Subgroups — C₆₇×#26041 #30705292

Order 17152.

862 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#478×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2552	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2087	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2059	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1862	C₂	1	2	yes	maximal
9	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#125×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
16	#478×C₆₇	—	2	8	no	—
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#106×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#398×C₆₇	—	2	2	no	—
27	#389×C₆₇	—	2	1	no	—
28	#389×C₆₇	C₂²	1	2	yes	—
29	#389×C₆₇	C₂²	1	2	yes	—
30	#385×C₆₇	—	2	2	no	—
31	#384×C₆₇	—	2	1	no	—
32	#384×C₆₇	C₂²	1	1	yes	characteristic
33	#384×C₆₇	C₂²	1	1	yes	characteristic
34	#380×C₆₇	—	2	1	no	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	#340×C₆₇	C₂²	1	1	yes	characteristic
38	#340×C₆₇	C₂²	1	1	yes	characteristic
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	—	2	1	no	—
44	C₂³×C₄×C₆₇	—	2	2	no	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	#125×C₆₇	—	2	4	no	—
49	#125×C₆₇	—	4	2	no	—
50	C₄×D₄×C₆₇	—	2	4	no	—
51	C₄×D₄×C₆₇	—	2	2	no	—
52	C₄×D₄×C₆₇	—	4	2	no	—
53	C₄×D₄×C₆₇	Q₈	1	4	yes	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
63	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	1	no	—
73	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	—	2	4	no	—
76	#107×C₆₇	—	4	2	no	—
77	#106×C₆₇	—	2	4	no	—
78	C₄⋊C₈×C₆₇	—	2	4	no	—
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	#102×C₆₇	—	2	4	no	—
81	#102×C₆₇	D₄	1	4	yes	—
82	#102×C₆₇	Q₈	1	4	yes	—
83	C₄×C₈×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	1	no	—
85	#95×C₆₇	—	2	2	no	—
86	#95×C₆₇	—	2	2	no	—
87	#95×C₆₇	D₄	1	2	yes	—
88	C₂⁴×C₆₇	—	2	1	no	—
89	C₂×D₄×C₆₇	—	2	1	no	—
90	C₂×D₄×C₆₇	—	2	1	no	—
91	C₂×D₄×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
94	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	4	4	no	—
102	C₂²×C₄×C₆₇	—	4	4	no	—
103	C₂²×C₄×C₆₇	—	4	2	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	—	8	2	no	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
111	C₂×C₈×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	4	4	no	—
130	C₄⋊C₄×C₆₇	—	4	4	no	—
131	C₄⋊C₄×C₆₇	—	4	4	no	—
132	C₄⋊C₄×C₆₇	—	4	2	no	—
133	C₄⋊C₄×C₆₇	—	4	2	no	—
134	C₄⋊C₄×C₆₇	—	8	1	no	—
135	C₄⋊C₄×C₆₇	—	8	1	no	—
136	C₄⋊C₄×C₆₇	—	8	2	no	—
137	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
139	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₂²⋊C₄×C₆₇	—	4	4	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	4	2	no	—
145	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
146	C₄²×C₆₇	#41	1	2	yes	—
147	C₂³×C₆₇	—	2	2	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	2	2	no	—
150	C₂³×C₆₇	—	4	1	no	—
151	C₂³×C₆₇	—	4	1	no	—
152	C₂³×C₆₇	—	4	1	no	—
153	C₂³×C₆₇	—	4	1	no	—
154	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
155	D₄×C₆₇	—	2	2	no	—
156	D₄×C₆₇	—	2	2	no	—
157	D₄×C₆₇	—	2	2	no	—
158	D₄×C₆₇	C₄⋊Q₈	1	4	yes	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	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	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	1	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	8	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	8	8	no	—
184	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
185	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
186	C₂×C₄×C₆₇	#135	1	2	yes	—
187	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
188	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
193	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
194	C₈×C₆₇	—	2	2	no	—
195	C₈×C₆₇	—	2	2	no	—
196	C₈×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	—	2	4	no	—
198	C₂²×C₆₇	—	4	2	no	—
199	C₂²×C₆₇	—	4	2	no	—
