Subgroups — C₂×C₆₇×#2656 #30713822

Order 17152.

4886 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₆₇	C₂	1	6	yes	maximal, complemented
2	C₆₇×#2656	C₂	1	16	yes	maximal, direct product factor
3	C₂×#399×C₆₇	C₂	1	4	yes	maximal, complemented
4	C₂×#392×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂²×#95×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×C₆₇	C₂²	1	4	yes	—
7	C₂³×D₄×C₆₇	C₂²	1	6	yes	—
8	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
9	C₂×#121×C₆₇	—	2	48	no	—
10	C₂×#121×C₆₇	C₂²	1	48	yes	complemented
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
12	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	12	yes	—
13	#399×C₆₇	C₂²	1	32	yes	complemented
14	#392×C₆₇	C₂²	1	32	yes	complemented
15	C₂×#95×C₆₇	C₂²	1	4	yes	—
16	C₂×#95×C₆₇	C₂²	1	8	yes	complemented
17	C₂⁵×C₆₇	C₂³	1	4	yes	—
18	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
19	C₂²×D₄×C₆₇	—	2	48	no	—
20	C₂²×D₄×C₆₇	—	2	24	no	—
21	C₂²×D₄×C₆₇	—	4	24	no	—
22	C₂²×D₄×C₆₇	C₂³	1	16	yes	—
23	C₂²×D₄×C₆₇	C₂³	1	24	yes	—
24	C₂³×C₄×C₆₇	—	4	12	no	—
25	C₂³×C₄×C₆₇	D₄	1	24	yes	complemented
26	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
29	#121×C₆₇	—	2	192	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	24	no	—
31	C₂×C₄⋊C₄×C₆₇	C₂³	1	24	yes	—
32	C₂×C₂²⋊C₄×C₆₇	—	2	48	no	—
33	C₂×C₂²⋊C₄×C₆₇	C₂³	1	48	yes	—
34	#95×C₆₇	C₂³	1	16	yes	—
35	C₂⁴×C₆₇	—	2	6	no	—
36	C₂⁴×C₆₇	—	2	2	no	—
37	C₂⁴×C₆₇	—	2	24	no	—
38	C₂⁴×C₆₇	—	2	24	no	—
39	C₂⁴×C₆₇	—	2	8	no	—
40	C₂⁴×C₆₇	C₂×D₄	1	12	yes	—
41	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
42	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
43	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
44	C₂×D₄×C₆₇	—	2	96	no	—
45	C₂×D₄×C₆₇	—	2	48	no	—
46	C₂×D₄×C₆₇	—	4	48	no	—
47	C₂×D₄×C₆₇	—	4	96	no	—
48	C₂×D₄×C₆₇	—	4	64	no	—
49	C₂²×C₄×C₆₇	—	2	24	no	—
50	C₂²×C₄×C₆₇	—	2	24	no	—
51	C₂²×C₄×C₆₇	—	4	48	no	—
52	C₂²×C₄×C₆₇	—	4	24	no	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	48	yes	complemented
55	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
56	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
57	C₄⋊C₄×C₆₇	—	2	48	no	—
58	C₂²⋊C₄×C₆₇	—	2	96	no	—
59	C₂³×C₆₇	—	2	24	no	—
60	C₂³×C₆₇	—	2	24	no	—
61	C₂³×C₆₇	—	2	6	no	—
62	C₂³×C₆₇	—	2	2	no	—
63	C₂³×C₆₇	—	2	24	no	—
64	C₂³×C₆₇	—	2	24	no	—
65	C₂³×C₆₇	—	2	8	no	—
66	C₂³×C₆₇	—	4	16	no	—
67	C₂³×C₆₇	—	4	48	no	—
68	C₂³×C₆₇	—	4	48	no	—
69	C₂³×C₆₇	#127	1	16	yes	—
70	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
71	C₂³×C₆₇	C₂²×D₄	1	6	yes	—
72	C₂³×C₆₇	H(5,𝔽₂)	1	6	yes	—
73	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
74	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
75	D₄×C₆₇	—	4	96	no	—
