Subgroups — C₆₇×#3082 #22557513

Order 8576.

362 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#544×C₆₇	C₂	1	1	yes	maximal, characteristic
2	#519×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#479×C₆₇	C₂	1	2	yes	maximal, complemented
4	#479×C₆₇	C₂	1	2	yes	maximal, complemented
5	#477×C₆₇	C₂	1	2	yes	maximal, complemented
6	#475×C₆₇	C₂	1	2	yes	maximal, complemented
7	#428×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#427×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#424×C₆₇	C₂	1	1	yes	maximal, characteristic
10	#419×C₆₇	C₂	1	1	yes	maximal, characteristic
11	C₂×#103×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₆₇	—	2	1	no	—
16	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
17	#125×C₆₇	—	2	1	no	—
18	#125×C₆₇	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₆₇	—	2	2	no	—
20	C₄×Q₈×C₆₇	—	2	1	no	—
21	C₄×Q₈×C₆₇	—	2	1	no	—
22	C₄×Q₈×C₆₇	C₂²	1	2	yes	—
23	C₄×Q₈×C₆₇	C₂²	1	2	yes	—
24	#117×C₆₇	—	2	2	no	—
25	#117×C₆₇	C₂²	1	1	yes	characteristic
26	#117×C₆₇	C₂²	1	1	yes	characteristic
27	#117×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#107×C₆₇	C₂²	1	1	yes	characteristic
30	#107×C₆₇	C₂²	1	2	yes	—
31	#107×C₆₇	C₂²	1	1	yes	characteristic
32	#106×C₆₇	C₂²	1	1	yes	characteristic
33	#106×C₆₇	C₂²	1	2	yes	—
34	#106×C₆₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
36	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
37	#103×C₆₇	C₂²	1	2	yes	—
38	#103×C₆₇	C₂²	1	2	yes	—
39	#103×C₆₇	C₂²	1	2	yes	—
40	#103×C₆₇	C₂²	1	2	yes	—
41	C₂×Q₈×C₆₇	—	2	1	no	—
42	C₂×Q₈×C₆₇	—	2	2	no	—
43	C₂×Q₈×C₆₇	—	2	2	no	—
44	C₂×Q₈×C₆₇	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₆₇	—	2	1	no	—
47	C₂²×C₄×C₆₇	—	2	1	no	—
48	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	M₁₆×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
63	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₆₇	—	2	1	no	—
68	C₂²⋊C₄×C₆₇	—	2	1	no	—
69	C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	—	2	1	no	—
73	C₄²×C₆₇	—	2	1	no	—
74	C₄²×C₆₇	C₂³	1	2	yes	—
75	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₆₇	—	2	4	no	—
77	Q₈×C₆₇	—	2	2	no	—
78	Q₈×C₆₇	—	8	2	no	—
79	Q₈×C₆₇	C₂×Q₈	1	4	yes	—
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	8	1	no	—
88	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₆₇	#41	1	2	yes	—
93	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
94	C₈×C₆₇	—	2	1	no	—
95	C₈×C₆₇	—	2	1	no	—
96	C₈×C₆₇	—	2	2	no	—
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	C₂²⋊Q₈	1	2	yes	—
100	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₆₇	—	2	4	no	—
102	C₄×C₆₇	—	2	2	no	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	4	1	no	—
105	C₄×C₆₇	—	4	2	no	—
106	C₄×C₆₇	—	4	1	no	—
107	C₄×C₆₇	—	4	1	no	—
108	C₄×C₆₇	—	4	1	no	—
109	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
110	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
111	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
112	C₂×C₆₇	—	2	2	no	—
113	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₆₇	#577	1	1	yes	characteristic
115	C₂×C₆₇	#578	1	1	yes	characteristic
116	#3082	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₆₇	#3082	1	1	yes	characteristic, direct product factor, Sylow
118	#544	C₂×C₆₇	1	1	yes	characteristic
119	#519	C₂×C₆₇	1	1	yes	characteristic
120	#479	C₂×C₆₇	1	2	yes	complemented
121	#479	C₂×C₆₇	1	2	yes	complemented
122	#477	C₂×C₆₇	1	2	yes	complemented
