Subgroups — C₆₇×#33256 #30712507

Order 17152.

718 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2859	C₂	1	2	yes	maximal
2	C₆₇×#1941	C₂	1	8	yes	maximal
3	C₆₇×#1844	C₂	1	2	yes	maximal
4	C₆₇×#1834	C₂	1	2	yes	maximal
5	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₆₇	—	2	2	no	—
7	C₂×#106×C₆₇	C₂²	1	8	yes	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	#389×C₆₇	—	2	4	no	—
12	#389×C₆₇	C₂²	1	8	yes	—
13	#384×C₆₇	C₂²	1	4	yes	—
14	#382×C₆₇	—	2	2	no	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	#340×C₆₇	C₂²	1	8	yes	—
17	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
18	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
19	#125×C₆₇	—	4	8	no	—
20	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
22	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
23	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
24	C₂×C₄²×C₆₇	D₄	1	2	yes	—
25	C₂×C₄²×C₆₇	D₄	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
28	#106×C₆₇	—	2	16	no	—
29	C₄×C₈×C₆₇	—	2	8	no	—
30	C₄×C₈×C₆₇	Q₈	1	4	yes	—
31	C₄×C₈×C₆₇	Q₈	1	4	yes	—
32	#95×C₆₇	—	2	8	no	—
33	C₂²×C₄×C₆₇	—	2	8	no	—
34	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
35	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
38	C₂×C₈×C₆₇	—	2	4	no	—
39	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
40	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
41	C₂×C₈×C₆₇	#41	1	8	yes	—
42	C₄⋊C₄×C₆₇	—	2	16	no	—
43	C₄⋊C₄×C₆₇	—	4	16	no	—
44	C₄⋊C₄×C₆₇	—	4	8	no	—
45	C₄⋊C₄×C₆₇	—	4	8	no	—
46	C₄²×C₆₇	—	2	4	no	—
47	C₄²×C₆₇	—	2	4	no	—
48	C₄²×C₆₇	SD₁₆	1	8	yes	—
49	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
50	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
51	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	2	4	no	—
53	C₂×C₄×C₆₇	—	2	4	no	—
54	C₂×C₄×C₆₇	—	4	8	no	—
55	C₂×C₄×C₆₇	—	4	8	no	—
56	C₂×C₄×C₆₇	—	4	8	no	—
57	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
58	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₆₇	C₂×SD₁₆	1	8	yes	—
60	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
61	C₂×C₄×C₆₇	#135	1	4	yes	—
62	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
65	C₈×C₆₇	—	2	4	no	—
66	C₈×C₆₇	#125	1	8	yes	complemented
67	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂²×C₆₇	#532	1	2	yes	—
69	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
70	C₂²×C₆₇	C₂×#135	1	2	yes	—
71	C₄×C₆₇	—	2	8	no	—
72	C₄×C₆₇	—	8	8	no	—
73	C₄×C₆₇	#498	1	4	yes	—
74	C₄×C₆₇	#499	1	4	yes	—
75	C₄×C₆₇	C₂×#125	1	2	yes	—
76	C₄×C₆₇	#533	1	2	yes	—
77	#33256	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₆₇	#2859	1	1	yes	characteristic
79	C₂×C₆₇	C₂×#498	1	1	yes	characteristic
80	C₂×C₆₇	C₂×#499	1	1	yes	characteristic
81	C₂×C₆₇	#3167	1	4	yes	—
82	#2859	C₂×C₆₇	1	2	yes	—
83	#1941	C₂×C₆₇	1	8	yes	—
84	#1844	C₂×C₆₇	1	2	yes	—
85	#1834	C₂×C₆₇	1	2	yes	—
86	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic
87	C₆₇	#33256	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×#125	—	2	2	no	—
89	C₂×#106	C₂²×C₆₇	1	8	yes	—
90	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
91	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
93	#389	—	2	4	no	—
94	#389	C₂²×C₆₇	1	8	yes	—
95	#384	C₂²×C₆₇	1	4	yes	—
96	#382	—	2	2	no	—
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	2	yes	—
101	#125	—	4	8	no	—
102	C₂×C₄⋊C₄	—	2	4	no	—
103	C₂×C₄⋊C₄	—	2	8	no	—
104	C₂×C₄⋊C₄	—	2	4	no	—
105	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
106	C₂×C₄²	D₄×C₆₇	1	2	yes	—
107	C₂×C₄²	D₄×C₆₇	1	2	yes	—
108	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
109	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
110	#106	—	2	16	no	—
111	C₄×C₈	—	2	8	no	—
112	C₄×C₈	Q₈×C₆₇	1	4	yes	—
113	C₄×C₈	Q₈×C₆₇	1	4	yes	—
114	#95	—	2	8	no	—
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
117	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
118	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
119	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
122	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
123	C₂×C₈	#41×C₆₇	1	8	yes	—
124	C₄⋊C₄	—	2	16	no	—
125	C₄⋊C₄	—	4	16	no	—
126	C₄⋊C₄	—	4	8	no	—
127	C₄⋊C₄	—	4	8	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	2	4	no	—
130	C₄²	SD₁₆×C₆₇	1	8	yes	—
131	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
132	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
133	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	8	no	—
137	C₂×C₄	—	4	8	no	—
138	C₂×C₄	—	4	8	no	—
139	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
140	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
141	C₂×C₄	C₂×SD₁₆×C₆₇	1	8	yes	—
142	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
143	C₂×C₄	#135×C₆₇	1	4	yes	—
144	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
145	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
146	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
147	C₈	—	2	4	no	—
148	C₈	#125×C₆₇	1	8	yes	complemented
149	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
150	C₂²	#532×C₆₇	1	2	yes	—
151	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
152	C₂²	C₂×#135×C₆₇	1	2	yes	—
153	C₄	—	2	8	no	—
154	C₄	—	8	8	no	—
155	C₄	#498×C₆₇	1	4	yes	—
156	C₄	#499×C₆₇	1	4	yes	—
157	C₄	C₂×#125×C₆₇	1	2	yes	—
158	C₄	#533×C₆₇	1	2	yes	—
159	C₂	C₆₇×#2859	1	1	yes	characteristic
160	C₂	C₂×#498×C₆₇	1	1	yes	characteristic
161	C₂	C₂×#499×C₆₇	1	1	yes	characteristic
162	C₂	C₆₇×#3167	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
66. C₈×C₆₇	#125	semidirect
77. #33256	C₆₇	direct (N × H)
87. C₆₇	#33256	direct (N × H)
148. C₈	#125×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2859	2
C₆₇×#1941	2
C₆₇×#1844	2
C₆₇×#1834	2
C₄²×C₈×C₆₇	2
#33256	67

Subgroups — C67×#33256 #30712507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33256 #30712507