Subgroups — C₆₇×#33454 #30712705

Order 17152.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#509×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2951	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2948	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2177	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2175	C₂	1	2	yes	maximal, complemented
6	C₄×SD₃₂×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₄×D₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2159	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#357×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	#520×C₆₇	—	2	1	no	—
15	#518×C₆₇	—	2	1	no	—
16	#509×C₆₇	C₄	1	8	yes	complemented
17	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
18	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	#442×C₆₇	—	2	2	no	—
22	#440×C₆₇	—	2	2	no	—
23	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
24	C₄×D₈×C₆₇	C₂²	1	2	yes	—
25	C₄×D₈×C₆₇	C₂²	1	2	yes	—
26	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
27	#427×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
29	#418×C₆₇	—	2	1	no	—
30	#417×C₆₇	—	2	1	no	—
31	#405×C₆₇	C₂²	1	1	yes	characteristic
32	#367×C₆₇	C₂²	1	2	yes	complemented
33	#366×C₆₇	C₂²	1	2	yes	complemented
34	#358×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	#358×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#357×C₆₇	C₂²	1	2	yes	—
37	#357×C₆₇	C₂²	1	2	yes	—
38	#357×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#357×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#346×C₆₇	C₂²	1	2	yes	complemented
41	C₄×C₁₆×C₆₇	C₂²	1	2	yes	complemented
42	C₂×#41×C₆₇	—	2	1	no	—
43	C₂²×D₄×C₆₇	—	2	1	no	—
44	#135×C₆₇	—	2	2	no	—
45	#135×C₆₇	C₂×C₄	1	4	yes	complemented
46	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×SD₁₆×C₆₇	—	2	1	no	—
48	C₂×D₈×C₆₇	—	2	1	no	—
49	C₂×D₈×C₆₇	—	2	2	no	—
50	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
51	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×D₈×C₆₇	C₂³	1	2	yes	—
53	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×D₄×C₆₇	—	2	2	no	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	C₄×D₄×C₆₇	—	4	1	no	—
60	C₄×D₄×C₆₇	—	4	1	no	—
61	C₄×D₄×C₆₇	—	4	2	no	—
62	#117×C₆₇	—	2	1	no	—
63	#117×C₆₇	—	4	1	no	—
64	#117×C₆₇	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
68	SD₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
69	D₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
70	M₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
71	C₂×C₁₆×C₆₇	—	2	2	no	—
72	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
75	#107×C₆₇	—	2	1	no	—
76	#107×C₆₇	C₂³	1	2	yes	—
77	#107×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	C₂³	1	1	yes	characteristic
79	#106×C₆₇	—	2	1	no	—
80	#103×C₆₇	—	2	2	no	—
81	#102×C₆₇	—	2	2	no	—
82	#102×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	2	no	—
84	#97×C₆₇	D₄	1	2	yes	—
85	C₄×C₈×C₆₇	C₂³	1	2	yes	—
86	C₂⁴×C₆₇	—	4	1	no	—
87	#41×C₆₇	—	2	4	no	—
88	#41×C₆₇	—	4	2	no	—
89	C₂×Q₈×C₆₇	—	2	1	no	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	—	4	1	no	—
94	C₂×D₄×C₆₇	—	4	1	no	—
95	C₂×D₄×C₆₇	—	4	2	no	—
96	C₂²×C₄×C₆₇	—	4	1	no	—
97	C₂²×C₄×C₆₇	—	4	1	no	—
98	C₂²×C₄×C₆₇	—	4	2	no	—
99	C₂²×C₄×C₆₇	—	4	2	no	—
100	C₂²×C₄×C₆₇	—	4	2	no	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	Q₁₆×C₆₇	—	2	1	no	—
103	Q₁₆×C₆₇	C₂²×C₄	1	2	yes	—
