Subgroups — C₆₇×#33626 #30712877

Order 17152.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2229	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2228	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2225	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2224	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2158	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2138	C₂	1	2	yes	maximal
12	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
13	#533×C₆₇	—	2	1	no	—
14	C₄×#41×C₆₇	—	2	1	no	—
15	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
16	#500×C₆₇	C₂²	1	1	yes	characteristic
17	#500×C₆₇	C₂²	1	1	yes	characteristic
18	#499×C₆₇	—	2	1	no	—
19	#499×C₆₇	C₂²	1	2	yes	—
20	#498×C₆₇	—	2	1	no	—
21	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×SD₁₆×C₆₇	—	2	1	no	—
24	C₄×SD₁₆×C₆₇	—	2	1	no	—
25	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
27	#427×C₆₇	—	2	1	no	—
28	#427×C₆₇	C₂²	1	2	yes	—
29	#427×C₆₇	C₂²	1	1	yes	characteristic
30	#416×C₆₇	—	2	1	no	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#367×C₆₇	C₂²	1	2	yes	—
33	#367×C₆₇	C₂²	1	2	yes	—
34	#366×C₆₇	C₂²	1	2	yes	—
35	#366×C₆₇	C₂²	1	2	yes	—
36	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
39	#358×C₆₇	C₂²	1	2	yes	—
40	#358×C₆₇	C₂²	1	2	yes	—
41	#357×C₆₇	C₂²	1	2	yes	—
42	#357×C₆₇	C₂²	1	2	yes	—
43	C₂×#41×C₆₇	—	2	1	no	—
44	#135×C₆₇	—	2	2	no	—
45	#135×C₆₇	Q₈	1	4	yes	complemented
46	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×SD₁₆×C₆₇	—	2	1	no	—
48	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₆₇	—	2	1	no	—
52	C₄⋊Q₈×C₆₇	—	2	1	no	—
53	#125×C₆₇	—	2	2	no	—
54	C₂²⋊Q₈×C₆₇	—	4	1	no	—
55	C₂²⋊Q₈×C₆₇	—	4	1	no	—
56	C₄×Q₈×C₆₇	—	2	1	no	—
57	C₄×Q₈×C₆₇	—	2	1	no	—
58	C₄×Q₈×C₆₇	—	4	1	no	—
59	C₄×Q₈×C₆₇	—	4	1	no	—
60	C₄×D₄×C₆₇	—	2	1	no	—
61	C₄×D₄×C₆₇	—	2	1	no	—
62	C₄×D₄×C₆₇	—	4	1	no	—
63	C₄×D₄×C₆₇	—	4	1	no	—
64	#117×C₆₇	—	2	1	no	—
65	#117×C₆₇	—	2	2	no	—
66	#117×C₆₇	—	4	1	no	—
67	C₂×C₄²×C₆₇	—	4	1	no	—
68	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
69	C₂×C₁₆×C₆₇	—	2	4	no	—
70	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
71	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
72	#107×C₆₇	—	2	1	no	—
73	#107×C₆₇	—	2	1	no	—
74	#107×C₆₇	C₂³	1	2	yes	—
75	#107×C₆₇	C₂³	1	2	yes	—
76	#107×C₆₇	C₂³	1	1	yes	characteristic
77	#107×C₆₇	C₂³	1	1	yes	characteristic
78	#106×C₆₇	—	2	2	no	—
79	#106×C₆₇	—	2	2	no	—
80	#106×C₆₇	—	2	1	no	—
81	#103×C₆₇	—	2	1	no	—
82	#103×C₆₇	—	2	1	no	—
83	#102×C₆₇	—	2	1	no	—
84	#102×C₆₇	—	2	1	no	—
85	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
86	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
87	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	#41×C₆₇	—	2	4	no	—
90	#41×C₆₇	—	4	2	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×Q₈×C₆₇	—	4	1	no	—
93	C₂×Q₈×C₆₇	—	4	1	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	4	1	no	—
96	C₂²×C₄×C₆₇	—	4	1	no	—
97	C₂²×C₄×C₆₇	—	4	1	no	—
98	C₂²×C₄×C₆₇	—	4	1	no	—
99	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	Q₁₆×C₆₇	—	2	1	no	—
103	Q₁₆×C₆₇	C₂×Q₈	1	2	yes	—
104	SD₁₆×C₆₇	—	2	4	no	—
105	D₈×C₆₇	—	2	1	no	—
106	D₈×C₆₇	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
112	C₂×C₈×C₆₇	#41	1	2	yes	—
113	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	4	1	no	—
119	C₄⋊C₄×C₆₇	—	4	1	no	—
120	C₄⋊C₄×C₆₇	—	4	2	no	—
121	C₄⋊C₄×C₆₇	—	4	2	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₄⋊C₄×C₆₇	—	4	1	no	—
124	C₂²⋊C₄×C₆₇	—	4	1	no	—
125	C₂²⋊C₄×C₆₇	—	4	2	no	—
126	C₂²⋊C₄×C₆₇	—	4	1	no	—
127	C₂²⋊C₄×C₆₇	—	4	1	no	—
128	C₄²×C₆₇	—	4	1	no	—
129	C₄²×C₆₇	—	4	1	no	—
130	C₄²×C₆₇	—	4	1	no	—
131	C₄²×C₆₇	—	4	2	no	—
132	C₄²×C₆₇	D₈	1	1	yes	characteristic
133	C₄²×C₆₇	D₈	1	1	yes	characteristic
134	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₁₆×C₆₇	—	2	2	no	—
137	C₁₆×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	4	1	no	—