76	D₄×C₆₇	—	4	64	no	—
77	C₂×C₄×C₆₇	—	2	48	no	—
78	C₂×C₄×C₆₇	—	2	24	no	—
79	C₂×C₄×C₆₇	—	2	24	no	—
80	C₂×C₄×C₆₇	—	4	48	no	—
81	C₂×C₄×C₆₇	—	4	16	no	—
82	C₂×C₄×C₆₇	C₂²×D₄	1	24	yes	—
83	C₂²×C₆₇	—	2	32	no	—
84	C₂²×C₆₇	—	2	12	no	—
85	C₂²×C₆₇	—	2	12	no	—
86	C₂²×C₆₇	—	4	8	no	—
87	C₂²×C₆₇	—	4	24	no	—
88	C₂²×C₆₇	—	4	24	no	—
89	C₂²×C₆₇	—	8	32	no	—
90	C₂²×C₆₇	C₂×#127	1	8	yes	—
91	C₂²×C₆₇	C₂×#127	1	4	yes	—
92	C₂²×C₆₇	#534	1	12	yes	—
93	C₂²×C₆₇	#537	1	12	yes	—
94	C₂²×C₆₇	C₂³×D₄	1	3	yes	—
95	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	3	yes	—
96	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
97	C₄×C₆₇	—	2	24	no	—
98	C₄×C₆₇	—	4	8	no	—
99	C₂×#2656	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₆₇	—	2	8	no	—
101	C₂×C₆₇	—	8	8	no	—
102	C₂×C₆₇	#2656	1	8	yes	direct product factor
103	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
104	C₂×C₆₇	C₂×#534	1	3	yes	—
105	C₂×C₆₇	C₂×#537	1	3	yes	—
106	C₂²×#121	C₂×C₆₇	1	6	yes	complemented
107	#2656	C₂×C₆₇	1	16	yes	direct product factor
108	C₂×#399	C₂×C₆₇	1	4	yes	complemented
109	C₂×#392	C₂×C₆₇	1	4	yes	complemented
110	C₂²×#95	C₂×C₆₇	1	1	yes	characteristic, complemented
111	C₆₇	C₂×#2656	1	1	yes	characteristic, direct product factor, Sylow
112	C₂³×D₄	C₂²×C₆₇	1	4	yes	—
113	C₂³×D₄	C₂²×C₆₇	1	6	yes	—
114	C₂⁴×C₄	C₂²×C₆₇	1	3	yes	—
115	C₂×#121	—	2	48	no	—
116	C₂×#121	C₂²×C₆₇	1	48	yes	complemented
117	C₂²×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
118	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	12	yes	—
119	#399	C₂²×C₆₇	1	32	yes	complemented
120	#392	C₂²×C₆₇	1	32	yes	complemented
121	C₂×#95	C₂²×C₆₇	1	4	yes	—
122	C₂×#95	C₂²×C₆₇	1	8	yes	complemented
123	C₂⁵	C₂³×C₆₇	1	4	yes	—
124	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
125	C₂²×D₄	—	2	48	no	—
126	C₂²×D₄	—	2	24	no	—
127	C₂²×D₄	—	4	24	no	—
128	C₂²×D₄	C₂³×C₆₇	1	16	yes	—
129	C₂²×D₄	C₂³×C₆₇	1	24	yes	—
130	C₂³×C₄	—	4	12	no	—
131	C₂³×C₄	D₄×C₆₇	1	24	yes	complemented
132	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
133	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
134	C₂³×C₄	C₂³×C₆₇	1	12	yes	—
135	#121	—	2	192	no	—
136	C₂×C₄⋊C₄	—	2	24	no	—
137	C₂×C₄⋊C₄	C₂³×C₆₇	1	24	yes	—
138	C₂×C₂²⋊C₄	—	2	48	no	—
139	C₂×C₂²⋊C₄	C₂³×C₆₇	1	48	yes	—
140	#95	C₂³×C₆₇	1	16	yes	—
141	C₂⁴	—	2	6	no	—
142	C₂⁴	—	2	2	no	—
143	C₂⁴	—	2	24	no	—
144	C₂⁴	—	2	24	no	—
145	C₂⁴	—	2	8	no	—
146	C₂⁴	C₂×D₄×C₆₇	1	12	yes	—
147	C₂⁴	C₂⁴×C₆₇	1	8	yes	—
148	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
149	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
150	C₂×D₄	—	2	96	no	—
151	C₂×D₄	—	2	48	no	—
152	C₂×D₄	—	4	48	no	—
153	C₂×D₄	—	4	96	no	—
154	C₂×D₄	—	4	64	no	—
155	C₂²×C₄	—	2	24	no	—
