Subgroups — C₂²×#493×C₆₇ #30739880

Order 17152.

4638 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₆₇	C₂	1	4	yes	maximal, complemented
2	C₂²×#127×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#493×C₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄×C₆₇	—	2	4	no	—
6	C₂³×D₄×C₆₇	C₂²	1	4	yes	—
7	C₂²×D₈×C₆₇	—	2	32	no	—
8	C₂²×D₈×C₆₇	C₂²	1	48	yes	complemented
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#127×C₆₇	—	2	12	no	—
11	C₂×#127×C₆₇	C₂²	1	24	yes	complemented
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	#493×C₆₇	C₂²	1	64	yes	direct product factor
14	C₂×C₄×C₈×C₆₇	C₂²	1	12	yes	complemented
15	C₂⁵×C₆₇	—	2	4	no	—
16	C₂²×D₄×C₆₇	—	2	32	no	—
17	C₂²×D₄×C₆₇	—	2	12	no	—
18	C₂²×D₄×C₆₇	—	2	48	no	—
19	C₂²×D₄×C₆₇	—	4	32	no	—
20	C₂²×D₄×C₆₇	C₂³	1	24	yes	—
21	C₂³×C₄×C₆₇	D₄	1	2	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₂×D₈×C₆₇	—	2	192	no	—
24	C₂×D₈×C₆₇	C₂³	1	64	yes	complemented
25	C₂²×C₈×C₆₇	D₄	1	16	yes	complemented
26	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
27	#127×C₆₇	—	2	48	no	—
28	#127×C₆₇	C₂³	1	32	yes	complemented
29	C₂×C₄²×C₆₇	D₄	1	6	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
31	C₄×C₈×C₆₇	C₂³	1	16	yes	complemented
32	C₂⁴×C₆₇	—	2	24	no	—
33	C₂⁴×C₆₇	—	4	16	no	—
34	C₂⁴×C₆₇	—	4	32	no	—
35	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×D₄×C₆₇	—	2	96	no	—
37	C₂×D₄×C₆₇	—	2	24	no	—
38	C₂×D₄×C₆₇	—	2	64	no	—
39	C₂×D₄×C₆₇	—	4	48	no	—
40	C₂×D₄×C₆₇	—	4	192	no	—
41	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
42	C₂²×C₄×C₆₇	D₈	1	16	yes	complemented
43	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
44	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
46	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
47	D₈×C₆₇	—	2	128	no	—
48	C₂×C₈×C₆₇	C₂×D₄	1	48	yes	complemented
49	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
50	C₄²×C₆₇	C₂×D₄	1	12	yes	—
51	C₄²×C₆₇	C₂⁴	1	4	yes	—
52	C₂³×C₆₇	—	2	16	no	—
53	C₂³×C₆₇	—	4	48	no	—
54	C₂³×C₆₇	—	4	96	no	—
55	C₂³×C₆₇	—	8	64	no	—
56	C₂³×C₆₇	#127	1	4	yes	—
57	C₂³×C₆₇	C₂×D₈	1	8	yes	—
58	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
59	D₄×C₆₇	—	2	32	no	—
60	D₄×C₆₇	—	4	48	no	—
61	D₄×C₆₇	—	4	128	no	—
62	C₂×C₄×C₆₇	#127	1	12	yes	—
63	C₂×C₄×C₆₇	C₂×D₈	1	48	yes	complemented
64	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
65	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
66	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
67	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
68	C₈×C₆₇	C₂²×D₄	1	16	yes	complemented
69	C₂²×C₆₇	—	4	16	no	—
70	C₂²×C₆₇	—	4	32	no	—
71	C₂²×C₆₇	—	8	96	no	—
72	C₂²×C₆₇	#493	1	16	yes	direct product factor
73	C₂²×C₆₇	C₂×#127	1	6	yes	—
74	C₂²×C₆₇	C₂²×D₈	1	12	yes	—
75	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
76	C₄×C₆₇	C₂×#127	1	6	yes	—
77	C₄×C₆₇	C₂²×D₈	1	16	yes	complemented
78	C₄×C₆₇	C₂³×D₄	1	2	yes	—
79	C₂²×#493	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₆₇	—	8	16	no	—
81	C₂×C₆₇	C₂×#493	1	12	yes	direct product factor
82	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
83	C₂×C₆₇	C₂³×D₈	1	2	yes	—
84	C₂³×D₈	C₂×C₆₇	1	4	yes	complemented
85	C₂²×#127	C₂×C₆₇	1	2	yes	complemented
86	C₂×#493	C₂×C₆₇	1	24	yes	direct product factor
87	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
88	C₆₇	C₂²×#493	1	1	yes	characteristic, direct product factor, Sylow
89	C₂³×D₄	—	2	4	no	—
90	C₂³×D₄	C₂²×C₆₇	1	4	yes	—
91	C₂²×D₈	—	2	32	no	—
92	C₂²×D₈	C₂²×C₆₇	1	48	yes	complemented
93	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
94	C₂×#127	—	2	12	no	—
95	C₂×#127	C₂²×C₆₇	1	24	yes	complemented
96	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
97	#493	C₂²×C₆₇	1	64	yes	direct product factor
98	C₂×C₄×C₈	C₂²×C₆₇	1	12	yes	complemented
99	C₂⁵	—	2	4	no	—
100	C₂²×D₄	—	2	32	no	—
101	C₂²×D₄	—	2	12	no	—
102	C₂²×D₄	—	2	48	no	—
