Subgroups — C₆₇×#2935 #22557366

Order 8576.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131×C₆₇	C₂	1	2	yes	maximal, complemented
2	#532×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#436×C₆₇	C₂	1	2	yes	maximal, complemented
4	#435×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₈×D₄×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#408×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₆₇	C₂²	1	2	yes	—
10	#131×C₆₇	—	2	8	no	—
11	C₂×M₁₆×C₆₇	C₂²	1	4	yes	—
12	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₆₇	C₂²	1	4	yes	—
16	C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
17	#127×C₆₇	C₄	1	1	yes	characteristic
18	#124×C₆₇	C₄	1	2	yes	—
19	#121×C₆₇	C₄	1	2	yes	—
20	#121×C₆₇	C₄	1	2	yes	—
21	C₄×D₄×C₆₇	C₂²	1	2	yes	—
22	C₄×D₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
28	C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
29	#97×C₆₇	C₂²	1	2	yes	—
30	C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	#41×C₆₇	—	2	4	no	—
32	#41×C₆₇	—	2	4	no	—
33	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
41	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	M₁₆×C₆₇	—	2	2	no	—
43	M₁₆×C₆₇	—	2	8	no	—
44	M₁₆×C₆₇	D₄	1	4	yes	complemented
45	C₂×C₈×C₆₇	—	2	4	no	—
46	C₂×C₈×C₆₇	—	2	8	no	—
47	C₂×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂×C₈×C₆₇	C₂³	1	2	yes	—
51	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
54	C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
55	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	Q₈×C₆₇	—	2	4	no	—
62	D₄×C₆₇	—	2	8	no	—
63	D₄×C₆₇	—	2	4	no	—
64	D₄×C₆₇	—	2	4	no	—
65	D₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	4	no	—
68	C₂×C₄×C₆₇	—	2	4	no	—
69	C₂×C₄×C₆₇	—	4	1	no	—
70	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₆₇	#41	1	2	yes	—
78	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
79	C₈×C₆₇	—	2	2	no	—
80	C₈×C₆₇	—	2	2	no	—
81	C₈×C₆₇	—	2	2	no	—
82	C₈×C₆₇	C₂×D₄	1	4	yes	—
83	C₂²×C₆₇	—	2	1	no	—
84	C₂²×C₆₇	—	2	1	no	—
85	C₂²×C₆₇	—	2	4	no	—
86	C₂²×C₆₇	—	2	4	no	—
87	C₂²×C₆₇	—	2	4	no	—
88	C₂²×C₆₇	C₄×D₄	1	2	yes	—
89	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₆₇	—	2	4	no	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	—	2	1	no	—
93	C₄×C₆₇	—	2	1	no	—
94	C₄×C₆₇	C₄×D₄	1	2	yes	—
95	C₄×C₆₇	#131	1	2	yes	complemented
96	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
98	C₂×C₆₇	—	2	2	no	—
99	C₂×C₆₇	—	4	4	no	—
100	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
101	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
102	C₂×C₆₇	#568	1	1	yes	characteristic
103	#2935	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₆₇	#2935	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×#131	C₂×C₆₇	1	2	yes	complemented
106	#532	C₂×C₆₇	1	1	yes	characteristic
107	#436	C₂×C₆₇	1	2	yes	complemented
108	#435	C₂×C₆₇	1	4	yes	complemented
109	C₈×D₄	C₂×C₆₇	1	2	yes	complemented
110	C₂×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
111	#408	C₂×C₆₇	1	2	yes	complemented
112	C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
113	C₂×#41	C₂²×C₆₇	1	2	yes	—
114	#131	—	2	8	no	—
115	C₂×M₁₆	C₂²×C₆₇	1	4	yes	—
116	C₂×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
117	C₂×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₆₇	1	2	yes	—
119	C₂²×C₈	C₂²×C₆₇	1	4	yes	—
120	C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
121	#127	C₄×C₆₇	1	1	yes	characteristic