104	SD₁₆×C₆₇	—	2	4	no	—
105	D₈×C₆₇	—	2	4	no	—
106	D₈×C₆₇	—	2	2	no	—
107	D₈×C₆₇	—	2	1	no	—
108	D₈×C₆₇	C₂²×C₄	1	2	yes	—
109	D₈×C₆₇	C₂²×C₄	1	4	yes	—
110	M₁₆×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	#41	1	2	yes	—
115	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₆₇	—	2	1	no	—
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	4	1	no	—
120	C₄⋊C₄×C₆₇	—	4	2	no	—
121	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
122	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
123	C₂²⋊C₄×C₆₇	—	4	1	no	—
124	C₂²⋊C₄×C₆₇	—	4	2	no	—
125	C₂²⋊C₄×C₆₇	—	4	2	no	—
126	C₂²⋊C₄×C₆₇	—	4	2	no	—
127	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
128	C₄²×C₆₇	—	4	1	no	—
129	C₄²×C₆₇	—	4	2	no	—
130	C₄²×C₆₇	C₂×D₄	1	2	yes	—
131	C₁₆×C₆₇	—	2	1	no	—
132	C₁₆×C₆₇	—	2	1	no	—
133	C₁₆×C₆₇	C₂²×C₄	1	4	yes	—
134	C₂³×C₆₇	—	4	1	no	—
135	C₂³×C₆₇	—	4	2	no	—
136	C₂³×C₆₇	—	4	4	no	—
137	C₂³×C₆₇	—	16	1	no	—
138	C₂³×C₆₇	—	16	1	no	—
139	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₆₇	—	2	2	no	—
141	Q₈×C₆₇	—	4	1	no	—
142	D₄×C₆₇	—	2	4	no	—
143	D₄×C₆₇	—	2	2	no	—
144	D₄×C₆₇	—	4	4	no	—
145	D₄×C₆₇	—	4	2	no	—
146	D₄×C₆₇	—	4	1	no	—
147	D₄×C₆₇	—	4	4	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	—	4	4	no	—
150	C₂×C₄×C₆₇	—	4	1	no	—
151	C₂×C₄×C₆₇	—	4	1	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	16	2	no	—
154	C₂×C₄×C₆₇	—	16	2	no	—
155	C₂×C₄×C₆₇	—	16	2	no	—
156	C₂×C₄×C₆₇	—	16	1	no	—
157	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
159	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₆₇	#135	1	2	yes	—
162	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₆₇	—	2	2	no	—
164	C₈×C₆₇	C₄×D₄	1	2	yes	—
165	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
166	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	—	4	2	no	—
170	C₂²×C₆₇	—	4	4	no	—
171	C₂²×C₆₇	—	8	1	no	—
172	C₂²×C₆₇	—	8	2	no	—
173	C₂²×C₆₇	—	8	2	no	—
174	C₂²×C₆₇	—	8	1	no	—
175	C₂²×C₆₇	—	16	4	no	—
176	C₂²×C₆₇	C₄×D₈	1	2	yes	—
177	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
178	C₄×C₆₇	—	2	2	no	—
179	C₄×C₆₇	—	2	2	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	C₄×D₈	1	2	yes	—
185	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
187	#33454	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂×C₆₇	—	2	2	no	—
189	C₂×C₆₇	—	8	2	no	—
190	C₂×C₆₇	—	8	4	no	—
191	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
192	C₂×C₆₇	#3415	1	1	yes	characteristic
193	C₂×C₆₇	#3416	1	1	yes	characteristic
194	C₂×#509	C₂×C₆₇	1	1	yes	characteristic
195	#2951	C₂×C₆₇	1	1	yes	characteristic, complemented
196	#2948	C₂×C₆₇	1	1	yes	characteristic, complemented
197	#2177	C₂×C₆₇	1	2	yes	complemented
198	#2175	C₂×C₆₇	1	2	yes	complemented
199	C₄×SD₃₂	C₂×C₆₇	1	2	yes	complemented
200	C₄×D₁₆	C₂×C₆₇	1	2	yes	complemented
201	#2159	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#2138	C₂×C₆₇	1	1	yes	characteristic, complemented
203	C₂×#357	C₂×C₆₇	1	1	yes	characteristic, complemented
204	#2108	C₂×C₆₇	1	1	yes	characteristic, complemented
205	C₆₇	#33454	1	1	yes	characteristic, direct product factor, Sylow
206	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
207	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
208	#520	—	2	1	no	—
209	#518	—	2	1	no	—
210	#509	C₄×C₆₇	1	8	yes	complemented
211	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
