Subgroups — C₆₇×#22675 #30701926

Order 17152.

898 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#441×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2351	C₂	1	2	yes	maximal
4	C₆₇×#1943	C₂	1	2	yes	maximal
5	C₆₇×#1863	C₂	1	4	yes	maximal
6	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1805	C₂	1	2	yes	maximal
8	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
12	#441×C₆₇	—	2	8	no	—
13	C₄×Q₁₆×C₆₇	C₄	1	8	yes	complemented
14	C₂×#107×C₆₇	C₂²	1	2	yes	—
15	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#103×C₆₇	C₂²	1	4	yes	—
19	C₂×#103×C₆₇	C₂²	1	2	yes	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#389×C₆₇	—	2	4	no	—
23	#389×C₆₇	C₂²	1	4	yes	—
24	#382×C₆₇	—	2	2	no	—
25	#382×C₆₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	#340×C₆₇	C₂²	1	4	yes	—
29	#336×C₆₇	C₂²	1	2	yes	—
30	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×Q₁₆×C₆₇	D₄	1	4	yes	—
33	C₂×Q₁₆×C₆₇	Q₈	1	4	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₆₇	—	2	4	no	—
38	C₄×Q₈×C₆₇	—	2	8	no	—
39	C₄×Q₈×C₆₇	—	4	4	no	—
40	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₆₇	—	2	2	no	—
48	C₂×C₄²×C₆₇	—	2	4	no	—
49	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	#107×C₆₇	—	4	2	no	—
52	#107×C₆₇	C₂×C₄	1	4	yes	—
53	#106×C₆₇	—	2	4	no	—
54	C₄⋊C₈×C₆₇	—	2	2	no	—
55	#103×C₆₇	—	2	8	no	—
56	#103×C₆₇	C₂×C₄	1	8	yes	—
57	#97×C₆₇	—	2	4	no	—
58	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
59	#95×C₆₇	—	2	2	no	—
60	#95×C₆₇	—	2	4	no	—
61	#95×C₆₇	—	2	2	no	—
62	#95×C₆₇	D₄	1	2	yes	—
63	C₂×Q₈×C₆₇	—	2	2	no	—
64	C₂×Q₈×C₆₇	—	2	2	no	—
65	C₂×Q₈×C₆₇	—	2	2	no	—
66	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
67	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
68	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	4	no	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
76	Q₁₆×C₆₇	C₄⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₆₇	—	2	1	no	—
78	C₂×C₈×C₆₇	—	2	1	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
86	C₄⋊C₄×C₆₇	—	2	8	no	—
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	4	2	no	—
91	C₄⋊C₄×C₆₇	—	4	8	no	—
92	C₄⋊C₄×C₆₇	—	4	8	no	—
93	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
94	C₄²×C₆₇	—	2	8	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	4	8	no	—
98	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₆₇	#41	1	2	yes	—
100	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₆₇	—	2	4	no	—
102	Q₈×C₆₇	—	2	4	no	—
103	Q₈×C₆₇	—	2	4	no	—
104	Q₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	—	4	4	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
126	C₈×C₆₇	—	2	2	no	—
127	C₈×C₆₇	—	4	2	no	—
128	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
129	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₆₇	D₄²	1	1	yes	characteristic
131	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
132	C₂²×C₆₇	#575	1	2	yes	—
133	C₂²×C₆₇	#578	1	2	yes	—
134	C₄×C₆₇	—	2	8	no	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	4	4	no	—
138	C₄×C₆₇	—	4	4	no	—
139	C₄×C₆₇	—	8	4	no	—
140	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₆₇	#520	1	1	yes	characteristic
142	C₄×C₆₇	#547	1	1	yes	characteristic
143	C₄×C₆₇	#554	1	1	yes	characteristic
144	#22675	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₆₇	#2943	1	1	yes	characteristic
147	C₂×C₆₇	#2950	1	1	yes	characteristic
148	C₂×C₆₇	#3277	1	2	yes	—
149	C₂×C₆₇	#3390	1	2	yes	—
150	C₂×#441	C₂×C₆₇	1	2	yes	—
151	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic
152	#2351	C₂×C₆₇	1	2	yes	—
153	#1943	C₂×C₆₇	1	2	yes	—
154	#1863	C₂×C₆₇	1	4	yes	—
155	#1846	C₂×C₆₇	1	1	yes	characteristic
156	#1805	C₂×C₆₇	1	2	yes	—
157	#1770	C₂×C₆₇	1	1	yes	characteristic
158	C₆₇	#22675	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
160	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
