Subgroups — C₆₇×#18004 #30697255

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3103	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2935	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1592	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1588	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1585	C₂	1	2	yes	maximal, complemented
7	C₆₇×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1571	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1492	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₆₇	—	2	1	no	—
13	#532×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#481×C₆₇	C₄	1	2	yes	—
16	#475×C₆₇	C₄	1	1	yes	characteristic
17	#475×C₆₇	C₄	1	1	yes	characteristic
18	#460×C₆₇	C₄	1	1	yes	characteristic
19	#460×C₆₇	C₄	1	1	yes	characteristic
20	#448×C₆₇	C₄	1	2	yes	—
21	#446×C₆₇	—	2	2	no	—
22	#446×C₆₇	C₂²	1	2	yes	—
23	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
25	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
26	#436×C₆₇	C₂²	1	2	yes	—
27	#435×C₆₇	—	2	2	no	—
28	C₈×D₄×C₆₇	C₂²	1	2	yes	—
29	#431×C₆₇	C₂²	1	1	yes	characteristic
30	#423×C₆₇	—	2	1	no	—
31	#423×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#408×C₆₇	—	2	1	no	—
34	C₄×M₁₆×C₆₇	—	2	1	no	—
35	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#335×C₆₇	C₂²	1	2	yes	—
38	#335×C₆₇	C₂²	1	2	yes	—
39	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
40	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
41	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
44	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
45	#322×C₆₇	C₂²	1	2	yes	—
46	#322×C₆₇	C₂²	1	2	yes	—
47	C₂×#41×C₆₇	—	2	1	no	—
48	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
51	#131×C₆₇	—	4	4	no	—
52	C₂×M₁₆×C₆₇	—	2	1	no	—
53	C₂×M₁₆×C₆₇	—	2	2	no	—
54	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₂²×C₈×C₆₇	—	2	1	no	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
61	#124×C₆₇	—	2	1	no	—
62	#121×C₆₇	—	2	1	no	—
63	#121×C₆₇	C₂×C₄	1	2	yes	—
64	C₄×Q₈×C₆₇	—	2	1	no	—
65	C₄×Q₈×C₆₇	—	2	1	no	—
66	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₄×D₄×C₆₇	—	2	1	no	—
71	C₄×D₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#106×C₆₇	C₂×C₄	1	2	yes	—
77	#106×C₆₇	C₂×C₄	1	2	yes	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	C₄⋊C₈×C₆₇	—	2	1	no	—
81	C₄⋊C₈×C₆₇	—	2	1	no	—
82	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
85	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
87	#103×C₆₇	C₂×C₄	1	2	yes	—
88	#103×C₆₇	C₂×C₄	1	2	yes	—
89	#102×C₆₇	C₂×C₄	1	2	yes	—
90	#102×C₆₇	C₂×C₄	1	2	yes	—
91	C₂²⋊C₈×C₆₇	—	2	2	no	—
92	C₂²⋊C₈×C₆₇	—	2	2	no	—
93	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
94	#97×C₆₇	—	2	2	no	—
95	#97×C₆₇	D₄	1	2	yes	—
96	C₄×C₈×C₆₇	—	2	2	no	—
97	C₄×C₈×C₆₇	—	2	1	no	—
98	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	2	yes	—
101	C₄×C₈×C₆₇	C₂³	1	2	yes	—
102	#41×C₆₇	—	4	2	no	—
103	#41×C₆₇	—	4	2	no	—
104	C₂×Q₈×C₆₇	—	2	1	no	—
105	C₂×Q₈×C₆₇	—	2	1	no	—
106	C₂×Q₈×C₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	4	1	no	—
108	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	1	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
120	SD₁₆×C₆₇	—	2	4	no	—
121	SD₁₆×C₆₇	—	4	2	no	—
122	M₁₆×C₆₇	—	2	2	no	—
123	M₁₆×C₆₇	—	2	2	no	—
124	M₁₆×C₆₇	—	4	4	no	—
125	M₁₆×C₆₇	—	4	2	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	2	1	no	—
131	C₂×C₈×C₆₇	—	4	2	no	—
132	C₂×C₈×C₆₇	—	4	4	no	—
133	C₂×C₈×C₆₇	—	4	2	no	—
134	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
135	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
136	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	2	1	no	—
140	C₄⋊C₄×C₆₇	—	2	1	no	—
141	C₄⋊C₄×C₆₇	—	2	1	no	—
142	C₄⋊C₄×C₆₇	—	2	1	no	—
143	C₄⋊C₄×C₆₇	—	2	1	no	—
144	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂²⋊C₄×C₆₇	—	2	2	no	—
148	C₂²⋊C₄×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	—	2	1	no	—
151	C₄²×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
154	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₆₇	#41	1	1	yes	characteristic
156	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
157	C₂³×C₆₇	—	2	2	no	—
158	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
159	Q₈×C₆₇	—	2	1	no	—
160	Q₈×C₆₇	—	2	1	no	—
161	Q₈×C₆₇	—	2	1	no	—
162	Q₈×C₆₇	—	4	2	no	—
163	Q₈×C₆₇	—	4	2	no	—
164	Q₈×C₆₇	#131	1	2	yes	complemented
165	D₄×C₆₇	—	2	2	no	—
166	D₄×C₆₇	—	2	2	no	—
167	D₄×C₆₇	—	4	4	no	—
