Subgroups — C₆₇×#21881 #30701132

Order 17152.

1018 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#542×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2921	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1997	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1967	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1901	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1900	C₂	1	2	yes	maximal
8	C₆₇×#1801	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1800	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
11	#542×C₆₇	—	2	4	no	—
12	#542×C₆₇	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#126×C₆₇	C₄	1	2	yes	—
15	C₂×#124×C₆₇	C₄	1	2	yes	—
16	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#123×C₆₇	C₄	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
24	#433×C₆₇	—	2	4	no	—
25	#424×C₆₇	—	2	4	no	—
26	#423×C₆₇	C₂²	1	2	yes	—
27	#423×C₆₇	C₂²	1	2	yes	—
28	#423×C₆₇	C₂²	1	2	yes	—
29	C₂×#101×C₆₇	C₂²	1	4	yes	—
30	C₂×#100×C₆₇	C₂²	1	4	yes	—
31	#407×C₆₇	C₂²	1	2	yes	—
32	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
33	#343×C₆₇	C₂²	1	4	yes	—
34	#343×C₆₇	C₂²	1	4	yes	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	4	4	no	—
43	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
44	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	#126×C₆₇	—	2	8	no	—
49	#124×C₆₇	—	2	8	no	—
50	#124×C₆₇	—	2	2	no	—
51	#124×C₆₇	—	2	2	no	—
52	#123×C₆₇	—	2	8	no	—
53	C₂²⋊Q₈×C₆₇	—	2	2	no	—
54	C₂²⋊Q₈×C₆₇	—	2	2	no	—
55	C₂²⋊Q₈×C₆₇	—	2	8	no	—
56	#121×C₆₇	—	2	2	no	—
57	#121×C₆₇	—	2	2	no	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	#117×C₆₇	D₄	1	4	yes	—
63	#117×C₆₇	D₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
73	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
74	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	—	2	4	no	—
82	C₄⋊C₈×C₆₇	—	2	4	no	—
83	C₄⋊C₈×C₆₇	—	2	4	no	—
84	#101×C₆₇	—	4	4	no	—
85	#100×C₆₇	—	4	4	no	—
86	C₂²⋊C₈×C₆₇	—	2	4	no	—
87	#97×C₆₇	—	2	2	no	—
88	C₄×C₈×C₆₇	—	2	2	no	—
89	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Q₈×C₆₇	—	2	4	no	—
92	C₂×Q₈×C₆₇	—	2	1	no	—
93	C₂×Q₈×C₆₇	—	2	1	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	—	8	2	no	—
97	C₂×D₄×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	2	1	no	—
99	C₂×D₄×C₆₇	—	2	1	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	8	2	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	8	2	no	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
118	M₁₆×C₆₇	—	2	4	no	—
119	M₁₆×C₆₇	—	2	4	no	—
120	M₁₆×C₆₇	—	2	4	no	—
121	M₁₆×C₆₇	—	2	4	no	—
122	M₁₆×C₆₇	—	4	4	no	—
123	C₂×C₈×C₆₇	—	2	4	no	—
124	C₂×C₈×C₆₇	—	2	4	no	—
125	C₂×C₈×C₆₇	—	4	4	no	—
126	C₄⋊C₄×C₆₇	—	2	4	no	—
127	C₄⋊C₄×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	1	no	—
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	—	2	1	no	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
134	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
135	C₂²⋊C₄×C₆₇	—	2	4	no	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	2	4	no	—
138	C₂²⋊C₄×C₆₇	—	2	4	no	—
139	C₂²⋊C₄×C₆₇	—	2	4	no	—
140	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
141	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
142	C₄²×C₆₇	—	2	4	no	—
143	C₄²×C₆₇	C₂×D₄	1	2	yes	—
144	C₄²×C₆₇	C₂×D₄	1	2	yes	—
145	C₄²×C₆₇	C₂×D₄	1	2	yes	—
146	C₄²×C₆₇	C₂×D₄	1	2	yes	—
147	C₂³×C₆₇	—	2	2	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	2	4	no	—
150	C₂³×C₆₇	—	4	1	no	—
151	C₂³×C₆₇	—	4	1	no	—
152	C₂³×C₆₇	—	4	2	no	—
153	C₂³×C₆₇	—	4	2	no	—
154	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
156	Q₈×C₆₇	—	4	1	no	—
157	Q₈×C₆₇	—	4	1	no	—
158	Q₈×C₆₇	—	4	2	no	—
159	Q₈×C₆₇	—	8	4	no	—
160	D₄×C₆₇	—	4	1	no	—
161	D₄×C₆₇	—	4	1	no	—
162	D₄×C₆₇	—	4	2	no	—
163	D₄×C₆₇	—	8	4	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	8	4	no	—
179	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
187	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