161	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
162	#441	—	2	8	no	—
163	C₄×Q₁₆	C₄×C₆₇	1	8	yes	complemented
164	C₂×#107	C₂²×C₆₇	1	2	yes	—
165	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
166	C₂×#106	C₂²×C₆₇	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
168	C₂×#103	C₂²×C₆₇	1	4	yes	—
169	C₂×#103	C₂²×C₆₇	1	2	yes	—
170	C₂×#97	C₂²×C₆₇	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
172	#389	—	2	4	no	—
173	#389	C₂²×C₆₇	1	4	yes	—
174	#382	—	2	2	no	—
175	#382	C₂²×C₆₇	1	2	yes	—
176	C₄×C₄⋊C₄	—	2	2	no	—
177	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
178	#340	C₂²×C₆₇	1	4	yes	—
179	#336	C₂²×C₆₇	1	2	yes	—
180	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
181	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
182	C₂×Q₁₆	D₄×C₆₇	1	4	yes	—
183	C₂×Q₁₆	Q₈×C₆₇	1	4	yes	—
184	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
185	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	—	2	8	no	—
189	C₄×Q₈	—	4	4	no	—
190	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
196	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
200	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
201	#107	—	4	2	no	—
202	#107	C₂×C₄×C₆₇	1	4	yes	—
203	#106	—	2	4	no	—
204	C₄⋊C₈	—	2	2	no	—
205	#103	—	2	8	no	—
206	#103	C₂×C₄×C₆₇	1	8	yes	—
207	#97	—	2	4	no	—
208	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
209	#95	—	2	2	no	—
210	#95	—	2	4	no	—
211	#95	—	2	2	no	—
212	#95	D₄×C₆₇	1	2	yes	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
217	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
218	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
223	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
226	Q₁₆	C₄⋊C₄×C₆₇	1	16	yes	complemented
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
235	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	8	no	—
242	C₄⋊C₄	—	4	8	no	—
243	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
244	C₄²	—	2	8	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	8	no	—
248	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
249	C₄²	#41×C₆₇	1	2	yes	—
250	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	2	4	no	—
253	Q₈	—	2	4	no	—
254	Q₈	C₂×C₄⋊C₄×C₆₇	1	8	yes	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
269	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
270	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
274	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
275	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	4	2	no	—
278	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
279	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
280	C₂²	D₄²×C₆₇	1	1	yes	characteristic
281	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
282	C₂²	#575×C₆₇	1	2	yes	—
283	C₂²	#578×C₆₇	1	2	yes	—
284	C₄	—	2	8	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	4	no	—
290	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
291	C₄	#520×C₆₇	1	1	yes	characteristic
292	C₄	#547×C₆₇	1	1	yes	characteristic
293	C₄	#554×C₆₇	1	1	yes	characteristic
294	C₂	D₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
295	C₂	C₆₇×#2943	1	1	yes	characteristic
296	C₂	C₆₇×#2950	1	1	yes	characteristic
297	C₂	C₆₇×#3277	1	2	yes	—
298	C₂	C₆₇×#3390	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₄⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×Q₁₆×C₆₇	C₄	semidirect
76. Q₁₆×C₆₇	C₄⋊C₄	semidirect
144. #22675	C₆₇	direct (N × H)
158. C₆₇	#22675	direct (N × H)
163. C₄×Q₁₆	C₄×C₆₇	semidirect
226. Q₁₆	C₄⋊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×#441×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₆₇×#2351	2
C₆₇×#1943	2
C₆₇×#1863	2
C₆₇×#1846	2
C₆₇×#1805	2
C₆₇×#1770	2
#22675	67

Subgroups — C67×#22675 #30701926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22675 #30701926