168	D₄×C₆₇	—	4	2	no	—
169	D₄×C₆₇	—	8	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	4	1	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	—	8	1	no	—
182	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
186	C₂×C₄×C₆₇	#137	1	2	yes	—
187	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
190	C₈×C₆₇	—	2	2	no	—
191	C₈×C₆₇	—	2	2	no	—
192	C₈×C₆₇	—	2	2	no	—
193	C₈×C₆₇	—	2	2	no	—
194	C₈×C₆₇	—	2	2	no	—
195	C₈×C₆₇	—	4	1	no	—
196	C₈×C₆₇	—	4	1	no	—
197	C₈×C₆₇	—	4	1	no	—
198	C₈×C₆₇	—	4	2	no	—
199	C₂²×C₆₇	—	2	1	no	—
200	C₂²×C₆₇	—	2	1	no	—
201	C₂²×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	4	2	no	—
204	C₂²×C₆₇	#440	1	2	yes	—
205	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
206	C₄×C₆₇	—	2	1	no	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	2	1	no	—
210	C₄×C₆₇	—	4	1	no	—
211	C₄×C₆₇	—	4	2	no	—
212	C₄×C₆₇	—	4	1	no	—
213	C₄×C₆₇	—	4	1	no	—
214	C₄×C₆₇	#440	1	2	yes	—
215	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
216	C₄×C₆₇	#568	1	1	yes	characteristic
217	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
219	#18004	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂×C₆₇	—	2	2	no	—
221	C₂×C₆₇	—	8	2	no	—
222	C₂×C₆₇	#2933	1	1	yes	characteristic
223	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
224	C₂×C₆₇	#2956	1	1	yes	characteristic
225	#3103	C₂×C₆₇	1	1	yes	characteristic
226	Q₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#2935	C₂×C₆₇	1	1	yes	characteristic
228	#1592	C₂×C₆₇	1	2	yes	complemented
229	#1588	C₂×C₆₇	1	2	yes	complemented
230	#1585	C₂×C₆₇	1	2	yes	complemented
231	SD₁₆⋊C₈	C₂×C₆₇	1	2	yes	complemented
232	#1571	C₂×C₆₇	1	1	yes	characteristic, complemented
233	#1492	C₂×C₆₇	1	1	yes	characteristic
234	C₂×Q₈⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#1451	C₂×C₆₇	1	1	yes	characteristic, complemented
236	C₆₇	#18004	1	1	yes	characteristic, direct product factor, Sylow
237	C₂×#131	—	2	1	no	—
238	#532	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
240	#481	C₄×C₆₇	1	2	yes	—
241	#475	C₄×C₆₇	1	1	yes	characteristic
242	#475	C₄×C₆₇	1	1	yes	characteristic
243	#460	C₄×C₆₇	1	1	yes	characteristic
244	#460	C₄×C₆₇	1	1	yes	characteristic
245	#448	C₄×C₆₇	1	2	yes	—
246	#446	—	2	2	no	—
247	#446	C₂²×C₆₇	1	2	yes	—
248	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
249	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
250	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
251	#436	C₂²×C₆₇	1	2	yes	—
252	#435	—	2	2	no	—
253	C₈×D₄	C₂²×C₆₇	1	2	yes	—
254	#431	C₂²×C₆₇	1	1	yes	characteristic
255	#423	—	2	1	no	—
256	#423	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
258	#408	—	2	1	no	—
259	C₄×M₁₆	—	2	1	no	—
260	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
262	#335	C₂²×C₆₇	1	2	yes	—
263	#335	C₂²×C₆₇	1	2	yes	—
264	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
265	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
266	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
267	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
268	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
269	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
270	#322	C₂²×C₆₇	1	2	yes	—
271	#322	C₂²×C₆₇	1	2	yes	—
272	C₂×#41	—	2	1	no	—
273	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
274	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
275	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
276	#131	—	4	4	no	—
277	C₂×M₁₆	—	2	1	no	—
278	C₂×M₁₆	—	2	2	no	—
279	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
280	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
281	C₂²×C₈	—	2	1	no	—
282	C₂²×C₈	—	2	1	no	—
283	C₂²×C₈	—	2	2	no	—
284	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
285	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
286	#124	—	2	1	no	—
287	#121	—	2	1	no	—
288	#121	C₂×C₄×C₆₇	1	2	yes	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
292	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
293	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
294	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
295	C₄×D₄	—	2	1	no	—
296	C₄×D₄	C₂³×C₆₇	1	2	yes	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
299	C₂×C₄²	—	2	1	no	—
300	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
301	#106	C₂×C₄×C₆₇	1	2	yes	—
302	#106	C₂×C₄×C₆₇	1	2	yes	—
303	C₄⋊C₈	—	2	2	no	—
304	C₄⋊C₈	—	2	2	no	—
305	C₄⋊C₈	—	2	1	no	—