188	C₈×C₆₇	—	4	4	no	—
189	C₈×C₆₇	—	4	4	no	—
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	—	2	2	no	—
193	C₂²×C₆₇	—	2	2	no	—
194	C₂²×C₆₇	—	2	2	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	8	2	no	—
199	C₂²×C₆₇	—	8	4	no	—
200	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
201	C₂²×C₆₇	#520	1	2	yes	—
202	C₄×C₆₇	—	2	2	no	—
203	C₄×C₆₇	—	2	2	no	—
204	C₄×C₆₇	—	2	2	no	—
205	C₄×C₆₇	—	2	2	no	—
206	C₄×C₆₇	—	2	2	no	—
207	C₄×C₆₇	—	4	2	no	—
208	C₄×C₆₇	—	4	2	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	#535	1	1	yes	characteristic
211	C₄×C₆₇	#536	1	1	yes	characteristic
212	C₄×C₆₇	#537	1	2	yes	—
213	#21881	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂×C₆₇	—	2	2	no	—
215	C₂×C₆₇	—	4	2	no	—
216	C₂×C₆₇	—	8	2	no	—
217	C₂×C₆₇	#2311	1	1	yes	characteristic
218	C₂×C₆₇	#2888	1	1	yes	characteristic
219	C₂×C₆₇	#2889	1	1	yes	characteristic
220	C₂×#542	C₂×C₆₇	1	1	yes	characteristic
221	#2921	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#2905	C₂×C₆₇	1	1	yes	characteristic, complemented
223	#1997	C₂×C₆₇	1	2	yes	complemented
224	#1967	C₂×C₆₇	1	2	yes	complemented
225	#1901	C₂×C₆₇	1	2	yes	complemented
226	#1900	C₂×C₆₇	1	2	yes	—
227	#1801	C₂×C₆₇	1	2	yes	complemented
228	#1800	C₂×C₆₇	1	2	yes	complemented
229	C₆₇	#21881	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
231	#542	—	2	4	no	—
232	#542	—	2	4	no	—
233	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#126	C₄×C₆₇	1	2	yes	—
235	C₂×#124	C₄×C₆₇	1	2	yes	—
236	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#123	C₄×C₆₇	1	2	yes	—
238	C₂×C₂²⋊Q₈	C₄×C₆₇	1	2	yes	—
239	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
244	#433	—	2	4	no	—
245	#424	—	2	4	no	—
246	#423	C₂²×C₆₇	1	2	yes	—
247	#423	C₂²×C₆₇	1	2	yes	—
248	#423	C₂²×C₆₇	1	2	yes	—
249	C₂×#101	C₂²×C₆₇	1	4	yes	—
250	C₂×#100	C₂²×C₆₇	1	4	yes	—
251	#407	C₂²×C₆₇	1	2	yes	—
252	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
253	#343	C₂²×C₆₇	1	4	yes	—
254	#343	C₂²×C₆₇	1	4	yes	—
255	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
256	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
257	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
258	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
259	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
260	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	4	4	no	—
263	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
264	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
265	C₂²×C₈	—	2	2	no	—
266	C₄⋊Q₈	—	2	2	no	—
267	C₄⋊Q₈	—	2	2	no	—
268	#126	—	2	8	no	—
269	#124	—	2	8	no	—
270	#124	—	2	2	no	—
271	#124	—	2	2	no	—
272	#123	—	2	8	no	—
273	C₂²⋊Q₈	—	2	2	no	—
274	C₂²⋊Q₈	—	2	2	no	—
275	C₂²⋊Q₈	—	2	8	no	—
276	#121	—	2	2	no	—
277	#121	—	2	2	no	—
278	C₄×Q₈	—	2	2	no	—
279	C₄×Q₈	—	2	2	no	—
280	C₄×D₄	—	2	2	no	—
281	C₄×D₄	—	2	2	no	—
282	#117	D₄×C₆₇	1	4	yes	—
283	#117	D₄×C₆₇	1	4	yes	—
284	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
285	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
286	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
287	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
291	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
292	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
293	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
294	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
295	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
296	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
297	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
298	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
299	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
301	C₄⋊C₈	—	2	4	no	—
302	C₄⋊C₈	—	2	4	no	—
303	C₄⋊C₈	—	2	4	no	—
304	#101	—	4	4	no	—
305	#100	—	4	4	no	—
306	C₂²⋊C₈	—	2	4	no	—
307	#97	—	2	2	no	—
308	C₄×C₈	—	2	2	no	—
309	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
310	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
