Subgroups — C₆₇×#20125 #30699376

Order 17152.

566 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3369	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3340	C₂	1	2	yes	maximal
3	C₆₇×#1716	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1692	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1683	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1652	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1634	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1633	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1585	C₂	1	2	yes	maximal, complemented
10	Q₈²×C₆₇	C₂²	1	1	yes	characteristic
11	#556×C₆₇	C₂²	1	2	yes	—
12	#555×C₆₇	C₂²	1	1	yes	characteristic
13	#498×C₆₇	C₂²	1	2	yes	—
14	#498×C₆₇	C₂²	1	1	yes	characteristic
15	#494×C₆₇	C₂²	1	1	yes	characteristic
16	#489×C₆₇	—	2	2	no	—
17	#487×C₆₇	C₂²	1	2	yes	—
18	#485×C₆₇	—	2	2	no	—
19	#482×C₆₇	—	2	2	no	—
20	#477×C₆₇	C₂²	1	2	yes	—
21	#476×C₆₇	C₂²	1	2	yes	—
22	#475×C₆₇	—	2	2	no	—
23	#474×C₆₇	C₂²	1	2	yes	—
24	#465×C₆₇	—	2	2	no	—
25	#464×C₆₇	—	2	2	no	—
26	#462×C₆₇	C₂²	1	2	yes	—
27	#446×C₆₇	C₂²	1	2	yes	—
28	#441×C₆₇	—	2	2	no	—
29	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
30	C₈×D₄×C₆₇	C₂²	1	2	yes	—
31	#427×C₆₇	—	2	2	no	—
32	#335×C₆₇	C₂²	1	2	yes	—
33	#333×C₆₇	C₂²	1	1	yes	characteristic
34	#333×C₆₇	C₂²	1	1	yes	characteristic
35	#332×C₆₇	C₂²	1	2	yes	—
36	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
37	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
38	#322×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₆₇	—	2	2	no	—
40	C₂×Q₁₆×C₆₇	—	4	1	no	—
41	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₄⋊Q₈×C₆₇	—	2	2	no	—
44	C₄⋊Q₈×C₆₇	—	4	1	no	—
45	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
48	#126×C₆₇	—	4	2	no	—
49	#124×C₆₇	—	2	2	no	—
50	#124×C₆₇	—	4	1	no	—
51	C₂²⋊Q₈×C₆₇	—	2	2	no	—
52	#121×C₆₇	—	2	2	no	—
53	C₄×Q₈×C₆₇	—	2	2	no	—
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
56	C₄×D₄×C₆₇	C₂³	1	2	yes	—
57	#117×C₆₇	—	2	2	no	—
58	#117×C₆₇	—	2	2	no	—
59	#107×C₆₇	—	2	2	no	—
60	#107×C₆₇	—	2	2	no	—
61	#106×C₆₇	—	2	2	no	—
62	#106×C₆₇	—	2	2	no	—
63	#106×C₆₇	—	2	2	no	—
64	#106×C₆₇	—	4	1	no	—
65	#106×C₆₇	D₄	1	2	yes	—
66	C₄⋊C₈×C₆₇	—	2	2	no	—
67	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
68	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
69	#103×C₆₇	—	2	2	no	—
70	#103×C₆₇	—	2	2	no	—
71	#103×C₆₇	—	2	2	no	—
72	#103×C₆₇	—	2	2	no	—
73	#103×C₆₇	D₄	1	2	yes	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	—	2	2	no	—
76	#102×C₆₇	D₄	1	2	yes	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	#97×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	—	2	1	no	—
83	C₂×Q₈×C₆₇	—	2	1	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₆₇	—	4	1	no	—
87	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	Q₁₆×C₆₇	—	4	4	no	—
90	Q₁₆×C₆₇	—	8	2	no	—
91	SD₁₆×C₆₇	—	4	2	no	—
