Subgroups — C₆₇×#19732 #30698983

Order 17152.

644 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3231	C₂	1	2	yes	maximal
2	C₆₇×#3230	C₂	1	2	yes	maximal
3	C₆₇×#3185	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#3139	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1718	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1718	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1710	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1453	C₂	1	1	yes	maximal, characteristic
10	#533×C₆₇	C₂²	1	1	yes	characteristic
11	#533×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#500×C₆₇	C₂²	1	2	yes	—
14	#500×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	C₂²	1	2	yes	—
16	#498×C₆₇	C₂²	1	1	yes	characteristic
17	#498×C₆₇	C₂²	1	2	yes	—
18	#494×C₆₇	C₂²	1	1	yes	characteristic
19	#491×C₆₇	—	2	2	no	—
20	#487×C₆₇	C₂²	1	2	yes	—
21	#487×C₆₇	C₂²	1	2	yes	—
22	#487×C₆₇	C₂²	1	2	yes	—
23	#487×C₆₇	C₂²	1	2	yes	—
24	#487×C₆₇	C₂²	1	2	yes	—
25	#487×C₆₇	C₂²	1	2	yes	—
26	#487×C₆₇	C₂²	1	2	yes	—
27	#485×C₆₇	—	2	4	no	—
28	#484×C₆₇	—	2	4	no	—
29	#479×C₆₇	—	2	4	no	—
30	#477×C₆₇	—	2	4	no	—
31	#470×C₆₇	—	2	2	no	—
32	#441×C₆₇	—	2	2	no	—
33	#431×C₆₇	C₂²	1	2	yes	—
34	#419×C₆₇	—	2	2	no	—
35	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	#322×C₆₇	C₂²	1	1	yes	characteristic
37	#322×C₆₇	C₂²	1	2	yes	—
38	C₈²×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₆₇	—	2	1	no	—
40	C₂×Q₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	C₄⋊Q₈×C₆₇	—	2	1	no	—
44	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
48	#125×C₆₇	—	2	2	no	—
49	#125×C₆₇	—	2	1	no	—
50	C₂²⋊Q₈×C₆₇	—	2	2	no	—
51	C₂²⋊Q₈×C₆₇	—	2	2	no	—
52	C₂²⋊Q₈×C₆₇	—	2	2	no	—
53	C₄×Q₈×C₆₇	—	2	2	no	—
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	#117×C₆₇	—	2	2	no	—
57	#117×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	#107×C₆₇	—	2	4	no	—
61	#107×C₆₇	—	2	4	no	—
62	#107×C₆₇	—	2	4	no	—
63	#106×C₆₇	—	2	2	no	—
64	#106×C₆₇	—	2	4	no	—
65	C₄⋊C₈×C₆₇	D₄	1	4	yes	—
66	#103×C₆₇	—	2	4	no	—
67	#103×C₆₇	—	2	4	no	—
68	#103×C₆₇	—	2	4	no	—
69	#103×C₆₇	—	2	4	no	—
70	#103×C₆₇	—	2	4	no	—
71	#103×C₆₇	—	2	4	no	—
72	#103×C₆₇	—	2	4	no	—
73	C₂²⋊C₈×C₆₇	D₄	1	4	yes	—
74	#97×C₆₇	D₄	1	2	yes	—
75	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₆₇	C₂³	1	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	2	yes	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	4	4	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂²×C₄×C₆₇	—	4	1	no	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₆₇	—	4	4	no	—
89	M₁₆×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	4	2	no	—
103	C₄⋊C₄×C₆₇	—	4	2	no	—
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₄⋊C₄×C₆₇	—	4	2	no	—
106	C₂²⋊C₄×C₆₇	—	4	2	no	—
107	C₂²⋊C₄×C₆₇	—	4	1	no	—
108	C₄²×C₆₇	—	4	2	no	—
109	C₄²×C₆₇	—	4	1	no	—
110	C₄²×C₆₇	C₂×D₄	1	2	yes	—
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₂³×C₆₇	#127	1	1	yes	characteristic
114	Q₈×C₆₇	—	4	4	no	—
115	Q₈×C₆₇	—	4	4	no	—
116	Q₈×C₆₇	—	4	4	no	—
117	Q₈×C₆₇	—	4	4	no	—
118	Q₈×C₆₇	—	8	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	—	8	2	no	—
126	C₂×C₄×C₆₇	#127	1	2	yes	—
127	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
130	C₈×C₆₇	—	2	2	no	—
131	C₈×C₆₇	—	2	2	no	—
132	C₈×C₆₇	—	2	4	no	—
133	C₈×C₆₇	—	2	4	no	—
134	C₂²×C₆₇	—	2	1	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
137	C₄×C₆₇	—	2	1	no	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	8	2	no	—
140	C₄×C₆₇	—	8	2	no	—
141	C₄×C₆₇	—	8	2	no	—
142	C₄×C₆₇	—	8	2	no	—
143	C₄×C₆₇	#575	1	2	yes	—
144	C₄×C₆₇	#578	1	4	yes	—
145	#19732	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₆₇	—	4	1	no	—
147	C₂×C₆₇	#3153	1	1	yes	characteristic
148	C₂×C₆₇	#3156	1	2	yes	—
149	#3231	C₂×C₆₇	1	2	yes	—
150	#3230	C₂×C₆₇	1	2	yes	—