139	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₆₇	—	2	2	no	—
141	Q₈×C₆₇	—	4	1	no	—
142	Q₈×C₆₇	—	8	2	no	—
143	Q₈×C₆₇	—	8	2	no	—
144	D₄×C₆₇	—	2	2	no	—
145	D₄×C₆₇	—	4	4	no	—
146	D₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	—	4	1	no	—
150	C₂×C₄×C₆₇	—	4	4	no	—
151	C₂×C₄×C₆₇	—	4	1	no	—
152	C₂×C₄×C₆₇	—	4	1	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	8	2	no	—
156	C₂×C₄×C₆₇	—	8	2	no	—
157	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₆₇	#137	1	2	yes	—
164	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₆₇	—	2	1	no	—
166	C₈×C₆₇	—	2	1	no	—
167	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
168	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
169	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	—	2	1	no	—
172	C₂²×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	—	8	1	no	—
174	C₂²×C₆₇	—	8	1	no	—
175	C₂²×C₆₇	#474	1	2	yes	—
176	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	4	2	no	—
181	C₄×C₆₇	—	8	1	no	—
182	C₄×C₆₇	—	8	2	no	—
183	C₄×C₆₇	—	8	1	no	—
184	C₄×C₆₇	—	8	2	no	—
185	C₄×C₆₇	#474	1	2	yes	—
186	C₄×C₆₇	#508	1	2	yes	—
187	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
188	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
189	#33626	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂×C₆₇	—	2	2	no	—
191	C₂×C₆₇	—	8	2	no	—
192	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
193	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
194	C₂×C₆₇	#3414	1	1	yes	characteristic
195	#3160	C₂×C₆₇	1	1	yes	characteristic, complemented
196	C₄×#135	C₂×C₆₇	1	1	yes	characteristic
197	#2229	C₂×C₆₇	1	2	yes	complemented
198	#2228	C₂×C₆₇	1	2	yes	complemented
199	#2227	C₂×C₆₇	1	1	yes	characteristic, complemented
200	#2226	C₂×C₆₇	1	1	yes	characteristic, complemented
201	#2225	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#2224	C₂×C₆₇	1	1	yes	characteristic, complemented
203	#2158	C₂×C₆₇	1	2	yes	complemented
204	C₂×C₄⋊C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
205	#2138	C₂×C₆₇	1	2	yes	—
206	C₆₇	#33626	1	1	yes	characteristic, direct product factor, Sylow
207	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
208	#533	—	2	1	no	—
209	C₄×#41	—	2	1	no	—
210	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
211	#500	C₂²×C₆₇	1	1	yes	characteristic
212	#500	C₂²×C₆₇	1	1	yes	characteristic
213	#499	—	2	1	no	—
214	#499	C₂²×C₆₇	1	2	yes	—
215	#498	—	2	1	no	—
216	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
217	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
218	C₄×SD₁₆	—	2	1	no	—
219	C₄×SD₁₆	—	2	1	no	—
220	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
221	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
222	#427	—	2	1	no	—
223	#427	C₂²×C₆₇	1	2	yes	—
224	#427	C₂²×C₆₇	1	1	yes	characteristic
225	#416	—	2	1	no	—
226	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
227	#367	C₂²×C₆₇	1	2	yes	—
228	#367	C₂²×C₆₇	1	2	yes	—
229	#366	C₂²×C₆₇	1	2	yes	—
230	#366	C₂²×C₆₇	1	2	yes	—
231	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
232	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
233	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
234	#358	C₂²×C₆₇	1	2	yes	—
235	#358	C₂²×C₆₇	1	2	yes	—
236	#357	C₂²×C₆₇	1	2	yes	—
237	#357	C₂²×C₆₇	1	2	yes	—
238	C₂×#41	—	2	1	no	—
239	#135	—	2	2	no	—
240	#135	Q₈×C₆₇	1	4	yes	complemented
241	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
242	C₂×SD₁₆	—	2	1	no	—
243	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
244	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
245	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	#125	—	2	2	no	—
249	C₂²⋊Q₈	—	4	1	no	—
250	C₂²⋊Q₈	—	4	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₄×Q₈	—	4	1	no	—
254	C₄×Q₈	—	4	1	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	—	4	1	no	—
258	C₄×D₄	—	4	1	no	—
259	#117	—	2	1	no	—