123	#475	C₂×C₆₇	1	2	yes	complemented
124	#428	C₂×C₆₇	1	1	yes	characteristic
125	#427	C₂×C₆₇	1	1	yes	characteristic
126	#424	C₂×C₆₇	1	1	yes	characteristic
127	#419	C₂×C₆₇	1	1	yes	characteristic
128	C₂×#103	C₂×C₆₇	1	1	yes	characteristic
129	C₂²×Q₈	C₂²×C₆₇	1	1	yes	characteristic
130	C₂×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
131	C₂²×C₈	C₂²×C₆₇	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	1	no	—
133	C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
134	#125	—	2	1	no	—
135	#125	C₂²×C₆₇	1	2	yes	—
136	C₂²⋊Q₈	—	2	2	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	—	2	1	no	—
139	C₄×Q₈	C₂²×C₆₇	1	2	yes	—
140	C₄×Q₈	C₂²×C₆₇	1	2	yes	—
141	#117	—	2	2	no	—
142	#117	C₂²×C₆₇	1	1	yes	characteristic
143	#117	C₂²×C₆₇	1	1	yes	characteristic
144	#117	C₂²×C₆₇	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
146	#107	C₂²×C₆₇	1	1	yes	characteristic
147	#107	C₂²×C₆₇	1	1	yes	characteristic
148	#107	C₂²×C₆₇	1	2	yes	—
149	#106	C₂²×C₆₇	1	1	yes	characteristic
150	#106	C₂²×C₆₇	1	1	yes	characteristic
151	#106	C₂²×C₆₇	1	2	yes	—
152	C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
153	C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
154	#103	C₂²×C₆₇	1	2	yes	—
155	#103	C₂²×C₆₇	1	2	yes	—
156	#103	C₂²×C₆₇	1	2	yes	—
157	#103	C₂²×C₆₇	1	2	yes	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	Q₈×C₆₇	1	4	yes	complemented
162	C₂×Q₈	C₂³×C₆₇	1	2	yes	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
166	M₁₆	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂³×C₆₇	1	1	yes	characteristic
169	C₂×C₈	C₂³×C₆₇	1	2	yes	—
170	C₂×C₈	C₂³×C₆₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
178	C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
180	C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
184	C₂²⋊C₄	—	2	1	no	—
185	C₂²⋊C₄	—	2	1	no	—
186	C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	C₂³×C₆₇	1	2	yes	—
192	C₂³	C₂×D₄×C₆₇	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	8	2	no	—
196	Q₈	C₂×Q₈×C₆₇	1	4	yes	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	8	1	no	—
205	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
206	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
208	C₂×C₄	C₂×Q₈×C₆₇	1	2	yes	—
209	C₂×C₄	#41×C₆₇	1	2	yes	—
210	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	2	2	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂²⋊Q₈×C₆₇	1	2	yes	—
217	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	1	no	—
226	C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
227	C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
228	C₄	C₂×#41×C₆₇	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
231	C₂	#577×C₆₇	1	1	yes	characteristic
232	C₂	#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₇
prime radical	C₂³×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #479×C₆₇	C₂	semidirect
4. #479×C₆₇	C₂	semidirect
5. #477×C₆₇	C₂	semidirect
6. #475×C₆₇	C₂	semidirect
44. C₂×Q₈×C₆₇	Q₈	semidirect
116. #3082	C₆₇	direct (N × H)
117. C₆₇	#3082	direct (N × H)
120. #479	C₂×C₆₇	semidirect
121. #479	C₂×C₆₇	semidirect
122. #477	C₂×C₆₇	semidirect
123. #475	C₂×C₆₇	semidirect
161. C₂×Q₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
#544×C₆₇	2
#519×C₆₇	2
#479×C₆₇	2
#477×C₆₇	2
#475×C₆₇	2
#428×C₆₇	2
#427×C₆₇	2
#424×C₆₇	2
#419×C₆₇	2
C₂×#103×C₆₇	2
#3082	67

Subgroups — C67×#3082 #22557513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#3082 #22557513