200	C₂²×C₆₇	—	4	2	no	—
201	C₂²×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	4	2	no	—
204	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
205	C₂²×C₆₇	D₄²	1	1	yes	characteristic
206	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
207	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
208	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	C₂²×C₆₇	#576	1	1	yes	characteristic
210	C₂²×C₆₇	#578	1	1	yes	characteristic
211	C₄×C₆₇	—	2	2	no	—
212	C₄×C₆₇	—	4	2	no	—
213	C₄×C₆₇	—	4	2	no	—
214	C₄×C₆₇	—	4	2	no	—
215	C₄×C₆₇	—	4	2	no	—
216	C₄×C₆₇	—	4	2	no	—
217	C₄×C₆₇	—	8	2	no	—
218	C₄×C₆₇	—	8	2	no	—
219	C₄×C₆₇	#478	1	4	yes	complemented
220	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
221	C₄×C₆₇	#544	1	1	yes	characteristic
222	C₄×C₆₇	#547	1	1	yes	characteristic
223	C₄×C₆₇	#554	1	1	yes	characteristic
224	#26041	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂×C₆₇	—	4	4	no	—
226	C₂×C₆₇	#2440	1	1	yes	characteristic
227	C₂×C₆₇	C₂×#478	1	1	yes	characteristic
228	C₂×C₆₇	#3083	1	1	yes	characteristic
229	C₂×C₆₇	#3274	1	1	yes	characteristic
230	C₂×C₆₇	#3283	1	1	yes	characteristic
231	C₂×C₆₇	#3380	1	1	yes	characteristic
232	C₂×C₆₇	#3386	1	1	yes	characteristic
233	C₂×#478	C₂×C₆₇	1	2	yes	—
234	#2552	C₂×C₆₇	1	1	yes	characteristic, complemented
235	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
236	#2087	C₂×C₆₇	1	2	yes	complemented
237	#2059	C₂×C₆₇	1	2	yes	complemented
238	#1948	C₂×C₆₇	1	1	yes	characteristic, complemented
239	#1939	C₂×C₆₇	1	1	yes	characteristic, complemented
240	#1862	C₂×C₆₇	1	2	yes	—
241	#1845	C₂×C₆₇	1	1	yes	characteristic, complemented
242	#1802	C₂×C₆₇	1	1	yes	characteristic
243	C₄×#102	C₂×C₆₇	1	1	yes	characteristic
244	C₆₇	#26041	1	1	yes	characteristic, direct product factor, Sylow
245	C₂×#125	C₂²×C₆₇	1	2	yes	—
246	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
247	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
248	C₂²×C₄⋊C₄	—	2	1	no	—
249	#478	—	2	8	no	—
250	C₂×#107	C₂²×C₆₇	1	2	yes	—
251	C₂×#107	C₂²×C₆₇	1	2	yes	—
252	C₂×#106	C₂²×C₆₇	1	2	yes	—
253	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
254	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×#102	C₂²×C₆₇	1	2	yes	—
256	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
259	#398	—	2	2	no	—
260	#389	—	2	1	no	—
261	#389	C₂²×C₆₇	1	2	yes	—
262	#389	C₂²×C₆₇	1	2	yes	—
263	#385	—	2	2	no	—
264	#384	—	2	1	no	—
265	#384	C₂²×C₆₇	1	1	yes	characteristic
266	#384	C₂²×C₆₇	1	1	yes	characteristic
267	#380	—	2	1	no	—
268	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
269	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
270	#340	C₂²×C₆₇	1	2	yes	—
271	#340	C₂²×C₆₇	1	1	yes	characteristic
272	#340	C₂²×C₆₇	1	1	yes	characteristic
273	#340	C₂²×C₆₇	1	2	yes	—
274	#336	C₂²×C₆₇	1	2	yes	—
275	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
276	C₂³×C₄	—	2	1	no	—
277	C₂³×C₄	—	2	2	no	—
278	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
279	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
281	#125	—	2	4	no	—
282	#125	—	4	2	no	—
283	C₄×D₄	—	2	4	no	—
284	C₄×D₄	—	2	2	no	—
285	C₄×D₄	—	4	2	no	—
286	C₄×D₄	Q₈×C₆₇	1	4	yes	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	4	4	no	—
296	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
297	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
303	C₂×C₂²⋊C₄	—	2	1	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₄²	—	2	1	no	—
306	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
307	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
308	#107	—	2	4	no	—
309	#107	—	4	2	no	—
310	#106	—	2	4	no	—
311	C₄⋊C₈	—	2	4	no	—
312	C₄⋊C₈	—	2	2	no	—
313	#102	—	2	4	no	—
314	#102	D₄×C₆₇	1	4	yes	—
315	#102	Q₈×C₆₇	1	4	yes	—
316	C₄×C₈	—	2	2	no	—
317	#95	—	2	1	no	—
318	#95	—	2	2	no	—
319	#95	—	2	2	no	—
320	#95	D₄×C₆₇	1	2	yes	—