151	#3185	C₂×C₆₇	1	1	yes	characteristic
152	#3139	C₂×C₆₇	1	1	yes	characteristic
153	#1723	C₂×C₆₇	1	2	yes	complemented
154	#1718	C₂×C₆₇	1	2	yes	complemented
155	#1718	C₂×C₆₇	1	2	yes	complemented
156	#1710	C₂×C₆₇	1	2	yes	complemented
157	#1453	C₂×C₆₇	1	1	yes	characteristic
158	C₆₇	#19732	1	1	yes	characteristic, direct product factor, Sylow
159	#533	C₂²×C₆₇	1	1	yes	characteristic
160	#533	C₂²×C₆₇	1	2	yes	—
161	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
162	#500	C₂²×C₆₇	1	2	yes	—
163	#500	C₂²×C₆₇	1	2	yes	—
164	#500	C₂²×C₆₇	1	2	yes	—
165	#498	C₂²×C₆₇	1	1	yes	characteristic
166	#498	C₂²×C₆₇	1	2	yes	—
167	#494	C₂²×C₆₇	1	1	yes	characteristic
168	#491	—	2	2	no	—
169	#487	C₂²×C₆₇	1	2	yes	—
170	#487	C₂²×C₆₇	1	2	yes	—
171	#487	C₂²×C₆₇	1	2	yes	—
172	#487	C₂²×C₆₇	1	2	yes	—
173	#487	C₂²×C₆₇	1	2	yes	—
174	#487	C₂²×C₆₇	1	2	yes	—
175	#487	C₂²×C₆₇	1	2	yes	—
176	#485	—	2	4	no	—
177	#484	—	2	4	no	—
178	#479	—	2	4	no	—
179	#477	—	2	4	no	—
180	#470	—	2	2	no	—
181	#441	—	2	2	no	—
182	#431	C₂²×C₆₇	1	2	yes	—
183	#419	—	2	2	no	—
184	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
185	#322	C₂²×C₆₇	1	1	yes	characteristic
186	#322	C₂²×C₆₇	1	2	yes	—
187	C₈²	C₂²×C₆₇	1	1	yes	characteristic
188	C₂²×Q₈	—	2	1	no	—
189	C₂×Q₁₆	—	2	2	no	—
190	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
194	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
195	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
196	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
197	#125	—	2	2	no	—
198	#125	—	2	1	no	—
199	C₂²⋊Q₈	—	2	2	no	—
200	C₂²⋊Q₈	—	2	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	#117	—	2	1	no	—
206	#117	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
209	#107	—	2	4	no	—
210	#107	—	2	4	no	—
211	#107	—	2	4	no	—
212	#106	—	2	2	no	—
213	#106	—	2	4	no	—
214	C₄⋊C₈	D₄×C₆₇	1	4	yes	—
215	#103	—	2	4	no	—
216	#103	—	2	4	no	—
217	#103	—	2	4	no	—
218	#103	—	2	4	no	—
219	#103	—	2	4	no	—
220	#103	—	2	4	no	—
221	#103	—	2	4	no	—
222	C₂²⋊C₈	D₄×C₆₇	1	4	yes	—
223	#97	D₄×C₆₇	1	2	yes	—
224	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₆₇	1	2	yes	—
226	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₆₇	1	2	yes	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	4	4	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
237	Q₁₆	—	4	4	no	—
238	M₁₆	—	4	2	no	—
239	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
240	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
242	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	4	1	no	—
259	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₆₇	1	2	yes	—
261	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
262	C₂³	#127×C₆₇	1	1	yes	characteristic
263	Q₈	—	4	4	no	—
264	Q₈	—	4	4	no	—
265	Q₈	—	4	4	no	—
266	Q₈	—	4	4	no	—
267	Q₈	—	8	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	#127×C₆₇	1	2	yes	—
276	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	8	2	no	—
289	C₄	—	8	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	#575×C₆₇	1	2	yes	—
293	C₄	#578×C₆₇	1	4	yes	—
294	C₂	—	4	1	no	—
295	C₂	C₆₇×#3153	1	1	yes	characteristic
296	C₂	C₆₇×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂³×C₆₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₇×#1723	C₂	semidirect
6. C₆₇×#1718	C₂	semidirect
7. C₆₇×#1718	C₂	semidirect
8. C₆₇×#1710	C₂	semidirect
145. #19732	C₆₇	direct (N × H)
153. #1723	C₂×C₆₇	semidirect
154. #1718	C₂×C₆₇	semidirect
155. #1718	C₂×C₆₇	semidirect
156. #1710	C₂×C₆₇	semidirect
158. C₆₇	#19732	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×#3231	2
C₆₇×#3230	2
C₆₇×#3185	2
C₆₇×#3139	2
C₆₇×#1723	2
C₆₇×#1718	2
C₆₇×#1710	2
C₆₇×#1453	2
#19732	67

Subgroups — C67×#19732 #30698983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19732 #30698983