260	#117	—	2	2	no	—
261	#117	—	4	1	no	—
262	C₂×C₄²	—	4	1	no	—
263	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
264	C₂×C₁₆	—	2	4	no	—
265	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
266	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
267	#107	—	2	1	no	—
268	#107	—	2	1	no	—
269	#107	C₂³×C₆₇	1	2	yes	—
270	#107	C₂³×C₆₇	1	2	yes	—
271	#107	C₂³×C₆₇	1	1	yes	characteristic
272	#107	C₂³×C₆₇	1	1	yes	characteristic
273	#106	—	2	2	no	—
274	#106	—	2	1	no	—
275	#106	—	2	2	no	—
276	#103	—	2	1	no	—
277	#103	—	2	1	no	—
278	#102	—	2	1	no	—
279	#102	—	2	1	no	—
280	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
281	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
282	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
283	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
284	#41	—	2	4	no	—
285	#41	—	4	2	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂×Q₈	—	4	1	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
295	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
297	Q₁₆	—	2	1	no	—
298	Q₁₆	C₂×Q₈×C₆₇	1	2	yes	—
299	SD₁₆	—	2	4	no	—
300	D₈	—	2	1	no	—
301	D₈	C₂×Q₈×C₆₇	1	2	yes	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
304	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
307	C₂×C₈	#41×C₆₇	1	2	yes	—
308	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₂²⋊C₄	—	4	1	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₂²⋊C₄	—	4	1	no	—
322	C₂²⋊C₄	—	4	1	no	—
323	C₄²	—	4	1	no	—
324	C₄²	—	4	1	no	—
325	C₄²	—	4	1	no	—
326	C₄²	—	4	2	no	—
327	C₄²	D₈×C₆₇	1	1	yes	characteristic
328	C₄²	D₈×C₆₇	1	1	yes	characteristic
329	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₁₆	—	2	2	no	—
332	C₁₆	—	2	2	no	—
333	C₂³	—	4	1	no	—
334	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
335	Q₈	—	2	2	no	—
336	Q₈	—	4	1	no	—
337	Q₈	—	8	2	no	—
338	Q₈	—	8	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	4	4	no	—
341	D₄	—	4	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	—	8	2	no	—
352	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
353	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
354	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
355	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
358	C₂×C₄	#137×C₆₇	1	2	yes	—
359	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	C₂²⋊Q₈×C₆₇	1	2	yes	—
363	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
364	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	8	1	no	—
369	C₂²	—	8	1	no	—
370	C₂²	#474×C₆₇	1	2	yes	—
371	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	8	1	no	—
379	C₄	—	8	2	no	—
380	C₄	#474×C₆₇	1	2	yes	—
381	C₄	#508×C₆₇	1	2	yes	—
382	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
383	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	8	2	no	—
386	C₂	C₂×#474×C₆₇	1	1	yes	characteristic
387	C₂	C₂×#508×C₆₇	1	1	yes	characteristic
388	C₂	C₆₇×#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3160	C₂	semidirect
3. C₆₇×#2229	C₂	semidirect
4. C₆₇×#2228	C₂	semidirect
5. C₆₇×#2227	C₂	semidirect
6. C₆₇×#2226	C₂	semidirect
7. C₆₇×#2225	C₂	semidirect
8. C₆₇×#2224	C₂	semidirect
9. C₆₇×#2158	C₂	semidirect
10. C₂×C₄⋊C₁₆×C₆₇	C₂	semidirect
45. #135×C₆₇	Q₈	semidirect
189. #33626	C₆₇	direct (N × H)
195. #3160	C₂×C₆₇	semidirect
197. #2229	C₂×C₆₇	semidirect
198. #2228	C₂×C₆₇	semidirect
199. #2227	C₂×C₆₇	semidirect
200. #2226	C₂×C₆₇	semidirect
201. #2225	C₂×C₆₇	semidirect
202. #2224	C₂×C₆₇	semidirect
203. #2158	C₂×C₆₇	semidirect
204. C₂×C₄⋊C₁₆	C₂×C₆₇	semidirect
206. C₆₇	#33626	direct (N × H)
240. #135	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3160	2
C₄×#135×C₆₇	2
C₆₇×#2229	2
C₆₇×#2228	2
C₆₇×#2227	2
C₆₇×#2226	2
C₆₇×#2225	2
C₆₇×#2224	2
C₆₇×#2158	2
C₂×C₄⋊C₁₆×C₆₇	2
C₆₇×#2138	2
#33626	67

Subgroups — C67×#33626 #30712877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33626 #30712877