Subgroups — C₆₇×#33686 #30712937

Order 17152.

544 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3176	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2234	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2229	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2207	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2185	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2175	C₂	1	2	yes	maximal, complemented
8	C₄×SD₃₂×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	#540×C₆₇	—	2	1	no	—
12	#519×C₆₇	—	2	1	no	—
13	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
15	#499×C₆₇	C₂²	1	1	yes	characteristic
16	#493×C₆₇	C₂²	1	1	yes	characteristic
17	#490×C₆₇	—	2	1	no	—
18	#469×C₆₇	—	2	1	no	—
19	#466×C₆₇	C₂²	1	2	yes	—
20	#442×C₆₇	—	2	1	no	—
21	#441×C₆₇	—	2	1	no	—
22	#441×C₆₇	—	2	1	no	—
23	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
25	C₄×D₈×C₆₇	C₂²	1	2	yes	—
26	#427×C₆₇	C₂²	1	1	yes	characteristic
27	#419×C₆₇	—	2	1	no	—
28	#405×C₆₇	C₂²	1	1	yes	characteristic
29	#367×C₆₇	C₂²	1	2	yes	—
30	#366×C₆₇	C₂²	1	2	yes	—
31	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
32	#358×C₆₇	C₂²	1	2	yes	—
33	#358×C₆₇	C₂²	1	2	yes	—
34	#357×C₆₇	C₂²	1	2	yes	—
35	#357×C₆₇	C₂²	1	2	yes	—
36	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
37	#346×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₆₇	—	2	1	no	—
40	C₂×Q₁₆×C₆₇	—	2	4	no	—
41	C₂×Q₁₆×C₆₇	—	2	1	no	—
42	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
43	C₂×D₈×C₆₇	—	2	1	no	—
44	C₂×D₈×C₆₇	C₂³	1	2	yes	—
45	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	#127×C₆₇	—	2	1	no	—
48	#125×C₆₇	—	2	1	no	—
49	#123×C₆₇	—	4	2	no	—
50	#121×C₆₇	—	2	2	no	—
51	#121×C₆₇	—	4	2	no	—
52	C₄×Q₈×C₆₇	—	2	2	no	—
53	C₄×Q₈×C₆₇	—	2	2	no	—
54	C₄×Q₈×C₆₇	—	4	2	no	—
55	C₄×D₄×C₆₇	—	2	2	no	—
56	#117×C₆₇	—	2	1	no	—
57	#117×C₆₇	—	4	2	no	—
58	#117×C₆₇	D₄	1	1	yes	characteristic
59	SD₃₂×C₆₇	—	2	4	no	—
60	SD₃₂×C₆₇	—	2	4	no	—
61	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
62	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
63	#107×C₆₇	C₂³	1	2	yes	—
64	#107×C₆₇	C₂³	1	2	yes	—
65	#106×C₆₇	—	2	1	no	—
66	#106×C₆₇	—	2	1	no	—
67	#103×C₆₇	—	2	2	no	—
68	#103×C₆₇	—	2	2	no	—
69	#102×C₆₇	—	2	2	no	—
70	#102×C₆₇	—	2	2	no	—
71	#97×C₆₇	D₄	1	1	yes	characteristic
72	#97×C₆₇	D₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₆₇	—	2	4	no	—
76	C₂×Q₈×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	—	4	2	no	—
78	C₂×Q₈×C₆₇	—	4	1	no	—
79	C₂×D₄×C₆₇	—	2	2	no	—
80	C₂×D₄×C₆₇	—	4	2	no	—
81	C₂×D₄×C₆₇	—	4	2	no	—
82	C₂²×C₄×C₆₇	—	4	1	no	—
83	C₂²×C₄×C₆₇	—	4	2	no	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	Q₁₆×C₆₇	—	2	4	no	—
86	Q₁₆×C₆₇	—	2	2	no	—
87	Q₁₆×C₆₇	#41	1	4	yes	—
88	D₈×C₆₇	—	2	2	no	—
89	D₈×C₆₇	—	2	2	no	—
90	D₈×C₆₇	—	4	2	no	—
91	M₁₆×C₆₇	—	4	1	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	4	2	no	—
100	C₄⋊C₄×C₆₇	—	4	2	no	—
101	C₄⋊C₄×C₆₇	—	4	2	no	—
102	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
103	C₂²⋊C₄×C₆₇	—	4	1	no	—
104	C₂²⋊C₄×C₆₇	—	4	2	no	—
105	C₂²⋊C₄×C₆₇	—	4	2	no	—
106	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
107	C₄²×C₆₇	—	4	1	no	—
108	C₄²×C₆₇	—	4	2	no	—
109	C₄²×C₆₇	—	4	2	no	—
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₆₇	—	2	2	no	—
113	C₁₆×C₆₇	—	2	2	no	—
114	C₂³×C₆₇	—	4	2	no	—
115	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₆₇	—	2	4	no	—
117	Q₈×C₆₇	—	4	4	no	—
118	Q₈×C₆₇	—	4	2	no	—
119	D₄×C₆₇	—	4	2	no	—
120	D₄×C₆₇	—	4	2	no	—
121	D₄×C₆₇	—	16	2	no	—
122	D₄×C₆₇	—	16	2	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	4	4	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	16	2	no	—
130	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₆₇	—	2	1	no	—
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	—	2	1	no	—
137	C₈×C₆₇	C₂×#41	1	2	yes	—
138	C₂²×C₆₇	—	2	1	no	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	—	8	2	no	—
141	C₂²×C₆₇	—	8	2	no	—
142	C₂²×C₆₇	—	8	2	no	—
143	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₆₇	—	2	1	no	—
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	—	4	4	no	—
148	C₄×C₆₇	—	8	2	no	—
149	C₄×C₆₇	—	8	2	no	—
150	C₄×C₆₇	#525	1	1	yes	characteristic
151	C₄×C₆₇	#575	1	1	yes	characteristic
