Subgroups — C₆₇×#33996 #30713247

Order 17152.

3630 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₆₇	C₂	1	3	yes	maximal
2	C₆₇×#2274	C₂	1	12	yes	maximal, complemented
3	C₆₇×#2273	C₂	1	4	yes	maximal
4	C₄×#117×C₆₇	C₂	1	12	yes	maximal, complemented
5	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
6	C₂×#117×C₆₇	C₄	1	48	yes	complemented
7	C₂×#117×C₆₇	C₂²	1	12	yes	—
8	C₂×#117×C₆₇	C₂²	1	12	yes	—
9	C₂²×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
10	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	6	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	6	yes	—
12	C₄×C₄⋊C₄×C₆₇	C₂²	1	48	yes	complemented
13	C₄×C₂²⋊C₄×C₆₇	C₂²	1	48	yes	—
14	#376×C₆₇	C₂²	1	8	yes	complemented
15	C₄³×C₆₇	C₂²	1	8	yes	complemented
16	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
17	C₂³×C₄×C₆₇	C₂×C₄	1	24	yes	—
18	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
19	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
20	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
21	#117×C₆₇	C₂×C₄	1	96	yes	complemented
22	#117×C₆₇	C₂×C₄	1	96	yes	complemented
23	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	48	yes	complemented
24	C₂×C₄⋊C₄×C₆₇	C₂³	1	24	yes	—
25	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	48	yes	—
26	C₂×C₂²⋊C₄×C₆₇	C₂³	1	24	yes	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	48	yes	complemented
28	C₂×C₄²×C₆₇	C₂³	1	32	yes	—
29	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
31	#95×C₆₇	C₂³	1	32	yes	—
32	C₂⁴×C₆₇	—	4	4	no	—
33	C₂⁴×C₆₇	C₂²×C₄	1	12	yes	—
34	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
35	C₂²×C₄×C₆₇	—	4	24	no	—
36	C₂²×C₄×C₆₇	C₄²	1	64	yes	complemented
37	C₂²×C₄×C₆₇	C₂²×C₄	1	48	yes	—
38	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
40	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
42	C₂²×C₄×C₆₇	C₂⁴	1	24	yes	—
43	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
44	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
45	C₄⋊C₄×C₆₇	C₂²×C₄	1	96	yes	—
46	C₂²⋊C₄×C₆₇	C₂²×C₄	1	96	yes	—
47	C₄²×C₆₇	—	4	32	no	—
48	C₄²×C₆₇	C₂²×C₄	1	48	yes	—
49	C₄²×C₆₇	C₂²×C₄	1	48	yes	—
50	C₂³×C₆₇	—	2	12	no	—
51	C₂³×C₆₇	—	4	24	no	—
52	C₂³×C₆₇	C₂×C₄²	1	16	yes	—
53	C₂³×C₆₇	C₂³×C₄	1	12	yes	—
54	C₂³×C₆₇	C₂³×C₄	1	6	yes	—
55	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	48	no	—
57	C₂×C₄×C₆₇	—	4	48	no	—
58	C₂×C₄×C₆₇	C₂×C₄²	1	48	yes	—
59	C₂×C₄×C₆₇	C₂×C₄²	1	48	yes	—
60	C₂×C₄×C₆₇	C₂³×C₄	1	48	yes	—
61	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
62	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
63	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
64	C₂²×C₆₇	—	2	12	no	—
65	C₂²×C₆₇	—	2	24	no	—
66	C₂²×C₆₇	—	4	16	no	—
67	C₂²×C₆₇	C₂²×C₄²	1	12	yes	—
68	C₂²×C₆₇	C₂⁴×C₄	1	3	yes	—
69	C₂²×C₆₇	#585	1	4	yes	—
70	C₄×C₆₇	—	2	48	no	—
71	C₄×C₆₇	C₂²×C₄²	1	12	yes	—
72	C₄×C₆₇	C₂²×C₄²	1	4	yes	—
73	#33996	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₆₇	—	2	12	no	—
75	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
76	C₂×C₆₇	#3428	1	6	yes	—
77	C₂²×#117	C₂×C₆₇	1	3	yes	—
78	#2274	C₂×C₆₇	1	12	yes	complemented
79	#2273	C₂×C₆₇	1	4	yes	—
80	C₄×#117	C₂×C₆₇	1	12	yes	complemented
81	C₆₇	#33996	1	1	yes	characteristic, direct product factor, Sylow
82	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
83	C₂×#117	C₄×C₆₇	1	48	yes	complemented
84	C₂×#117	C₂²×C₆₇	1	12	yes	—
85	C₂×#117	C₂²×C₆₇	1	12	yes	—
86	C₂²×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
87	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	6	yes	—
88	C₂²×C₄²	C₂²×C₆₇	1	6	yes	—
