C₂²×D₄	—	2	2	no	—
272	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
273	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
274	C₂³×C₄	—	2	2	no	—
275	C₂³×C₄	—	2	2	no	—
276	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
277	C₂×D₈	—	2	4	no	—
278	C₂×D₈	C₂³×C₆₇	1	4	yes	—
279	C₂×M₁₆	—	2	8	no	—
280	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
281	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
282	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
284	#126	—	2	16	no	—
285	#125	—	2	4	no	—
286	#125	C₂³×C₆₇	1	8	yes	—
287	#123	—	2	16	no	—
288	#121	—	2	16	no	—
289	#121	—	2	4	no	—
290	#121	—	2	4	no	—
291	#121	C₂³×C₆₇	1	8	yes	—
292	#121	C₂³×C₆₇	1	8	yes	—
293	C₄×D₄	—	2	16	no	—
294	C₄×D₄	—	2	4	no	—
295	C₄×D₄	D₄×C₆₇	1	4	yes	—
296	C₄×D₄	C₂³×C₆₇	1	8	yes	—
297	C₄×D₄	C₂³×C₆₇	1	4	yes	—
298	#117	—	2	4	no	—
299	#117	C₂³×C₆₇	1	8	yes	—
300	C₂×C₄⋊C₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	4	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	2	2	no	—
304	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
306	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
308	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
314	C₂×C₄²	—	2	2	no	—
315	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
316	#107	C₂³×C₆₇	1	4	yes	—
317	#107	C₂³×C₆₇	1	8	yes	—
318	#107	C₂³×C₆₇	1	4	yes	—
319	#107	C₂³×C₆₇	1	4	yes	—
320	#106	C₂³×C₆₇	1	8	yes	—
321	#106	C₂³×C₆₇	1	8	yes	—
322	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
323	#102	C₂³×C₆₇	1	8	yes	—
324	#102	C₂³×C₆₇	1	8	yes	—
325	#102	C₂³×C₆₇	1	8	yes	—
326	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
327	C₄×C₈	C₂³×C₆₇	1	4	yes	—
328	C₂⁴	—	2	2	no	—
329	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
330	C₂×D₄	—	2	8	no	—
331	C₂×D₄	—	2	8	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	4	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	4	2	no	—
336	C₂×D₄	—	4	8	no	—
337	C₂×D₄	—	8	4	no	—
338	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
339	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	8	no	—
346	C₂²×C₄	—	2	8	no	—
347	C₂²×C₄	—	8	4	no	—
348	C₂²×C₄	—	8	4	no	—
349	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
353	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
354	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
355	D₈	—	2	8	no	—
356	M₁₆	—	2	16	no	—
357	C₂×C₈	—	2	4	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	#41×C₆₇	1	4	yes	—
360	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
361	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
362	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	8	no	—
366	C₄⋊C₄	—	2	8	no	—
367	C₄⋊C₄	—	2	8	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
370	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
371	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
372	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
373	C₂²⋊C₄	—	2	8	no	—
374	C₂²⋊C₄	—	2	8	no	—
375	C₂²⋊C₄	—	2	8	no	—
376	C₂²⋊C₄	—	2	8	no	—
377	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
378	C₄²	—	2	8	no	—
379	C₄²	C₂×D₄×C₆₇	1	2	yes	—
380	C₄²	C₂⁴×C₆₇	1	2	yes	—
381	C₂³	—	2	4	no	—
382	C₂³	—	2	4	no	—
383	C₂³	—	2	4	no	—
384	C₂³	—	2	4	no	—
385	C₂³	—	4	4	no	—
386	C₂³	—	4	4	no	—