152	#33686	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂×C₆₇	—	4	1	no	—
154	C₂×C₆₇	—	16	2	no	—
155	C₂×C₆₇	#3100	1	1	yes	characteristic
156	C₂×C₆₇	#3416	1	2	yes	—
157	#3176	C₂×C₆₇	1	1	yes	characteristic
158	#2950	C₂×C₆₇	1	1	yes	characteristic, complemented
159	#2234	C₂×C₆₇	1	2	yes	complemented
160	#2229	C₂×C₆₇	1	2	yes	complemented
161	#2207	C₂×C₆₇	1	2	yes	complemented
162	#2185	C₂×C₆₇	1	2	yes	complemented
163	#2175	C₂×C₆₇	1	2	yes	complemented
164	C₄×SD₃₂	C₂×C₆₇	1	2	yes	complemented
165	#2164	C₂×C₆₇	1	1	yes	characteristic, complemented
166	C₆₇	#33686	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
168	#540	—	2	1	no	—
169	#519	—	2	1	no	—
170	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
171	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
172	#499	C₂²×C₆₇	1	1	yes	characteristic
173	#493	C₂²×C₆₇	1	1	yes	characteristic
174	#490	—	2	1	no	—
175	#469	—	2	1	no	—
176	#466	C₂²×C₆₇	1	2	yes	—
177	#442	—	2	1	no	—
178	#441	—	2	1	no	—
179	#441	—	2	1	no	—
180	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
181	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
182	C₄×D₈	C₂²×C₆₇	1	2	yes	—
183	#427	C₂²×C₆₇	1	1	yes	characteristic
184	#419	—	2	1	no	—
185	#405	C₂²×C₆₇	1	1	yes	characteristic
186	#367	C₂²×C₆₇	1	2	yes	—
187	#366	C₂²×C₆₇	1	2	yes	—
188	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
189	#358	C₂²×C₆₇	1	2	yes	—
190	#358	C₂²×C₆₇	1	2	yes	—
191	#357	C₂²×C₆₇	1	2	yes	—
192	#357	C₂²×C₆₇	1	2	yes	—
193	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
194	#346	C₂²×C₆₇	1	1	yes	characteristic
195	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
196	C₂²×Q₈	—	2	1	no	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	C₂³×C₆₇	1	2	yes	—
202	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
204	#127	—	2	1	no	—
205	#125	—	2	1	no	—
206	#123	—	4	2	no	—
207	#121	—	2	2	no	—
208	#121	—	4	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	4	2	no	—
212	C₄×D₄	—	2	2	no	—
213	#117	—	2	1	no	—
214	#117	—	4	2	no	—
215	#117	D₄×C₆₇	1	1	yes	characteristic
216	SD₃₂	—	2	4	no	—
217	SD₃₂	—	2	4	no	—
218	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
219	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
220	#107	C₂³×C₆₇	1	2	yes	—
221	#107	C₂³×C₆₇	1	2	yes	—
222	#106	—	2	1	no	—
223	#106	—	2	1	no	—
224	#103	—	2	2	no	—
225	#103	—	2	2	no	—
226	#102	—	2	2	no	—
227	#102	—	2	2	no	—
228	#97	D₄×C₆₇	1	1	yes	characteristic
229	#97	D₄×C₆₇	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
242	Q₁₆	—	2	4	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	#41×C₆₇	1	4	yes	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	—	4	2	no	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
269	C₁₆	—	2	2	no	—
270	C₁₆	—	2	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
273	Q₈	—	2	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	16	2	no	—
279	D₄	—	16	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	16	2	no	—
287	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
288	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
289	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	C₂×#41×C₆₇	1	2	yes	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	#525×C₆₇	1	1	yes	characteristic
308	C₄	#575×C₆₇	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	—	16	2	no	—
311	C₂	C₆₇×#3100	1	1	yes	characteristic
312	C₂	C₆₇×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#3176	C₂
LCS	G ▷ C₂×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×#2950	C₂	semidirect
3. C₆₇×#2234	C₂	semidirect
4. C₆₇×#2229	C₂	semidirect
5. C₆₇×#2207	C₂	semidirect
6. C₆₇×#2185	C₂	semidirect
7. C₆₇×#2175	C₂	semidirect
8. C₄×SD₃₂×C₆₇	C₂	semidirect
9. C₆₇×#2164	C₂	semidirect
152. #33686	C₆₇	direct (N × H)
158. #2950	C₂×C₆₇	semidirect
159. #2234	C₂×C₆₇	semidirect
160. #2229	C₂×C₆₇	semidirect
161. #2207	C₂×C₆₇	semidirect
162. #2185	C₂×C₆₇	semidirect
163. #2175	C₂×C₆₇	semidirect
164. C₄×SD₃₂	C₂×C₆₇	semidirect
165. #2164	C₂×C₆₇	semidirect
166. C₆₇	#33686	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3176	2
C₆₇×#2950	2
C₆₇×#2234	2
C₆₇×#2229	2
C₆₇×#2207	2
C₆₇×#2185	2
C₆₇×#2175	2
C₄×SD₃₂×C₆₇	2
C₆₇×#2164	2
#33686	67

Subgroups — C67×#33686 #30712937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33686 #30712937