Subgroups — C₆₇×#8238 #30687489

Order 17152.

534 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2929	C₂	1	1	yes	maximal, characteristic
2	C₈×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1501	C₂	1	2	yes	maximal
4	C₆₇×#1298	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1297	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#131×C₆₇	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	#525×C₆₇	C₂²	1	1	yes	characteristic
9	#435×C₆₇	—	2	2	no	—
10	C₈×D₄×C₆₇	—	2	2	no	—
11	#424×C₆₇	C₄	1	1	yes	characteristic, complemented
12	#408×C₆₇	C₄	1	1	yes	characteristic, complemented
13	#407×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
15	#405×C₆₇	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×C₈×C₆₇	—	2	2	no	—
17	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
18	#336×C₆₇	—	2	2	no	—
19	#328×C₆₇	C₄	1	4	yes	complemented
20	#327×C₆₇	C₄	1	4	yes	complemented
21	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	#131×C₆₇	—	4	2	no	—
24	C₂×M₁₆×C₆₇	—	2	1	no	—
25	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	1	no	—
29	C₂²×C₈×C₆₇	—	4	2	no	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	D₄	1	2	yes	—
33	C₂²×C₈×C₆₇	D₄	1	2	yes	—
34	#123×C₆₇	—	2	1	no	—
35	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
36	#121×C₆₇	C₂×C₄	1	2	yes	—
37	#120×C₆₇	—	2	1	no	—
38	C₄×D₄×C₆₇	—	2	2	no	—
39	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₂²⋊C₈×C₆₇	—	2	2	no	—
44	C₂²⋊C₈×C₆₇	—	2	2	no	—
45	C₂²⋊C₈×C₆₇	—	2	2	no	—
46	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
47	#97×C₆₇	—	2	1	no	—
48	C₄×C₈×C₆₇	—	2	1	no	—
49	C₄×C₈×C₆₇	—	4	4	no	—
50	#95×C₆₇	C₂×C₄	1	4	yes	—
51	C₂⁴×C₆₇	#41	1	1	yes	characteristic
52	#41×C₆₇	—	4	1	no	—
53	#41×C₆₇	—	4	1	no	—
54	C₂×Q₈×C₆₇	C₄²	1	1	yes	characteristic
55	C₂×D₄×C₆₇	—	2	1	no	—
56	C₂×D₄×C₆₇	—	2	2	no	—
57	C₂×D₄×C₆₇	C₄²	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	2	1	no	—
60	C₂²×C₄×C₆₇	—	4	1	no	—
61	C₂²×C₄×C₆₇	—	4	1	no	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
66	M₁₆×C₆₇	—	4	2	no	—
67	M₁₆×C₆₇	—	4	1	no	—
68	M₁₆×C₆₇	—	4	1	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	4	1	no	—
75	C₂×C₈×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	—	4	2	no	—
77	C₂×C₈×C₆₇	—	4	2	no	—
78	C₂×C₈×C₆₇	—	4	2	no	—
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₄⋊C₄×C₆₇	C₄²	1	2	yes	—
84	C₂²⋊C₄×C₆₇	—	2	8	no	—
85	C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₄²×C₆₇	—	2	1	no	—
89	C₄²×C₆₇	—	4	2	no	—
90	C₄²×C₆₇	—	4	2	no	—
91	C₂³×C₆₇	—	2	4	no	—
92	C₂³×C₆₇	—	2	1	no	—
93	C₂³×C₆₇	—	4	1	no	—
94	C₂³×C₆₇	#117	1	1	yes	characteristic
95	C₂³×C₆₇	C₄×D₄	1	2	yes	—
96	Q₈×C₆₇	—	4	1	no	—
97	D₄×C₆₇	—	4	1	no	—
98	D₄×C₆₇	—	4	2	no	—
99	D₄×C₆₇	—	4	4	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	8	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	4	1	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	1	no	—
112	C₂×C₄×C₆₇	—	4	1	no	—
113	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
116	C₈×C₆₇	—	2	2	no	—
117	C₈×C₆₇	—	2	1	no	—
118	C₈×C₆₇	—	4	4	no	—
119	C₈×C₆₇	—	4	1	no	—
120	C₈×C₆₇	—	4	1	no	—
121	C₈×C₆₇	#99	1	2	yes	—
122	C₂²×C₆₇	—	2	2	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	1	no	—
125	C₂²×C₆₇	—	2	2	no	—
126	C₂²×C₆₇	—	4	4	no	—
127	C₂²×C₆₇	—	4	1	no	—
128	C₂²×C₆₇	—	4	2	no	—
129	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₆₇	#443	1	4	yes	—
131	C₄×C₆₇	—	2	2	no	—
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	4	8	no	—
134	C₄×C₆₇	—	4	1	no	—
135	C₄×C₆₇	—	4	2	no	—
136	C₄×C₆₇	C₂×#99	1	1	yes	characteristic
137	C₄×C₆₇	#410	1	1	yes	characteristic
138	#8238	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₆₇	—	2	4	no	—
140	C₂×C₆₇	—	4	1	no	—
141	C₂×C₆₇	—	8	1	no	—
142	C₂×C₆₇	C₄×#99	1	1	yes	characteristic
143	C₂×C₆₇	#1764	1	2	yes	—
144	#2929	C₂×C₆₇	1	1	yes	characteristic
145	C₈×C₂²⋊C₄	C₂×C₆₇	1	2	yes	complemented
