Subgroups — C₆₇×#33501 #30712752

Order 17152.

444 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2183	C₂	1	8	yes	maximal, complemented
3	C₆₇×#2164	C₂	1	4	yes	maximal
4	C₆₇×#2163	C₂	1	2	yes	maximal
5	#533×C₆₇	C₄	1	1	yes	characteristic
6	#446×C₆₇	C₄	1	4	yes	—
7	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
8	#424×C₆₇	C₄	1	2	yes	—
9	#423×C₆₇	C₄	1	1	yes	characteristic
10	#405×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
13	C₄⋊C₁₆×C₆₇	C₂²	1	8	yes	—
14	C₂²⋊C₁₆×C₆₇	C₂²	1	4	yes	—
15	#346×C₆₇	C₂²	1	4	yes	—
16	#346×C₆₇	C₂²	1	4	yes	—
17	C₄×C₁₆×C₆₇	C₂²	1	4	yes	—
18	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈×C₆₇	C₈	1	2	yes	—
22	#125×C₆₇	C₈	1	2	yes	—
23	C₂²⋊Q₈×C₆₇	C₈	1	4	yes	—
24	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
25	#117×C₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	C₂³	1	8	yes	—
28	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
29	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
30	#97×C₆₇	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₆₇	C₂³	1	4	yes	—
33	C₄×C₈×C₆₇	C₂³	1	2	yes	—
34	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	M₁₆×C₆₇	—	4	2	no	—
38	C₂×C₈×C₆₇	—	2	2	no	—
39	C₂×C₈×C₆₇	—	4	1	no	—
40	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
41	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
42	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
44	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
45	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
46	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
47	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
48	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
51	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
52	C₁₆×C₆₇	—	2	8	no	—
53	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
54	Q₈×C₆₇	—	4	4	no	—
55	C₂×C₄×C₆₇	—	2	1	no	—
56	C₂×C₄×C₆₇	—	2	1	no	—
57	C₂×C₄×C₆₇	—	4	2	no	—
58	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
61	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
62	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₆₇	—	2	1	no	—
64	C₈×C₆₇	—	2	2	no	—
65	C₈×C₆₇	—	2	4	no	—
66	C₈×C₆₇	2₋¹⁺⁴	1	2	yes	—
67	C₂²×C₆₇	—	2	2	no	—
68	C₂²×C₆₇	—	2	1	no	—
69	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
70	C₄×C₆₇	—	2	2	no	—
71	C₄×C₆₇	—	2	2	no	—
72	C₄×C₆₇	—	2	8	no	—
73	C₄×C₆₇	—	2	1	no	—
74	C₄×C₆₇	#519	1	1	yes	characteristic
75	C₄×C₆₇	#568	1	1	yes	characteristic
76	#33501	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₆₇	—	4	1	no	—
78	C₂×C₆₇	#2982	1	1	yes	characteristic
79	C₂×C₆₇	#3407	1	2	yes	—
80	#2960	C₂×C₆₇	1	1	yes	characteristic
81	#2183	C₂×C₆₇	1	8	yes	complemented
82	#2164	C₂×C₆₇	1	4	yes	—
83	#2163	C₂×C₆₇	1	2	yes	—
84	C₆₇	#33501	1	1	yes	characteristic, direct product factor, Sylow
85	#533	C₄×C₆₇	1	1	yes	characteristic
86	#446	C₄×C₆₇	1	4	yes	—
87	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
88	#424	C₄×C₆₇	1	2	yes	—
89	#423	C₄×C₆₇	1	1	yes	characteristic
90	#405	C₂²×C₆₇	1	2	yes	—
91	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
92	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	—
93	C₄⋊C₁₆	C₂²×C₆₇	1	8	yes	—
94	C₂²⋊C₁₆	C₂²×C₆₇	1	4	yes	—
95	#346	C₂²×C₆₇	1	4	yes	—
96	#346	C₂²×C₆₇	1	4	yes	—
97	C₄×C₁₆	C₂²×C₆₇	1	4	yes	—
98	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
99	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
100	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
101	C₄⋊Q₈	C₈×C₆₇	1	2	yes	—
102	#125	C₈×C₆₇	1	2	yes	—
103	C₂²⋊Q₈	C₈×C₆₇	1	4	yes	—
104	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
105	#117	C₂×C₄×C₆₇	1	2	yes	—
106	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
107	C₂×C₁₆	C₂³×C₆₇	1	8	yes	—
108	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
109	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
110	#97	C₂×C₄×C₆₇	1	4	yes	—
111	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
112	C₄×C₈	C₂³×C₆₇	1	4	yes	—
113	C₄×C₈	C₂³×C₆₇	1	2	yes	—
114	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	—
115	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
116	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
121	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
122	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
123	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
124	C₄⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
125	C₄⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
126	C₄⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
127	C₂²⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
128	C₄²	C₂×C₈×C₆₇	1	1	yes	characteristic
129	C₄²	C₂×C₈×C₆₇	1	1	yes	characteristic
130	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
131	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
132	C₁₆	—	2	8	no	—
133	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
134	Q₈	—	4	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
140	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
141	C₂×C₄	C₂²×C₈×C₆₇	1	8	yes	—
142	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	4	no	—
146	C₈	2₋¹⁺⁴×C₆₇	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	8	no	—
153	C₄	—	2	1	no	—
154	C₄	#519×C₆₇	1	1	yes	characteristic
155	C₄	#568×C₆₇	1	1	yes	characteristic
156	C₂	—	4	1	no	—
157	C₂	C₆₇×#2982	1	1	yes	characteristic
158	C₂	C₆₇×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂³×C₆₇	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2183	C₂	semidirect
76. #33501	C₆₇	direct (N × H)
81. #2183	C₂×C₆₇	semidirect
84. C₆₇	#33501	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2960	2
C₆₇×#2183	2
C₆₇×#2164	2
C₆₇×#2163	2
#33501	67

Subgroups — C67×#33501 #30712752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33501 #30712752