306	C₄⋊C₈	—	2	1	no	—
307	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
308	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
309	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
310	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
311	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
312	#103	C₂×C₄×C₆₇	1	2	yes	—
313	#103	C₂×C₄×C₆₇	1	2	yes	—
314	#102	C₂×C₄×C₆₇	1	2	yes	—
315	#102	C₂×C₄×C₆₇	1	2	yes	—
316	C₂²⋊C₈	—	2	2	no	—
317	C₂²⋊C₈	—	2	2	no	—
318	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
319	#97	—	2	2	no	—
320	#97	D₄×C₆₇	1	2	yes	—
321	C₄×C₈	—	2	2	no	—
322	C₄×C₈	—	2	1	no	—
323	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
324	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
325	C₄×C₈	C₂³×C₆₇	1	2	yes	—
326	C₄×C₈	C₂³×C₆₇	1	2	yes	—
327	#41	—	4	2	no	—
328	#41	—	4	2	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	4	1	no	—
333	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
334	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
345	SD₁₆	—	2	4	no	—
346	SD₁₆	—	4	2	no	—
347	M₁₆	—	2	2	no	—
348	M₁₆	—	2	2	no	—
349	M₁₆	—	4	4	no	—
350	M₁₆	—	4	2	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	4	4	no	—
357	C₂×C₈	—	4	2	no	—
358	C₂×C₈	—	4	2	no	—
359	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
360	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
361	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
362	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
370	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
371	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
372	C₂²⋊C₄	—	2	2	no	—
373	C₂²⋊C₄	—	2	2	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	1	no	—
376	C₄²	—	2	1	no	—
377	C₄²	—	2	1	no	—
378	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
379	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
380	C₄²	#41×C₆₇	1	1	yes	characteristic
381	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
382	C₂³	—	2	2	no	—
383	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
384	Q₈	—	2	1	no	—
385	Q₈	—	2	1	no	—
386	Q₈	—	2	1	no	—
387	Q₈	—	4	2	no	—
388	Q₈	—	4	2	no	—
389	Q₈	#131×C₆₇	1	2	yes	complemented
390	D₄	—	2	2	no	—
391	D₄	—	2	2	no	—
392	D₄	—	4	4	no	—
393	D₄	—	4	2	no	—
394	D₄	—	8	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	8	1	no	—
407	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
409	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
410	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
411	C₂×C₄	#137×C₆₇	1	2	yes	—
412	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
415	C₈	—	2	2	no	—
416	C₈	—	2	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	4	1	no	—
421	C₈	—	4	1	no	—
422	C₈	—	4	1	no	—
423	C₈	—	4	2	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	#440×C₆₇	1	2	yes	—
430	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
431	C₄	—	2	1	no	—
432	C₄	—	2	2	no	—
433	C₄	—	2	1	no	—
434	C₄	—	2	1	no	—
435	C₄	—	4	1	no	—
436	C₄	—	4	2	no	—
437	C₄	—	4	1	no	—
438	C₄	—	4	1	no	—
439	C₄	#440×C₆₇	1	2	yes	—
440	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
441	C₄	#568×C₆₇	1	1	yes	characteristic
442	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
443	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
444	C₂	—	2	2	no	—
445	C₂	—	8	2	no	—
446	C₂	C₆₇×#2933	1	1	yes	characteristic
447	C₂	C₂×#440×C₆₇	1	1	yes	characteristic
448	C₂	C₆₇×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#532×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₄×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1592	C₂	semidirect
5. C₆₇×#1588	C₂	semidirect
6. C₆₇×#1585	C₂	semidirect
7. C₆₇×SD₁₆⋊C₈	C₂	semidirect
8. C₆₇×#1571	C₂	semidirect
10. C₂×Q₈⋊C₈×C₆₇	C₂	semidirect
11. C₆₇×#1451	C₂	semidirect
164. Q₈×C₆₇	#131	semidirect
219. #18004	C₆₇	direct (N × H)
226. Q₈×M₁₆	C₂×C₆₇	semidirect
228. #1592	C₂×C₆₇	semidirect
229. #1588	C₂×C₆₇	semidirect
230. #1585	C₂×C₆₇	semidirect
231. SD₁₆⋊C₈	C₂×C₆₇	semidirect
232. #1571	C₂×C₆₇	semidirect
234. C₂×Q₈⋊C₈	C₂×C₆₇	semidirect
235. #1451	C₂×C₆₇	semidirect
236. C₆₇	#18004	direct (N × H)
389. Q₈	#131×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3103	2
Q₈×M₁₆×C₆₇	2
C₆₇×#2935	2
C₆₇×#1592	2
C₆₇×#1588	2
C₆₇×#1585	2
C₆₇×SD₁₆⋊C₈	2
C₆₇×#1571	2
C₆₇×#1492	2
C₂×Q₈⋊C₈×C₆₇	2
C₆₇×#1451	2
#18004	67

Subgroups — C67×#18004 #30697255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18004 #30697255