321	C₂⁴	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
326	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
327	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	4	4	no	—
335	C₂²×C₄	—	4	4	no	—
336	C₂²×C₄	—	4	2	no	—
337	C₂²×C₄	—	4	2	no	—
338	C₂²×C₄	—	8	2	no	—
339	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
340	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
352	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	4	4	no	—
363	C₄⋊C₄	—	4	4	no	—
364	C₄⋊C₄	—	4	4	no	—
365	C₄⋊C₄	—	4	2	no	—
366	C₄⋊C₄	—	4	2	no	—
367	C₄⋊C₄	—	8	1	no	—
368	C₄⋊C₄	—	8	1	no	—
369	C₄⋊C₄	—	8	2	no	—
370	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
371	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
372	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₂²⋊C₄	—	4	4	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	4	2	no	—
378	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
379	C₄²	#41×C₆₇	1	2	yes	—
380	C₂³	—	2	2	no	—
381	C₂³	—	2	2	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	4	1	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	1	no	—
386	C₂³	—	4	1	no	—
387	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
388	D₄	—	2	2	no	—
389	D₄	—	2	2	no	—
390	D₄	—	2	2	no	—
391	D₄	C₄⋊Q₈×C₆₇	1	4	yes	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	8	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	8	8	no	—
417	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
418	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
419	C₂×C₄	#135×C₆₇	1	2	yes	—
420	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
421	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
422	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
423	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
425	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
426	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
427	C₈	—	2	2	no	—
428	C₈	—	2	2	no	—
429	C₈	—	4	2	no	—
430	C₂²	—	2	4	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
438	C₂²	D₄²×C₆₇	1	1	yes	characteristic
439	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
440	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
441	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
442	C₂²	#576×C₆₇	1	1	yes	characteristic
443	C₂²	#578×C₆₇	1	1	yes	characteristic
444	C₄	—	2	2	no	—
445	C₄	—	4	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	4	2	no	—
449	C₄	—	4	2	no	—
450	C₄	—	8	2	no	—
451	C₄	—	8	2	no	—
452	C₄	#478×C₆₇	1	4	yes	complemented
453	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
454	C₄	#544×C₆₇	1	1	yes	characteristic
455	C₄	#547×C₆₇	1	1	yes	characteristic
456	C₄	#554×C₆₇	1	1	yes	characteristic
457	C₂	—	4	4	no	—
458	C₂	C₆₇×#2440	1	1	yes	characteristic
459	C₂	C₂×#478×C₆₇	1	1	yes	characteristic
460	C₂	C₆₇×#3083	1	1	yes	characteristic
461	C₂	C₆₇×#3274	1	1	yes	characteristic
462	C₂	C₆₇×#3283	1	1	yes	characteristic
463	C₂	C₆₇×#3380	1	1	yes	characteristic
464	C₂	C₆₇×#3386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₄⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2552	C₂	semidirect
4. C₆₇×#2087	C₂	semidirect
5. C₆₇×#2059	C₂	semidirect
6. C₆₇×#1948	C₂	semidirect
7. C₆₇×#1939	C₂	semidirect
9. C₆₇×#1845	C₂	semidirect
219. C₄×C₆₇	#478	semidirect
224. #26041	C₆₇	direct (N × H)
234. #2552	C₂×C₆₇	semidirect
236. #2087	C₂×C₆₇	semidirect
237. #2059	C₂×C₆₇	semidirect
238. #1948	C₂×C₆₇	semidirect
239. #1939	C₂×C₆₇	semidirect
241. #1845	C₂×C₆₇	semidirect
244. C₆₇	#26041	direct (N × H)
452. C₄	#478×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#478×C₆₇	2
C₆₇×#2552	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#2087	2
C₆₇×#2059	2
C₆₇×#1948	2
C₆₇×#1939	2
C₆₇×#1862	2
C₆₇×#1845	2
C₆₇×#1802	2
C₄×#102×C₆₇	2
#26041	67

Subgroups — C67×#26041 #30705292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#26041 #30705292