89	C₄×C₄⋊C₄	C₂²×C₆₇	1	48	yes	complemented
90	C₄×C₂²⋊C₄	C₂²×C₆₇	1	48	yes	—
91	#376	C₂²×C₆₇	1	8	yes	complemented
92	C₄³	C₂²×C₆₇	1	8	yes	complemented
93	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
94	C₂³×C₄	C₂×C₄×C₆₇	1	24	yes	—
95	C₂³×C₄	C₂³×C₆₇	1	12	yes	—
96	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
97	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
98	#117	C₂×C₄×C₆₇	1	96	yes	complemented
99	#117	C₂×C₄×C₆₇	1	96	yes	complemented
100	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	48	yes	complemented
101	C₂×C₄⋊C₄	C₂³×C₆₇	1	24	yes	—
102	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	48	yes	—
103	C₂×C₂²⋊C₄	C₂³×C₆₇	1	24	yes	—
104	C₂×C₄²	C₂×C₄×C₆₇	1	48	yes	complemented
105	C₂×C₄²	C₂³×C₆₇	1	32	yes	—
106	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
107	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
108	#95	C₂³×C₆₇	1	32	yes	—
109	C₂⁴	—	4	4	no	—
110	C₂⁴	C₂²×C₄×C₆₇	1	12	yes	—
111	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
112	C₂²×C₄	—	4	24	no	—
113	C₂²×C₄	C₄²×C₆₇	1	64	yes	complemented
114	C₂²×C₄	C₂²×C₄×C₆₇	1	48	yes	—
115	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
116	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
117	C₂²×C₄	C₂²×C₄×C₆₇	1	24	yes	—
118	C₂²×C₄	C₂²×C₄×C₆₇	1	24	yes	—
119	C₂²×C₄	C₂⁴×C₆₇	1	24	yes	—
120	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
121	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
122	C₄⋊C₄	C₂²×C₄×C₆₇	1	96	yes	—
123	C₂²⋊C₄	C₂²×C₄×C₆₇	1	96	yes	—
124	C₄²	—	4	32	no	—
125	C₄²	C₂²×C₄×C₆₇	1	48	yes	—
126	C₄²	C₂²×C₄×C₆₇	1	48	yes	—
127	C₂³	—	2	12	no	—
128	C₂³	—	4	24	no	—
129	C₂³	C₂×C₄²×C₆₇	1	16	yes	—
130	C₂³	C₂³×C₄×C₆₇	1	12	yes	—
131	C₂³	C₂³×C₄×C₆₇	1	6	yes	—
132	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
133	C₂×C₄	—	2	48	no	—
134	C₂×C₄	—	4	48	no	—
135	C₂×C₄	C₂×C₄²×C₆₇	1	48	yes	—
136	C₂×C₄	C₂×C₄²×C₆₇	1	48	yes	—
137	C₂×C₄	C₂³×C₄×C₆₇	1	48	yes	—
138	C₂×C₄	C₂³×C₄×C₆₇	1	6	yes	—
139	C₂×C₄	C₂³×C₄×C₆₇	1	6	yes	—
140	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
141	C₂²	—	2	12	no	—
142	C₂²	—	2	24	no	—
143	C₂²	—	4	16	no	—
144	C₂²	C₂²×C₄²×C₆₇	1	12	yes	—
145	C₂²	C₂⁴×C₄×C₆₇	1	3	yes	—
146	C₂²	#585×C₆₇	1	4	yes	—
147	C₄	—	2	48	no	—
148	C₄	C₂²×C₄²×C₆₇	1	4	yes	—
149	C₄	C₂²×C₄²×C₆₇	1	12	yes	—
150	C₂	—	2	12	no	—
151	C₂	C₂³×C₄²×C₆₇	1	1	yes	characteristic
152	C₂	C₆₇×#3428	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂	C₂³×C₄²×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂⁵×C₆₇	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2274	C₂	semidirect
4. C₄×#117×C₆₇	C₂	semidirect
6. C₂×#117×C₆₇	C₄	semidirect
12. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
14. #376×C₆₇	C₂²	semidirect
15. C₄³×C₆₇	C₂²	semidirect
21. #117×C₆₇	C₂×C₄	semidirect
22. #117×C₆₇	C₂×C₄	semidirect
23. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
27. C₂×C₄²×C₆₇	C₂×C₄	semidirect
36. C₂²×C₄×C₆₇	C₄²	semidirect
73. #33996	C₆₇	direct (N × H)
78. #2274	C₂×C₆₇	semidirect
80. C₄×#117	C₂×C₆₇	semidirect
81. C₆₇	#33996	direct (N × H)
83. C₂×#117	C₄×C₆₇	semidirect
89. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
91. #376	C₂²×C₆₇	semidirect
92. C₄³	C₂²×C₆₇	semidirect
98. #117	C₂×C₄×C₆₇	semidirect
99. #117	C₂×C₄×C₆₇	semidirect
100. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
104. C₂×C₄²	C₂×C₄×C₆₇	semidirect
113. C₂²×C₄	C₄²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#117×C₆₇	2
C₆₇×#2274	2
C₆₇×#2273	2
C₄×#117×C₆₇	2
#33996	67

Subgroups — C67×#33996 #30713247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33996 #30713247