122	#124	C₄×C₆₇	1	2	yes	—
123	#121	C₄×C₆₇	1	2	yes	—
124	#121	C₄×C₆₇	1	2	yes	—
125	C₄×D₄	C₂²×C₆₇	1	2	yes	—
126	C₄×D₄	C₂²×C₆₇	1	2	yes	—
127	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
128	C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
129	C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
130	C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
131	C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
132	C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
133	#97	C₂²×C₆₇	1	2	yes	—
134	C₄×C₈	C₂²×C₆₇	1	2	yes	—
135	#41	—	2	4	no	—
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×C₄×C₆₇	1	2	yes	—
138	C₂×D₄	C₂×C₄×C₆₇	1	2	yes	—
139	C₂×D₄	C₂×C₄×C₆₇	1	4	yes	—
140	C₂×D₄	C₂×C₄×C₆₇	1	2	yes	—
141	C₂×D₄	C₂×C₄×C₆₇	1	2	yes	—
142	C₂²×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
145	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
146	M₁₆	—	2	2	no	—
147	M₁₆	—	2	8	no	—
148	M₁₆	D₄×C₆₇	1	4	yes	complemented
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	C₂³×C₆₇	1	2	yes	—
152	C₂×C₈	C₂³×C₆₇	1	2	yes	—
153	C₂×C₈	C₂³×C₆₇	1	2	yes	—
154	C₂×C₈	C₂³×C₆₇	1	2	yes	—
155	C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
156	C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
157	C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
158	C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
159	C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
160	C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
161	C₄²	C₂³×C₆₇	1	1	yes	characteristic
162	C₄²	C₂³×C₆₇	1	1	yes	characteristic
163	C₂³	C₂²×C₄×C₆₇	1	4	yes	—
164	C₂³	C₂²×C₄×C₆₇	1	1	yes	characteristic
165	Q₈	—	2	4	no	—
166	D₄	—	2	8	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	4	no	—
169	D₄	—	4	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₄×C₆₇	1	4	yes	—
178	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
181	C₂×C₄	#41×C₆₇	1	2	yes	—
182	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₈	C₂×D₄×C₆₇	1	4	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	C₄×D₄×C₆₇	1	2	yes	—
193	C₂²	C₂³×C₄×C₆₇	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	C₄×D₄×C₆₇	1	2	yes	—
199	C₄	#131×C₆₇	1	2	yes	complemented
200	C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
201	C₄	C₂×#41×C₆₇	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	4	4	no	—
204	C₂	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
205	C₂	C₂×#131×C₆₇	1	1	yes	characteristic
206	C₂	#568×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂³×C₄
prime radical	#532×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#131×C₆₇	C₂	semidirect
3. #436×C₆₇	C₂	semidirect
4. #435×C₆₇	C₂	semidirect
5. C₈×D₄×C₆₇	C₂	semidirect
6. C₂×C₄⋊C₈×C₆₇	C₂	semidirect
7. #408×C₆₇	C₂	semidirect
8. C₄×M₁₆×C₆₇	C₂	semidirect
44. M₁₆×C₆₇	D₄	semidirect
95. C₄×C₆₇	#131	semidirect
103. #2935	C₆₇	direct (N × H)
104. C₆₇	#2935	direct (N × H)
105. C₂×#131	C₂×C₆₇	semidirect
107. #436	C₂×C₆₇	semidirect
108. #435	C₂×C₆₇	semidirect
109. C₈×D₄	C₂×C₆₇	semidirect
110. C₂×C₄⋊C₈	C₂×C₆₇	semidirect
111. #408	C₂×C₆₇	semidirect
112. C₄×M₁₆	C₂×C₆₇	semidirect
148. M₁₆	D₄×C₆₇	semidirect
199. C₄	#131×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131×C₆₇	2
#532×C₆₇	2
#436×C₆₇	2
#435×C₆₇	2
C₈×D₄×C₆₇	2
C₂×C₄⋊C₈×C₆₇	2
#408×C₆₇	2
C₄×M₁₆×C₆₇	2
#2935	67

Subgroups — C67×#2935 #22557366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2935 #22557366