212	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
213	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
214	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
215	#442	—	2	2	no	—
216	#440	—	2	2	no	—
217	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
218	C₄×D₈	C₂²×C₆₇	1	2	yes	—
219	C₄×D₈	C₂²×C₆₇	1	2	yes	—
220	C₄×D₈	C₂²×C₆₇	1	2	yes	complemented
221	#427	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
223	#418	—	2	1	no	—
224	#417	—	2	1	no	—
225	#405	C₂²×C₆₇	1	1	yes	characteristic
226	#367	C₂²×C₆₇	1	2	yes	complemented
227	#366	C₂²×C₆₇	1	2	yes	complemented
228	#358	C₂²×C₆₇	1	1	yes	characteristic, complemented
229	#358	C₂²×C₆₇	1	1	yes	characteristic, complemented
230	#357	C₂²×C₆₇	1	2	yes	—
231	#357	C₂²×C₆₇	1	2	yes	—
232	#357	C₂²×C₆₇	1	1	yes	characteristic, complemented
233	#357	C₂²×C₆₇	1	1	yes	characteristic, complemented
234	#346	C₂²×C₆₇	1	2	yes	complemented
235	C₄×C₁₆	C₂²×C₆₇	1	2	yes	complemented
236	C₂×#41	—	2	1	no	—
237	C₂²×D₄	—	2	1	no	—
238	#135	—	2	2	no	—
239	#135	C₂×C₄×C₆₇	1	4	yes	complemented
240	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
241	C₂×SD₁₆	—	2	1	no	—
242	C₂×D₈	—	2	1	no	—
243	C₂×D₈	—	2	2	no	—
244	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
245	C₂×D₈	C₂³×C₆₇	1	2	yes	—
246	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
247	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
248	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
249	C₄×Q₈	—	2	2	no	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	—	2	2	no	—
253	C₄×D₄	—	4	1	no	—
254	C₄×D₄	—	4	1	no	—
255	C₄×D₄	—	4	2	no	—
256	#117	—	2	1	no	—
257	#117	—	4	1	no	—
258	#117	D₄×C₆₇	1	1	yes	characteristic
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	—	4	1	no	—
261	C₂×C₂²⋊C₄	—	4	2	no	—
262	SD₃₂	C₂×C₄×C₆₇	1	8	yes	complemented
263	D₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
264	M₃₂	C₂×C₄×C₆₇	1	4	yes	complemented
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
267	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
268	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
269	#107	—	2	1	no	—
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	1	no	—
274	#103	—	2	2	no	—
275	#102	—	2	2	no	—
276	#102	—	2	2	no	—
277	#102	—	2	2	no	—
278	#97	D₄×C₆₇	1	2	yes	—
279	C₄×C₈	C₂³×C₆₇	1	2	yes	—
280	C₂⁴	—	4	1	no	—
281	#41	—	2	4	no	—
282	#41	—	4	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×D₄	—	2	4	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
296	Q₁₆	—	2	1	no	—
297	Q₁₆	C₂²×C₄×C₆₇	1	2	yes	—
298	SD₁₆	—	2	4	no	—
299	D₈	—	2	4	no	—
300	D₈	—	2	2	no	—
301	D₈	—	2	1	no	—
302	D₈	C₂²×C₄×C₆₇	1	2	yes	—
303	D₈	C₂²×C₄×C₆₇	1	4	yes	—
304	M₁₆	—	2	2	no	—
305	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
306	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
307	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₂×C₈	#41×C₆₇	1	2	yes	—
309	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
316	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
317	C₂²⋊C₄	—	4	1	no	—
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
322	C₄²	—	4	1	no	—
323	C₄²	—	4	2	no	—
324	C₄²	C₂×D₄×C₆₇	1	2	yes	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	2	1	no	—
327	C₁₆	C₂²×C₄×C₆₇	1	4	yes	—
328	C₂³	—	4	1	no	—
329	C₂³	—	4	2	no	—
330	C₂³	—	4	4	no	—
331	C₂³	—	16	1	no	—
332	C₂³	—	16	1	no	—