146	#1501	C₂×C₆₇	1	2	yes	—
147	#1298	C₂×C₆₇	1	1	yes	characteristic, complemented
148	#1297	C₂×C₆₇	1	1	yes	characteristic, complemented
149	C₆₇	#8238	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×#131	C₄×C₆₇	1	1	yes	characteristic, complemented
151	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
152	#525	C₂²×C₆₇	1	1	yes	characteristic
153	#435	—	2	2	no	—
154	C₈×D₄	—	2	2	no	—
155	#424	C₄×C₆₇	1	1	yes	characteristic, complemented
156	#408	C₄×C₆₇	1	1	yes	characteristic, complemented
157	#407	C₂²×C₆₇	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
159	#405	C₄×C₆₇	1	1	yes	characteristic, complemented
160	C₂×C₄×C₈	—	2	2	no	—
161	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
162	#336	—	2	2	no	—
163	#328	C₄×C₆₇	1	4	yes	complemented
164	#327	C₄×C₆₇	1	4	yes	complemented
165	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
166	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
167	#131	—	4	2	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
170	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	4	2	no	—
174	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
175	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
176	C₂²×C₈	D₄×C₆₇	1	2	yes	—
177	C₂²×C₈	D₄×C₆₇	1	2	yes	—
178	#123	—	2	1	no	—
179	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
180	#121	C₂×C₄×C₆₇	1	2	yes	—
181	#120	—	2	1	no	—
182	C₄×D₄	—	2	2	no	—
183	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	2	no	—
189	C₂²⋊C₈	—	2	2	no	—
190	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
191	#97	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	4	4	no	—
194	#95	C₂×C₄×C₆₇	1	4	yes	—
195	C₂⁴	#41×C₆₇	1	1	yes	characteristic
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	C₄²×C₆₇	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₄²×C₆₇	1	1	yes	characteristic
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
209	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
225	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	C₄²×C₆₇	1	2	yes	—
228	C₂²⋊C₄	—	2	8	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	4	2	no	—
234	C₄²	—	4	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	#117×C₆₇	1	1	yes	characteristic
239	C₂³	C₄×D₄×C₆₇	1	2	yes	—
240	Q₈	—	4	1	no	—
241	D₄	—	4	1	no	—
242	D₄	—	4	2	no	—
243	D₄	—	4	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
260	C₈	—	2	1	no	—
261	C₈	—	2	2	no	—
262	C₈	—	4	4	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₈	#99×C₆₇	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	2	no	—
273	C₂²	C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
274	C₂²	#443×C₆₇	1	4	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	8	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	2	no	—
280	C₄	C₂×#99×C₆₇	1	1	yes	characteristic
281	C₄	#410×C₆₇	1	1	yes	characteristic
282	C₂	—	2	4	no	—
283	C₂	—	4	1	no	—
284	C₂	—	8	1	no	—
285	C₂	C₄×#99×C₆₇	1	1	yes	characteristic
286	C₂	C₆₇×#1764	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂²×C₆₇	C₄×C₂²⋊C₄
prime radical	#525×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×C₂²⋊C₄×C₆₇	C₂	semidirect
4. C₆₇×#1298	C₂	semidirect
5. C₆₇×#1297	C₂	semidirect
6. C₂×#131×C₆₇	C₄	semidirect
11. #424×C₆₇	C₄	semidirect
12. #408×C₆₇	C₄	semidirect
15. #405×C₆₇	C₄	semidirect
19. #328×C₆₇	C₄	semidirect
20. #327×C₆₇	C₄	semidirect
138. #8238	C₆₇	direct (N × H)
145. C₈×C₂²⋊C₄	C₂×C₆₇	semidirect
147. #1298	C₂×C₆₇	semidirect
148. #1297	C₂×C₆₇	semidirect
149. C₆₇	#8238	direct (N × H)
150. C₂×#131	C₄×C₆₇	semidirect
155. #424	C₄×C₆₇	semidirect
156. #408	C₄×C₆₇	semidirect
159. #405	C₄×C₆₇	semidirect
163. #328	C₄×C₆₇	semidirect
164. #327	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2929	2
C₈×C₂²⋊C₄×C₆₇	2
C₆₇×#1501	2
C₆₇×#1298	2
C₆₇×#1297	2
#8238	67

Subgroups — C67×#8238 #30687489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#8238 #30687489