92	C₂×C₈×C₆₇	—	2	1	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	4	4	no	—
96	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
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	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	4	2	no	—
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²⋊C₄×C₆₇	—	2	2	no	—
108	C₂²⋊C₄×C₆₇	—	4	2	no	—
109	C₂²⋊C₄×C₆₇	—	4	2	no	—
110	C₂²⋊C₄×C₆₇	—	4	2	no	—
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	—	4	1	no	—
113	C₄²×C₆₇	—	4	2	no	—
114	C₄²×C₆₇	—	4	1	no	—
115	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂³×C₆₇	—	2	2	no	—
117	Q₈×C₆₇	—	2	2	no	—
118	Q₈×C₆₇	—	2	2	no	—
119	Q₈×C₆₇	—	4	2	no	—
120	Q₈×C₆₇	—	4	2	no	—
121	Q₈×C₆₇	—	8	1	no	—
122	Q₈×C₆₇	—	8	1	no	—
123	Q₈×C₆₇	—	16	1	no	—
124	D₄×C₆₇	—	2	2	no	—
125	D₄×C₆₇	—	2	2	no	—
126	D₄×C₆₇	—	16	1	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	4	1	no	—
135	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₆₇	—	2	2	no	—
138	C₈×C₆₇	—	2	2	no	—
139	C₈×C₆₇	—	2	2	no	—
140	C₈×C₆₇	—	4	2	no	—
141	C₈×C₆₇	—	4	2	no	—
142	C₂²×C₆₇	—	4	2	no	—
143	C₂²×C₆₇	—	4	2	no	—
144	C₂²×C₆₇	—	4	2	no	—
145	C₂²×C₆₇	D₄²	1	1	yes	characteristic
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	4	2	no	—
148	C₄×C₆₇	—	4	2	no	—
149	C₄×C₆₇	—	4	2	no	—
150	C₄×C₆₇	—	8	2	no	—
151	C₄×C₆₇	—	8	1	no	—
152	C₄×C₆₇	—	8	1	no	—
153	C₄×C₆₇	#575	1	2	yes	—
154	C₄×C₆₇	#578	1	2	yes	—
155	#20125	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₆₇	—	8	2	no	—
157	C₂×C₆₇	#3269	1	2	yes	—
158	C₂×C₆₇	#3291	1	1	yes	characteristic
159	#3369	C₂×C₆₇	1	2	yes	complemented
160	#3340	C₂×C₆₇	1	2	yes	—
161	#1716	C₂×C₆₇	1	1	yes	characteristic, complemented
162	#1692	C₂×C₆₇	1	2	yes	complemented
163	#1683	C₂×C₆₇	1	2	yes	complemented
164	#1652	C₂×C₆₇	1	2	yes	complemented
165	#1634	C₂×C₆₇	1	1	yes	characteristic
166	#1633	C₂×C₆₇	1	1	yes	characteristic, complemented
167	#1585	C₂×C₆₇	1	2	yes	complemented
168	C₆₇	#20125	1	1	yes	characteristic, direct product factor, Sylow
169	Q₈²	C₂²×C₆₇	1	1	yes	characteristic
170	#556	C₂²×C₆₇	1	2	yes	—
171	#555	C₂²×C₆₇	1	1	yes	characteristic
172	#498	C₂²×C₆₇	1	2	yes	—
173	#498	C₂²×C₆₇	1	1	yes	characteristic
174	#494	C₂²×C₆₇	1	1	yes	characteristic
175	#489	—	2	2	no	—
176	#487	C₂²×C₆₇	1	2	yes	—
177	#485	—	2	2	no	—
178	#482	—	2	2	no	—
179	#477	C₂²×C₆₇	1	2	yes	—
180	#476	C₂²×C₆₇	1	2	yes	—
181	#475	—	2	2	no	—
182	#474	C₂²×C₆₇	1	2	yes	—
183	#465	—	2	2	no	—
184	#464	—	2	2	no	—
185	#462	C₂²×C₆₇	1	2	yes	—
186	#446	C₂²×C₆₇	1	2	yes	—
187	#441	—	2	2	no	—
188	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
189	C₈×D₄	C₂²×C₆₇	1	2	yes	—
190	#427	—	2	2	no	—
191	#335	C₂²×C₆₇	1	2	yes	—
192	#333	C₂²×C₆₇	1	1	yes	characteristic
193	#333	C₂²×C₆₇	1	1	yes	characteristic
194	#332	C₂²×C₆₇	1	2	yes	—
195	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
196	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
197	#322	C₂²×C₆₇	1	1	yes	characteristic