156	C₂²×C₄	—	2	24	no	—
157	C₂²×C₄	—	4	48	no	—
158	C₂²×C₄	—	4	24	no	—
159	C₂²×C₄	C₂×D₄×C₆₇	1	24	yes	—
160	C₂²×C₄	C₂×D₄×C₆₇	1	48	yes	complemented
161	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
162	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
163	C₄⋊C₄	—	2	48	no	—
164	C₂²⋊C₄	—	2	96	no	—
165	C₂³	—	2	24	no	—
166	C₂³	—	2	24	no	—
167	C₂³	—	2	6	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	24	no	—
170	C₂³	—	2	24	no	—
171	C₂³	—	2	8	no	—
172	C₂³	—	4	16	no	—
173	C₂³	—	4	48	no	—
174	C₂³	—	4	48	no	—
175	C₂³	#127×C₆₇	1	16	yes	—
176	C₂³	C₂²×D₄×C₆₇	1	12	yes	—
177	C₂³	C₂²×D₄×C₆₇	1	6	yes	—
178	C₂³	H(5,𝔽₂)×C₆₇	1	6	yes	—
179	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
180	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
181	D₄	—	4	96	no	—
182	D₄	—	4	64	no	—
183	C₂×C₄	—	2	48	no	—
184	C₂×C₄	—	2	24	no	—
185	C₂×C₄	—	2	24	no	—
186	C₂×C₄	—	4	48	no	—
187	C₂×C₄	—	4	16	no	—
188	C₂×C₄	C₂²×D₄×C₆₇	1	24	yes	—
189	C₂²	—	2	32	no	—
190	C₂²	—	2	12	no	—
191	C₂²	—	2	12	no	—
192	C₂²	—	4	8	no	—
193	C₂²	—	4	24	no	—
194	C₂²	—	4	24	no	—
195	C₂²	—	8	32	no	—
196	C₂²	C₂×#127×C₆₇	1	8	yes	—
197	C₂²	C₂×#127×C₆₇	1	4	yes	—
198	C₂²	#534×C₆₇	1	12	yes	—
199	C₂²	#537×C₆₇	1	12	yes	—
200	C₂²	C₂³×D₄×C₆₇	1	3	yes	—
201	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	3	yes	—
202	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
203	C₄	—	2	24	no	—
204	C₄	—	4	8	no	—
205	C₂	—	2	8	no	—
206	C₂	—	8	8	no	—
207	C₂	C₆₇×#2656	1	8	yes	direct product factor
208	C₂	C₂²×#127×C₆₇	1	1	yes	characteristic
209	C₂	C₂×#534×C₆₇	1	3	yes	—
210	C₂	C₂×#537×C₆₇	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#121×C₆₇	C₂	semidirect
2. C₆₇×#2656	C₂	direct (N × H)
3. C₂×#399×C₆₇	C₂	semidirect
4. C₂×#392×C₆₇	C₂	semidirect
5. C₂²×#95×C₆₇	C₂	semidirect
10. C₂×#121×C₆₇	C₂²	semidirect
13. #399×C₆₇	C₂²	semidirect
14. #392×C₆₇	C₂²	semidirect
16. C₂×#95×C₆₇	C₂²	semidirect
25. C₂³×C₄×C₆₇	D₄	semidirect
54. C₂²×C₄×C₆₇	C₂×D₄	semidirect
99. C₂×#2656	C₆₇	direct (N × H)
102. C₂×C₆₇	#2656	direct (N × H)
106. C₂²×#121	C₂×C₆₇	semidirect
107. #2656	C₂×C₆₇	direct (N × H)
108. C₂×#399	C₂×C₆₇	semidirect
109. C₂×#392	C₂×C₆₇	semidirect
110. C₂²×#95	C₂×C₆₇	semidirect
111. C₆₇	C₂×#2656	direct (N × H)
116. C₂×#121	C₂²×C₆₇	semidirect
119. #399	C₂²×C₆₇	semidirect
120. #392	C₂²×C₆₇	semidirect
122. C₂×#95	C₂²×C₆₇	semidirect
131. C₂³×C₄	D₄×C₆₇	semidirect
160. C₂²×C₄	C₂×D₄×C₆₇	semidirect
207. C₂	C₆₇×#2656	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#121×C₆₇	2
C₆₇×#2656	2
C₂×#399×C₆₇	2
C₂×#392×C₆₇	2
C₂²×#95×C₆₇	2
C₂×#2656	67

Subgroups — C2×C67×#2656 #30713822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2656 #30713822