333	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	D₄	—	2	4	no	—
337	D₄	—	2	2	no	—
338	D₄	—	4	4	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	1	no	—
341	D₄	—	4	4	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	16	2	no	—
348	C₂×C₄	—	16	2	no	—
349	C₂×C₄	—	16	2	no	—
350	C₂×C₄	—	16	1	no	—
351	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
352	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
353	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
354	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
355	C₂×C₄	#135×C₆₇	1	2	yes	—
356	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	C₄×D₄×C₆₇	1	2	yes	—
359	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
360	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	4	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	8	1	no	—
369	C₂²	—	16	4	no	—
370	C₂²	C₄×D₈×C₆₇	1	2	yes	—
371	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	4	2	no	—
375	C₄	—	8	1	no	—
376	C₄	—	8	2	no	—
377	C₄	—	8	1	no	—
378	C₄	C₄×D₈×C₆₇	1	2	yes	—
379	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
380	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	8	4	no	—
383	C₂	—	8	2	no	—
384	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
385	C₂	C₆₇×#3415	1	1	yes	characteristic
386	C₂	C₆₇×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#509×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #117×C₆₇ ◁ #405×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2951	C₂	semidirect
3. C₆₇×#2948	C₂	semidirect
4. C₆₇×#2177	C₂	semidirect
5. C₆₇×#2175	C₂	semidirect
6. C₄×SD₃₂×C₆₇	C₂	semidirect
7. C₄×D₁₆×C₆₇	C₂	semidirect
8. C₆₇×#2159	C₂	semidirect
9. C₆₇×#2138	C₂	semidirect
10. C₂×#357×C₆₇	C₂	semidirect
11. C₆₇×#2108	C₂	semidirect
16. #509×C₆₇	C₄	semidirect
23. C₄×Q₁₆×C₆₇	C₂²	semidirect
26. C₄×D₈×C₆₇	C₂²	semidirect
32. #367×C₆₇	C₂²	semidirect
33. #366×C₆₇	C₂²	semidirect
34. #358×C₆₇	C₂²	semidirect
35. #358×C₆₇	C₂²	semidirect
38. #357×C₆₇	C₂²	semidirect
39. #357×C₆₇	C₂²	semidirect
40. #346×C₆₇	C₂²	semidirect
41. C₄×C₁₆×C₆₇	C₂²	semidirect
45. #135×C₆₇	C₂×C₄	semidirect
50. C₂×D₈×C₆₇	C₂×C₄	semidirect
68. SD₃₂×C₆₇	C₂×C₄	semidirect
69. D₁₆×C₆₇	C₂×C₄	semidirect
70. M₃₂×C₆₇	C₂×C₄	semidirect
187. #33454	C₆₇	direct (N × H)
195. #2951	C₂×C₆₇	semidirect
196. #2948	C₂×C₆₇	semidirect
197. #2177	C₂×C₆₇	semidirect
198. #2175	C₂×C₆₇	semidirect
199. C₄×SD₃₂	C₂×C₆₇	semidirect
200. C₄×D₁₆	C₂×C₆₇	semidirect
201. #2159	C₂×C₆₇	semidirect
202. #2138	C₂×C₆₇	semidirect
203. C₂×#357	C₂×C₆₇	semidirect
204. #2108	C₂×C₆₇	semidirect
205. C₆₇	#33454	direct (N × H)
210. #509	C₄×C₆₇	semidirect
217. C₄×Q₁₆	C₂²×C₆₇	semidirect
220. C₄×D₈	C₂²×C₆₇	semidirect
226. #367	C₂²×C₆₇	semidirect
227. #366	C₂²×C₆₇	semidirect
228. #358	C₂²×C₆₇	semidirect
229. #358	C₂²×C₆₇	semidirect
232. #357	C₂²×C₆₇	semidirect
233. #357	C₂²×C₆₇	semidirect
234. #346	C₂²×C₆₇	semidirect
235. C₄×C₁₆	C₂²×C₆₇	semidirect
239. #135	C₂×C₄×C₆₇	semidirect
244. C₂×D₈	C₂×C₄×C₆₇	semidirect
262. SD₃₂	C₂×C₄×C₆₇	semidirect
263. D₁₆	C₂×C₄×C₆₇	semidirect
264. M₃₂	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#509×C₆₇	2
C₆₇×#2951	2
C₆₇×#2948	2
C₆₇×#2177	2
C₆₇×#2175	2
C₄×SD₃₂×C₆₇	2
C₄×D₁₆×C₆₇	2
C₆₇×#2159	2
C₆₇×#2138	2
C₂×#357×C₆₇	2
C₆₇×#2108	2
#33454	67

Subgroups — C67×#33454 #30712705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33454 #30712705