Subgroups — C₆₇×#2541 #22556972

Order 8576.

948 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#127×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#127×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#121×C₆₇	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#392×C₆₇	C₂	1	4	yes	maximal, complemented
6	#390×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
9	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
10	C₂²×D₄×C₆₇	C₂²	1	8	yes	—
11	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
13	C₂³×C₄×C₆₇	C₂²	1	2	yes	—
14	#127×C₆₇	—	2	8	no	—
15	#127×C₆₇	—	2	4	no	—
16	#121×C₆₇	—	2	16	no	—
17	C₄×D₄×C₆₇	—	2	4	no	—
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
22	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#95×C₆₇	C₂²	1	2	yes	—
25	C₂⁴×C₆₇	C₂³	1	4	yes	—
26	C₂⁴×C₆₇	C₂³	1	2	yes	—
27	C₂×D₄×C₆₇	—	2	16	no	—
28	C₂×D₄×C₆₇	—	2	2	no	—
29	C₂×D₄×C₆₇	—	2	16	no	—
30	C₂×D₄×C₆₇	—	2	8	no	—
31	C₂×D₄×C₆₇	—	2	4	no	—
32	C₂×D₄×C₆₇	—	2	4	no	—
33	C₂×D₄×C₆₇	—	2	4	no	—
34	C₂×D₄×C₆₇	—	2	4	no	—
35	C₂×D₄×C₆₇	—	2	2	no	—
36	C₂×D₄×C₆₇	—	2	4	no	—
37	C₂×D₄×C₆₇	—	4	8	no	—
38	C₂×D₄×C₆₇	—	4	4	no	—
39	C₂×D₄×C₆₇	D₄	1	4	yes	complemented
40	C₂²×C₄×C₆₇	—	2	4	no	—
41	C₂²×C₄×C₆₇	—	2	8	no	—
42	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
44	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₄⋊C₄×C₆₇	D₄	1	8	yes	complemented
49	C₂²⋊C₄×C₆₇	—	2	4	no	—
50	C₂²⋊C₄×C₆₇	—	2	16	no	—
51	C₄²×C₆₇	—	2	4	no	—
52	C₄²×C₆₇	D₄	1	4	yes	complemented
53	C₂³×C₆₇	—	2	2	no	—
54	C₂³×C₆₇	—	2	4	no	—
55	C₂³×C₆₇	—	2	2	no	—
56	C₂³×C₆₇	—	2	4	no	—
57	C₂³×C₆₇	—	2	8	no	—
58	C₂³×C₆₇	—	2	4	no	—
59	C₂³×C₆₇	—	2	4	no	—
60	C₂³×C₆₇	—	2	4	no	—
61	C₂³×C₆₇	—	2	8	no	—
62	C₂³×C₆₇	C₂×D₄	1	4	yes	—
63	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
64	D₄×C₆₇	—	2	8	no	—
65	D₄×C₆₇	—	4	32	no	—
66	D₄×C₆₇	—	4	8	no	—
67	D₄×C₆₇	—	4	8	no	—
68	D₄×C₆₇	—	4	8	no	—
69	C₂×C₄×C₆₇	—	2	4	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	4	8	no	—
75	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×D₄	1	8	yes	—
77	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
81	C₂×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₆₇	—	2	4	no	—
83	C₂²×C₆₇	—	2	8	no	—
84	C₂²×C₆₇	—	4	2	no	—
85	C₂²×C₆₇	—	4	2	no	—
86	C₂²×C₆₇	—	4	4	no	—
87	C₂²×C₆₇	—	4	4	no	—
88	C₂²×C₆₇	—	4	4	no	—
89	C₂²×C₆₇	—	4	4	no	—
90	C₂²×C₆₇	—	4	8	no	—
91	C₂²×C₆₇	—	4	8	no	—
92	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	C₂²×C₆₇	C₂²×D₄	1	2	yes	—
95	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
96	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
97	C₄×C₆₇	—	2	2	no	—
98	C₄×C₆₇	—	2	8	no	—
99	C₄×C₆₇	—	4	2	no	—
100	C₄×C₆₇	#121	1	4	yes	complemented
101	C₄×C₆₇	#127	1	4	yes	complemented
102	C₂×C₆₇	—	4	4	no	—
103	C₂×C₆₇	—	8	4	no	—
104	C₂×C₆₇	C₂×#121	1	1	yes	characteristic
105	C₂×C₆₇	C₂×#127	1	1	yes	characteristic
106	C₂×C₆₇	#535	1	1	yes	characteristic
107	C₂×C₆₇	D₄²	1	2	yes	—
108	C₂×C₆₇	#550	1	2	yes	—
109	#2541	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₆₇	#2541	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×#127	C₂×C₆₇	1	2	yes	complemented
112	C₂×#127	C₂×C₆₇	1	1	yes	characteristic, complemented
113	C₂×#121	C₂×C₆₇	1	4	yes	complemented
114	C₂×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
115	#392	C₂×C₆₇	1	4	yes	complemented
116	#390	C₂×C₆₇	1	2	yes	complemented
117	C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
118	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
119	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
120	C₂²×D₄	C₂²×C₆₇	1	8	yes	—
121	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