387	C₂³	—	4	4	no	—
388	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
389	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
390	D₄	—	2	4	no	—
391	D₄	—	2	4	no	—
392	D₄	—	4	4	no	—
393	D₄	—	4	16	no	—
394	D₄	—	4	4	no	—
395	D₄	—	8	8	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	8	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	—	4	4	no	—
407	C₂×C₄	—	4	4	no	—
408	C₂×C₄	—	8	8	no	—
409	C₂×C₄	—	8	8	no	—
410	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
411	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
412	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
413	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
415	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
416	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
417	C₈	—	2	4	no	—
418	C₈	—	2	2	no	—
419	C₈	C₂×#41×C₆₇	1	4	yes	—
420	C₂²	—	2	4	no	—
421	C₂²	—	2	4	no	—
422	C₂²	—	2	4	no	—
423	C₂²	—	4	8	no	—
424	C₂²	—	4	4	no	—
425	C₂²	—	4	4	no	—
426	C₂²	—	4	4	no	—
427	C₂²	—	8	8	no	—
428	C₂²	#547×C₆₇	1	2	yes	—
429	C₂²	#575×C₆₇	1	2	yes	—
430	C₂²	#576×C₆₇	1	2	yes	—
431	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
432	C₄	—	2	2	no	—
433	C₄	—	2	2	no	—
434	C₄	—	2	4	no	—
435	C₄	—	4	4	no	—
436	C₄	—	4	4	no	—
437	C₄	—	4	4	no	—
438	C₄	#547×C₆₇	1	2	yes	—
439	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
440	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
441	C₂	—	4	4	no	—
442	C₂	—	8	4	no	—
443	C₂	C₆₇×#3355	1	4	yes	direct product factor
444	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
445	C₂	C₂×#575×C₆₇	1	1	yes	characteristic
446	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. C₂×#553×C₆₇	C₂	semidirect
2. C₆₇×#3355	C₂	direct (N × H)
3. C₂×#499×C₆₇	C₂	semidirect
4. C₂×#482×C₆₇	C₂	semidirect
5. C₂×#480×C₆₇	C₂	semidirect
6. C₂×#478×C₆₇	C₂	semidirect
7. C₂×#469×C₆₇	C₂	semidirect
8. C₂×C₄×D₈×C₆₇	C₂	semidirect
9. C₂×#436×C₆₇	C₂	semidirect
10. C₂×#428×C₆₇	C₂	semidirect
14. #553×C₆₇	C₂²	semidirect
26. #499×C₆₇	C₂²	semidirect
27. #482×C₆₇	C₂²	semidirect
28. #480×C₆₇	C₂²	semidirect
29. #478×C₆₇	C₂²	semidirect
30. #469×C₆₇	C₂²	semidirect
31. C₄×D₈×C₆₇	C₂²	semidirect
32. #436×C₆₇	C₂²	semidirect
33. #428×C₆₇	C₂²	semidirect
217. C₂×#3355	C₆₇	direct (N × H)
220. C₂×C₆₇	#3355	direct (N × H)
224. C₂×#553	C₂×C₆₇	semidirect
225. #3355	C₂×C₆₇	direct (N × H)
226. C₂×#499	C₂×C₆₇	semidirect
227. C₂×#482	C₂×C₆₇	semidirect
228. C₂×#480	C₂×C₆₇	semidirect
229. C₂×#478	C₂×C₆₇	semidirect
230. C₂×#469	C₂×C₆₇	semidirect
231. C₂×C₄×D₈	C₂×C₆₇	semidirect
232. C₂×#436	C₂×C₆₇	semidirect
233. C₂×#428	C₂×C₆₇	semidirect
234. C₆₇	C₂×#3355	direct (N × H)
238. #553	C₂²×C₆₇	semidirect
250. #499	C₂²×C₆₇	semidirect
251. #482	C₂²×C₆₇	semidirect
252. #480	C₂²×C₆₇	semidirect
253. #478	C₂²×C₆₇	semidirect
254. #469	C₂²×C₆₇	semidirect
255. C₄×D₈	C₂²×C₆₇	semidirect
256. #436	C₂²×C₆₇	semidirect
257. #428	C₂²×C₆₇	semidirect
443. C₂	C₆₇×#3355	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#553×C₆₇	2
C₆₇×#3355	2
C₂×#499×C₆₇	2
C₂×#482×C₆₇	2
C₂×#480×C₆₇	2
C₂×#478×C₆₇	2
C₂×#469×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂×#436×C₆₇	2
C₂×#428×C₆₇	2
C₂×#3355	67

Subgroups — C2×C67×#3355 #30740499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3355 #30740499