198	C₂×Q₁₆	—	2	2	no	—
199	C₂×Q₁₆	—	4	1	no	—
200	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
201	C₂²×C₈	—	2	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	4	1	no	—
204	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
205	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
207	#126	—	4	2	no	—
208	#124	—	2	2	no	—
209	#124	—	4	1	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	#121	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
215	C₄×D₄	C₂³×C₆₇	1	2	yes	—
216	#117	—	2	2	no	—
217	#117	—	2	2	no	—
218	#107	—	2	2	no	—
219	#107	—	2	2	no	—
220	#106	—	2	2	no	—
221	#106	—	2	2	no	—
222	#106	—	2	2	no	—
223	#106	—	4	1	no	—
224	#106	D₄×C₆₇	1	2	yes	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
227	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
228	#103	—	2	2	no	—
229	#103	—	2	2	no	—
230	#103	—	2	2	no	—
231	#103	—	2	2	no	—
232	#103	D₄×C₆₇	1	2	yes	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#102	D₄×C₆₇	1	2	yes	—
236	C₂²⋊C₈	—	2	2	no	—
237	#97	—	2	2	no	—
238	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₆₇	1	2	yes	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
247	C₂²×C₄	—	2	2	no	—
248	Q₁₆	—	4	4	no	—
249	Q₁₆	—	8	2	no	—
250	SD₁₆	—	4	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	4	no	—
255	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
256	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
265	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	2	no	—
273	C₄²	—	4	1	no	—
274	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
275	C₂³	—	2	2	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	8	1	no	—
281	Q₈	—	8	1	no	—
282	Q₈	—	16	1	no	—
283	D₄	—	2	2	no	—
284	D₄	—	2	2	no	—
285	D₄	—	16	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
295	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	D₄²×C₆₇	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	1	no	—
311	C₄	—	8	1	no	—
312	C₄	#575×C₆₇	1	2	yes	—
313	C₄	#578×C₆₇	1	2	yes	—
314	C₂	—	8	2	no	—
315	C₂	C₆₇×#3269	1	2	yes	—
316	C₂	C₆₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	#555×C₆₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3369	C₂	semidirect
3. C₆₇×#1716	C₂	semidirect
4. C₆₇×#1692	C₂	semidirect
5. C₆₇×#1683	C₂	semidirect
6. C₆₇×#1652	C₂	semidirect
8. C₆₇×#1633	C₂	semidirect
9. C₆₇×#1585	C₂	semidirect
155. #20125	C₆₇	direct (N × H)
159. #3369	C₂×C₆₇	semidirect
161. #1716	C₂×C₆₇	semidirect
162. #1692	C₂×C₆₇	semidirect
163. #1683	C₂×C₆₇	semidirect
164. #1652	C₂×C₆₇	semidirect
166. #1633	C₂×C₆₇	semidirect
167. #1585	C₂×C₆₇	semidirect
168. C₆₇	#20125	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×#3369	2
C₆₇×#3340	2
C₆₇×#1716	2
C₆₇×#1692	2
C₆₇×#1683	2
C₆₇×#1652	2
C₆₇×#1634	2
C₆₇×#1633	2
C₆₇×#1585	2
#20125	67

Subgroups — C67×#20125 #30699376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20125 #30699376