122	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
123	C₂³×C₄	C₂²×C₆₇	1	2	yes	—
124	#127	—	2	8	no	—
125	#127	—	2	4	no	—
126	#121	—	2	16	no	—
127	C₄×D₄	—	2	4	no	—
128	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
132	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
133	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
134	#95	C₂²×C₆₇	1	2	yes	—
135	C₂⁴	C₂³×C₆₇	1	4	yes	—
136	C₂⁴	C₂³×C₆₇	1	2	yes	—
137	C₂×D₄	—	2	16	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	16	no	—
140	C₂×D₄	—	2	8	no	—
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	2	4	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	4	8	no	—
148	C₂×D₄	—	4	4	no	—
149	C₂×D₄	D₄×C₆₇	1	4	yes	complemented
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
153	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
155	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	D₄×C₆₇	1	8	yes	complemented
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	2	16	no	—
161	C₄²	—	2	4	no	—
162	C₄²	D₄×C₆₇	1	4	yes	complemented
163	C₂³	—	2	2	no	—
164	C₂³	—	2	4	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	4	no	—
167	C₂³	—	2	8	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	—	2	4	no	—
171	C₂³	—	2	8	no	—
172	C₂³	C₂×D₄×C₆₇	1	4	yes	—
173	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
174	D₄	—	2	8	no	—
175	D₄	—	4	32	no	—
176	D₄	—	4	8	no	—
177	D₄	—	4	8	no	—
178	D₄	—	4	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
186	C₂×C₄	C₂×D₄×C₆₇	1	8	yes	—
187	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
188	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
189	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
190	C₂×C₄	C₂×D₄×C₆₇	1	4	yes	—
191	C₂×C₄	#41×C₆₇	1	2	yes	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	8	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	4	no	—
197	C₂²	—	4	4	no	—
198	C₂²	—	4	4	no	—
199	C₂²	—	4	4	no	—
200	C₂²	—	4	8	no	—
201	C₂²	—	4	8	no	—
202	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
203	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
204	C₂²	C₂²×D₄×C₆₇	1	2	yes	—
205	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
206	C₂²	H(5,𝔽₂)×C₆₇	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	8	no	—
209	C₄	—	4	2	no	—
210	C₄	#121×C₆₇	1	4	yes	complemented
211	C₄	#127×C₆₇	1	4	yes	complemented
212	C₂	—	4	4	no	—
213	C₂	—	8	4	no	—
214	C₂	C₂×#121×C₆₇	1	1	yes	characteristic
215	C₂	C₂×#127×C₆₇	1	1	yes	characteristic
216	C₂	#535×C₆₇	1	1	yes	characteristic
217	C₂	D₄²×C₆₇	1	2	yes	—
218	C₂	#550×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#127×C₆₇	C₂	semidirect
2. C₂×#127×C₆₇	C₂	semidirect
3. C₂×#121×C₆₇	C₂	semidirect
4. C₂×C₄×D₄×C₆₇	C₂	semidirect
5. #392×C₆₇	C₂	semidirect
6. #390×C₆₇	C₂	semidirect
7. C₄×C₄⋊C₄×C₆₇	C₂	semidirect
39. C₂×D₄×C₆₇	D₄	semidirect
48. C₄⋊C₄×C₆₇	D₄	semidirect
52. C₄²×C₆₇	D₄	semidirect
100. C₄×C₆₇	#121	semidirect
101. C₄×C₆₇	#127	semidirect
109. #2541	C₆₇	direct (N × H)
110. C₆₇	#2541	direct (N × H)
111. C₂×#127	C₂×C₆₇	semidirect
112. C₂×#127	C₂×C₆₇	semidirect
113. C₂×#121	C₂×C₆₇	semidirect
114. C₂×C₄×D₄	C₂×C₆₇	semidirect
115. #392	C₂×C₆₇	semidirect
116. #390	C₂×C₆₇	semidirect
117. C₄×C₄⋊C₄	C₂×C₆₇	semidirect
149. C₂×D₄	D₄×C₆₇	semidirect
158. C₄⋊C₄	D₄×C₆₇	semidirect
162. C₄²	D₄×C₆₇	semidirect
210. C₄	#121×C₆₇	semidirect
211. C₄	#127×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#127×C₆₇	2
C₂×#121×C₆₇	2
C₂×C₄×D₄×C₆₇	2
#392×C₆₇	2
#390×C₆₇	2
C₄×C₄⋊C₄×C₆₇	2
#2541	67

Subgroups — C67×#2541 #22556972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2541 #22556972