103	C₂²×D₄	—	4	32	no	—
104	C₂²×D₄	C₂³×C₆₇	1	24	yes	—
105	C₂³×C₄	D₄×C₆₇	1	2	yes	—
106	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
107	C₂×D₈	—	2	192	no	—
108	C₂×D₈	C₂³×C₆₇	1	64	yes	complemented
109	C₂²×C₈	D₄×C₆₇	1	16	yes	complemented
110	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
111	#127	—	2	48	no	—
112	#127	C₂³×C₆₇	1	32	yes	complemented
113	C₂×C₄²	D₄×C₆₇	1	6	yes	—
114	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
115	C₄×C₈	C₂³×C₆₇	1	16	yes	complemented
116	C₂⁴	—	2	24	no	—
117	C₂⁴	—	4	16	no	—
118	C₂⁴	—	4	32	no	—
119	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
120	C₂×D₄	—	2	96	no	—
121	C₂×D₄	—	2	24	no	—
122	C₂×D₄	—	2	64	no	—
123	C₂×D₄	—	4	48	no	—
124	C₂×D₄	—	4	192	no	—
125	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
126	C₂²×C₄	D₈×C₆₇	1	16	yes	complemented
127	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
128	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
129	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
130	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
131	D₈	—	2	128	no	—
132	C₂×C₈	C₂×D₄×C₆₇	1	48	yes	complemented
133	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
134	C₄²	C₂×D₄×C₆₇	1	12	yes	—
135	C₄²	C₂⁴×C₆₇	1	4	yes	—
136	C₂³	—	2	16	no	—
137	C₂³	—	4	48	no	—
138	C₂³	—	4	96	no	—
139	C₂³	—	8	64	no	—
140	C₂³	#127×C₆₇	1	4	yes	—
141	C₂³	C₂×D₈×C₆₇	1	8	yes	—
142	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
143	D₄	—	2	32	no	—
144	D₄	—	4	48	no	—
145	D₄	—	4	128	no	—
146	C₂×C₄	#127×C₆₇	1	12	yes	—
147	C₂×C₄	C₂×D₈×C₆₇	1	48	yes	complemented
148	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
149	C₂×C₄	C₂²×D₄×C₆₇	1	12	yes	—
150	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
151	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
152	C₈	C₂²×D₄×C₆₇	1	16	yes	complemented
153	C₂²	—	4	32	no	—
154	C₂²	—	4	16	no	—
155	C₂²	—	8	96	no	—
156	C₂²	#493×C₆₇	1	16	yes	direct product factor
157	C₂²	C₂×#127×C₆₇	1	6	yes	—
158	C₂²	C₂²×D₈×C₆₇	1	12	yes	—
159	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
160	C₄	C₂×#127×C₆₇	1	6	yes	—
161	C₄	C₂²×D₈×C₆₇	1	16	yes	complemented
162	C₄	C₂³×D₄×C₆₇	1	2	yes	—
163	C₂	—	8	16	no	—
164	C₂	C₂×#493×C₆₇	1	12	yes	direct product factor
165	C₂	C₂²×#127×C₆₇	1	1	yes	characteristic
166	C₂	C₂³×D₈×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×D₈×C₆₇	C₂	semidirect
2. C₂²×#127×C₆₇	C₂	semidirect
3. C₂×#493×C₆₇	C₂	direct (N × H)
4. C₂²×C₄×C₈×C₆₇	C₂	semidirect
8. C₂²×D₈×C₆₇	C₂²	semidirect
11. C₂×#127×C₆₇	C₂²	semidirect
13. #493×C₆₇	C₂²	direct (N × H)
14. C₂×C₄×C₈×C₆₇	C₂²	semidirect
24. C₂×D₈×C₆₇	C₂³	semidirect
25. C₂²×C₈×C₆₇	D₄	semidirect
28. #127×C₆₇	C₂³	semidirect
31. C₄×C₈×C₆₇	C₂³	semidirect
42. C₂²×C₄×C₆₇	D₈	semidirect
48. C₂×C₈×C₆₇	C₂×D₄	semidirect
63. C₂×C₄×C₆₇	C₂×D₈	semidirect
68. C₈×C₆₇	C₂²×D₄	semidirect
72. C₂²×C₆₇	#493	direct (N × H)
77. C₄×C₆₇	C₂²×D₈	semidirect
79. C₂²×#493	C₆₇	direct (N × H)
81. C₂×C₆₇	C₂×#493	direct (N × H)
84. C₂³×D₈	C₂×C₆₇	semidirect
85. C₂²×#127	C₂×C₆₇	semidirect
86. C₂×#493	C₂×C₆₇	direct (N × H)
87. C₂²×C₄×C₈	C₂×C₆₇	semidirect
88. C₆₇	C₂²×#493	direct (N × H)
92. C₂²×D₈	C₂²×C₆₇	semidirect
95. C₂×#127	C₂²×C₆₇	semidirect
97. #493	C₂²×C₆₇	direct (N × H)
98. C₂×C₄×C₈	C₂²×C₆₇	semidirect
108. C₂×D₈	C₂³×C₆₇	semidirect
109. C₂²×C₈	D₄×C₆₇	semidirect
112. #127	C₂³×C₆₇	semidirect
115. C₄×C₈	C₂³×C₆₇	semidirect
126. C₂²×C₄	D₈×C₆₇	semidirect
132. C₂×C₈	C₂×D₄×C₆₇	semidirect
147. C₂×C₄	C₂×D₈×C₆₇	semidirect
152. C₈	C₂²×D₄×C₆₇	semidirect
156. C₂²	#493×C₆₇	direct (N × H)
161. C₄	C₂²×D₈×C₆₇	semidirect
164. C₂	C₂×#493×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₆₇	2
C₂²×#127×C₆₇	2
C₂×#493×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂²×#493	67

Subgroups — C22×#493×C67 #30739880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#493×C₆₇ #30739880