311	C₂×Q₈	—	2	4	no	—
312	C₂×Q₈	—	2	1	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	8	2	no	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	8	2	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	8	2	no	—
327	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
328	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
329	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
330	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
331	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
332	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
333	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
334	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
338	M₁₆	—	2	4	no	—
339	M₁₆	—	2	4	no	—
340	M₁₆	—	2	4	no	—
341	M₁₆	—	2	4	no	—
342	M₁₆	—	4	4	no	—
343	C₂×C₈	—	2	4	no	—
344	C₂×C₈	—	2	4	no	—
345	C₂×C₈	—	4	4	no	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	—	2	4	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
354	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
355	C₂²⋊C₄	—	2	4	no	—
356	C₂²⋊C₄	—	2	4	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₂²⋊C₄	—	2	4	no	—
359	C₂²⋊C₄	—	2	4	no	—
360	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
361	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
362	C₄²	—	2	4	no	—
363	C₄²	C₂×D₄×C₆₇	1	2	yes	—
364	C₄²	C₂×D₄×C₆₇	1	2	yes	—
365	C₄²	C₂×D₄×C₆₇	1	2	yes	—
366	C₄²	C₂×D₄×C₆₇	1	2	yes	—
367	C₂³	—	2	2	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	4	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	4	2	no	—
374	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
375	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
376	Q₈	—	4	1	no	—
377	Q₈	—	4	1	no	—
378	Q₈	—	4	2	no	—
379	Q₈	—	8	4	no	—
380	D₄	—	4	1	no	—
381	D₄	—	4	1	no	—
382	D₄	—	4	2	no	—
383	D₄	—	8	4	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	8	4	no	—
399	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
400	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
401	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
402	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
403	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
404	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
405	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
406	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
407	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
408	C₈	—	4	4	no	—
409	C₈	—	4	4	no	—
410	C₂²	—	2	1	no	—
411	C₂²	—	2	1	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	2	2	no	—
414	C₂²	—	2	2	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	8	2	no	—
419	C₂²	—	8	4	no	—
420	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
421	C₂²	#520×C₆₇	1	2	yes	—
422	C₄	—	2	2	no	—
423	C₄	—	2	2	no	—
424	C₄	—	2	2	no	—
425	C₄	—	2	2	no	—
426	C₄	—	2	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	4	2	no	—
429	C₄	—	4	2	no	—
430	C₄	#535×C₆₇	1	1	yes	characteristic
431	C₄	#536×C₆₇	1	1	yes	characteristic
432	C₄	#537×C₆₇	1	2	yes	—
433	C₂	—	2	2	no	—
434	C₂	—	4	2	no	—
435	C₂	—	8	2	no	—
436	C₂	C₆₇×#2311	1	1	yes	characteristic
437	C₂	C₆₇×#2888	1	1	yes	characteristic
438	C₂	C₆₇×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2921	C₂	semidirect
3. C₆₇×#2905	C₂	semidirect
4. C₆₇×#1997	C₂	semidirect
5. C₆₇×#1967	C₂	semidirect
6. C₆₇×#1901	C₂	semidirect
8. C₆₇×#1801	C₂	semidirect
9. C₆₇×#1800	C₂	semidirect
213. #21881	C₆₇	direct (N × H)
221. #2921	C₂×C₆₇	semidirect
222. #2905	C₂×C₆₇	semidirect
223. #1997	C₂×C₆₇	semidirect
224. #1967	C₂×C₆₇	semidirect
225. #1901	C₂×C₆₇	semidirect
227. #1801	C₂×C₆₇	semidirect
228. #1800	C₂×C₆₇	semidirect
229. C₆₇	#21881	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×#542×C₆₇	2
C₆₇×#2921	2
C₆₇×#2905	2
C₆₇×#1997	2
C₆₇×#1967	2
C₆₇×#1901	2
C₆₇×#1900	2
C₆₇×#1801	2
C₆₇×#1800	2
#21881	67

Subgroups — C67×#21881 #30701132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21881 #30701132