Subgroups — C₆₇×#7141 #30686392

Order 17152.

254 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₆₇	C₄	1	2	yes	—
5	C₂²×C₁₆×C₆₇	C₄	1	2	yes	—
6	#431×C₆₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	#346×C₆₇	—	2	1	no	—
9	C₄×C₁₆×C₆₇	—	2	1	no	—
10	C₄×C₁₆×C₆₇	—	2	1	no	—
11	C₄×C₁₆×C₆₇	—	2	1	no	—
12	#322×C₆₇	—	2	1	no	—
13	C₈²×C₆₇	—	2	1	no	—
14	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
17	M₃₂×C₆₇	—	2	2	no	—
18	M₃₂×C₆₇	C₈	1	4	yes	complemented
19	C₂×C₁₆×C₆₇	—	2	2	no	—
20	C₂×C₁₆×C₆₇	—	2	2	no	—
21	C₂×C₁₆×C₆₇	—	2	2	no	—
22	C₂×C₁₆×C₆₇	C₈	1	4	yes	complemented
23	C₄⋊C₈×C₆₇	C₈	1	4	yes	—
24	C₂²⋊C₈×C₆₇	—	2	2	no	—
25	C₄×C₈×C₆₇	—	2	2	no	—
26	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
32	C₂×C₈×C₆₇	—	2	4	no	—
33	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
40	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
41	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₆₇	C₂×C₈	1	2	yes	—
44	C₁₆×C₆₇	—	2	4	no	—
45	C₁₆×C₆₇	—	2	4	no	—
46	C₂³×C₆₇	#97	1	1	yes	characteristic
47	C₂×C₄×C₆₇	—	2	4	no	—
48	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
56	C₈×C₆₇	—	4	4	no	—
57	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
58	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
59	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
60	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
61	C₂²×C₆₇	—	2	2	no	—
62	C₂²×C₆₇	#322	1	1	yes	characteristic
63	C₂²×C₆₇	#322	1	1	yes	characteristic
64	C₂²×C₆₇	#336	1	1	yes	characteristic
65	C₄×C₆₇	—	2	2	no	—
66	C₄×C₆₇	C₈²	1	1	yes	characteristic
67	C₄×C₆₇	#322	1	1	yes	characteristic
68	C₄×C₆₇	#336	1	1	yes	characteristic
69	C₄×C₆₇	#336	1	1	yes	characteristic
70	C₄×C₆₇	#350	1	4	yes	—
71	#7141	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₆₇	—	2	1	no	—
73	C₂×C₆₇	—	2	1	no	—
74	C₂×C₆₇	#1273	1	1	yes	characteristic
75	C₂×C₆₇	#1379	1	2	yes	—
76	C₄×M₃₂	C₂×C₆₇	1	1	yes	characteristic
77	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic
78	#1450	C₂×C₆₇	1	1	yes	characteristic
79	C₆₇	#7141	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×M₃₂	C₄×C₆₇	1	2	yes	—
81	C₂²×C₁₆	C₄×C₆₇	1	2	yes	—
82	#431	C₄×C₆₇	1	2	yes	—
83	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
84	#346	—	2	1	no	—
85	C₄×C₁₆	—	2	1	no	—
86	C₄×C₁₆	—	2	1	no	—
87	C₄×C₁₆	—	2	1	no	—
88	#322	—	2	1	no	—
89	C₈²	—	2	1	no	—
90	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
91	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
93	M₃₂	—	2	2	no	—
94	M₃₂	C₈×C₆₇	1	4	yes	complemented
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	C₈×C₆₇	1	4	yes	complemented
99	C₄⋊C₈	C₈×C₆₇	1	4	yes	—
100	C₂²⋊C₈	—	2	2	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
103	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
104	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
105	C₄×C₈	Q₈×C₆₇	1	1	yes	characteristic
106	C₂²×C₄	C₄²×C₆₇	1	1	yes	characteristic
107	C₂²×C₄	M₁₆×C₆₇	1	2	yes	—
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
111	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
112	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
113	C₂×C₈	C₂×C₈×C₆₇	1	2	yes	—
114	C₂×C₈	C₂×C₈×C₆₇	1	2	yes	—
115	C₂×C₈	M₁₆×C₆₇	1	2	yes	—
116	C₂×C₈	M₁₆×C₆₇	1	2	yes	—
117	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
118	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
119	C₄²	C₂×C₈×C₆₇	1	2	yes	—
120	C₁₆	—	2	4	no	—
121	C₁₆	—	2	4	no	—
122	C₂³	#97×C₆₇	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	#95×C₆₇	1	1	yes	characteristic
125	C₂×C₄	C₄×C₈×C₆₇	1	1	yes	characteristic
126	C₂×C₄	C₄×C₈×C₆₇	1	1	yes	characteristic
127	C₂×C₄	C₄×C₈×C₆₇	1	1	yes	characteristic
128	C₂×C₄	#97×C₆₇	1	1	yes	characteristic
129	C₂×C₄	#97×C₆₇	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊C₈×C₆₇	1	2	yes	—
131	C₂×C₄	C₄⋊C₈×C₆₇	1	2	yes	—
132	C₈	—	4	4	no	—
133	C₈	C₂²⋊C₈×C₆₇	1	2	yes	—
134	C₈	C₂²⋊C₈×C₆₇	1	2	yes	—
135	C₈	C₄⋊C₈×C₆₇	1	2	yes	—
136	C₈	C₄⋊C₈×C₆₇	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	#322×C₆₇	1	1	yes	characteristic
139	C₂²	#322×C₆₇	1	1	yes	characteristic
140	C₂²	#336×C₆₇	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	C₈²×C₆₇	1	1	yes	characteristic
143	C₄	#322×C₆₇	1	1	yes	characteristic
144	C₄	#336×C₆₇	1	1	yes	characteristic
145	C₄	#336×C₆₇	1	1	yes	characteristic
146	C₄	#350×C₆₇	1	4	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	C₆₇×#1273	1	1	yes	characteristic
150	C₂	C₆₇×#1379	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₆₇	D₄
commutator	C₄	C₈²×C₆₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₇
socle	C₂²×C₆₇	#336
prime radical	C₂³×C₆₇	#97
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
18. M₃₂×C₆₇	C₈	semidirect
22. C₂×C₁₆×C₆₇	C₈	semidirect
71. #7141	C₆₇	direct (N × H)
79. C₆₇	#7141	direct (N × H)
94. M₃₂	C₈×C₆₇	semidirect
98. C₂×C₁₆	C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₆₇×#1450	2
#7141	67

Subgroups — C67×#7141 #30686392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#7141 #30686392