Subgroups — C₆₇×#33257 #30712508

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2864	C₂	1	2	yes	maximal
2	C₆₇×#1942	C₂	1	4	yes	maximal
3	C₆₇×#1941	C₂	1	4	yes	maximal
4	C₆₇×#1846	C₂	1	2	yes	maximal
5	C₆₇×#1835	C₂	1	2	yes	maximal
6	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₆₇	C₂²	1	4	yes	—
8	C₂×#106×C₆₇	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#389×C₆₇	C₂²	1	8	yes	—
14	#387×C₆₇	—	2	8	no	—
15	#382×C₆₇	C₂²	1	4	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	#340×C₆₇	C₂²	1	8	yes	—
18	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
19	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
22	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
23	C₂×C₄²×C₆₇	D₄	1	4	yes	—
24	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
25	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	#107×C₆₇	—	2	8	no	—
27	#106×C₆₇	—	2	8	no	—
28	C₄×C₈×C₆₇	—	2	4	no	—
29	C₄×C₈×C₆₇	—	2	4	no	—
30	C₄×C₈×C₆₇	Q₈	1	4	yes	—
31	C₄×C₈×C₆₇	Q₈	1	4	yes	—
32	#95×C₆₇	—	2	8	no	—
33	#95×C₆₇	—	2	8	no	—
34	C₂²×C₄×C₆₇	—	2	8	no	—
35	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
37	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
38	C₂×C₈×C₆₇	—	2	2	no	—
39	C₂×C₈×C₆₇	—	2	2	no	—
40	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
41	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
42	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
43	C₂×C₈×C₆₇	#41	1	4	yes	—
44	C₂×C₈×C₆₇	#41	1	4	yes	—
45	C₄⋊C₄×C₆₇	—	2	8	no	—
46	C₄⋊C₄×C₆₇	—	2	8	no	—
47	C₄⋊C₄×C₆₇	—	4	16	no	—
48	C₄²×C₆₇	—	2	8	no	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	2	2	no	—
51	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
52	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
53	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	—	2	4	no	—
55	C₂×C₄×C₆₇	—	2	4	no	—
56	C₂×C₄×C₆₇	—	2	4	no	—
57	C₂×C₄×C₆₇	—	4	8	no	—
58	C₂×C₄×C₆₇	—	4	16	no	—
59	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
60	C₂×C₄×C₆₇	#135	1	8	yes	—
61	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
65	C₈×C₆₇	—	2	4	no	—
66	C₈×C₆₇	#125	1	4	yes	—
67	C₈×C₆₇	#125	1	4	yes	—
68	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂²×C₆₇	#532	1	2	yes	—
70	C₂²×C₆₇	C₂×#135	1	4	yes	—
71	C₄×C₆₇	—	2	4	no	—
72	C₄×C₆₇	—	2	8	no	—
73	C₄×C₆₇	—	8	8	no	—
74	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
75	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
76	C₄×C₆₇	#533	1	2	yes	—
77	#33257	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₆₇	#2859	1	1	yes	characteristic
79	C₂×C₆₇	#3160	1	2	yes	—
80	C₂×C₆₇	#3168	1	4	yes	—
81	#2864	C₂×C₆₇	1	2	yes	—
82	#1942	C₂×C₆₇	1	4	yes	—
83	#1941	C₂×C₆₇	1	4	yes	—
84	#1846	C₂×C₆₇	1	2	yes	—
85	#1835	C₂×C₆₇	1	2	yes	—
86	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic
87	C₆₇	#33257	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×#107	C₂²×C₆₇	1	4	yes	—
89	C₂×#106	C₂²×C₆₇	1	4	yes	—
90	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
91	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
94	#389	C₂²×C₆₇	1	8	yes	—
95	#387	—	2	8	no	—
96	#382	C₂²×C₆₇	1	4	yes	—
97	C₄³	C₂²×C₆₇	1	1	yes	characteristic
98	#340	C₂²×C₆₇	1	8	yes	—
99	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
100	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
101	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	2	8	no	—
103	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
104	C₂×C₄²	D₄×C₆₇	1	4	yes	—
105	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
106	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
107	#107	—	2	8	no	—
108	#106	—	2	8	no	—
109	C₄×C₈	—	2	4	no	—
110	C₄×C₈	—	2	4	no	—
111	C₄×C₈	Q₈×C₆₇	1	4	yes	—
112	C₄×C₈	Q₈×C₆₇	1	4	yes	—
113	#95	—	2	8	no	—
114	#95	—	2	8	no	—
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
117	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
118	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
122	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
123	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
124	C₂×C₈	#41×C₆₇	1	4	yes	—
125	C₂×C₈	#41×C₆₇	1	4	yes	—
126	C₄⋊C₄	—	2	8	no	—
127	C₄⋊C₄	—	2	8	no	—
128	C₄⋊C₄	—	4	16	no	—
129	C₄²	—	2	8	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
133	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
134	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	4	8	no	—
139	C₂×C₄	—	4	16	no	—
140	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
141	C₂×C₄	#135×C₆₇	1	8	yes	—
142	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
143	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
144	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
145	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
146	C₈	—	2	4	no	—
147	C₈	#125×C₆₇	1	4	yes	—
148	C₈	#125×C₆₇	1	4	yes	—
149	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
150	C₂²	#532×C₆₇	1	2	yes	—
151	C₂²	C₂×#135×C₆₇	1	4	yes	—
152	C₄	—	2	4	no	—
153	C₄	—	2	8	no	—
154	C₄	—	8	8	no	—
155	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
156	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
157	C₄	#533×C₆₇	1	2	yes	—
158	C₂	C₆₇×#2859	1	1	yes	characteristic
159	C₂	C₆₇×#3160	1	2	yes	—
160	C₂	C₆₇×#3168	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
77. #33257	C₆₇	direct (N × H)
87. C₆₇	#33257	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×#2864	2
C₆₇×#1942	2
C₆₇×#1941	2
C₆₇×#1846	2
C₆₇×#1835	2
C₄²×C₈×C₆₇	2
#33257	67

Subgroups — C67×#33257 #30712508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33257 #30712508