Subgroups — C₆₇×#33313 #30712564

Order 17152.

862 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	2	yes	maximal
2	C₂×#498×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#2862	C₂	1	2	yes	maximal
4	C₆₇×#1846	C₂	1	8	yes	maximal
5	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
7	#500×C₆₇	—	2	8	no	—
8	#498×C₆₇	—	2	8	no	—
9	C₂×#107×C₆₇	C₂²	1	8	yes	—
10	C₂×#106×C₆₇	C₂²	1	8	yes	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
14	#391×C₆₇	—	2	8	no	—
15	#382×C₆₇	C₂²	1	8	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₆₇	—	2	4	no	—
18	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
19	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
20	C₄⋊Q₈×C₆₇	—	2	8	no	—
21	C₄⋊Q₈×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	4	yes	—
25	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
27	#107×C₆₇	—	2	16	no	—
28	#106×C₆₇	—	2	16	no	—
29	C₄×C₈×C₆₇	D₄	1	4	yes	—
30	C₄×C₈×C₆₇	D₄	1	4	yes	—
31	C₄×C₈×C₆₇	Q₈	1	16	yes	complemented
32	#95×C₆₇	—	2	16	no	—
33	C₂×Q₈×C₆₇	—	4	8	no	—
34	C₂×Q₈×C₆₇	—	4	16	no	—
35	C₂²×C₄×C₆₇	—	2	8	no	—
36	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
37	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
40	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
41	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
42	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
43	C₄⋊C₄×C₆₇	—	2	8	no	—
44	C₄⋊C₄×C₆₇	—	2	8	no	—
45	C₄⋊C₄×C₆₇	—	4	8	no	—
46	C₄²×C₆₇	—	2	8	no	—
47	C₄²×C₆₇	C₂×D₄	1	2	yes	—
48	C₄²×C₆₇	C₂×D₄	1	2	yes	—
49	C₄²×C₆₇	C₂×Q₈	1	8	yes	—
50	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
51	Q₈×C₆₇	—	8	16	no	—
52	C₂×C₄×C₆₇	—	2	8	no	—
53	C₂×C₄×C₆₇	—	4	8	no	—
54	C₂×C₄×C₆₇	—	4	16	no	—
55	C₂×C₄×C₆₇	#127	1	4	yes	—
56	C₂×C₄×C₆₇	C₄⋊Q₈	1	8	yes	—
57	C₂×C₄×C₆₇	#135	1	8	yes	—
58	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
61	C₈×C₆₇	C₄⋊Q₈	1	8	yes	—
62	C₈×C₆₇	C₄⋊Q₈	1	8	yes	—
63	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
64	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
65	C₂²×C₆₇	C₂×#135	1	4	yes	—
66	C₄×C₆₇	—	2	8	no	—
67	C₄×C₆₇	—	8	8	no	—
68	C₄×C₆₇	#495	1	8	yes	—
69	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
70	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
71	#33313	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₆₇	#2868	1	1	yes	characteristic
73	C₂×C₆₇	C₂×#495	1	2	yes	—
74	C₂×C₆₇	#3160	1	4	yes	—
75	C₂×#500	C₂×C₆₇	1	2	yes	—
76	C₂×#498	C₂×C₆₇	1	2	yes	—
77	#2862	C₂×C₆₇	1	2	yes	—
78	#1846	C₂×C₆₇	1	8	yes	—
79	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic
80	C₆₇	#33313	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
82	#500	—	2	8	no	—
83	#498	—	2	8	no	—
84	C₂×#107	C₂²×C₆₇	1	8	yes	—
85	C₂×#106	C₂²×C₆₇	1	8	yes	—
86	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
89	#391	—	2	8	no	—
90	#382	C₂²×C₆₇	1	8	yes	—
91	C₄³	C₂²×C₆₇	1	1	yes	characteristic
92	C₂²×Q₈	—	2	4	no	—
93	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
94	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
95	C₄⋊Q₈	—	2	8	no	—
96	C₄⋊Q₈	—	2	8	no	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
99	C₂×C₄²	D₄×C₆₇	1	4	yes	—
100	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
101	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
102	#107	—	2	16	no	—
103	#106	—	2	16	no	—
104	C₄×C₈	D₄×C₆₇	1	4	yes	—
105	C₄×C₈	D₄×C₆₇	1	4	yes	—
106	C₄×C₈	Q₈×C₆₇	1	16	yes	complemented
107	#95	—	2	16	no	—
108	C₂×Q₈	—	4	8	no	—
109	C₂×Q₈	—	4	16	no	—
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
112	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
113	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
114	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
115	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
116	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
117	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
118	C₄⋊C₄	—	2	8	no	—
119	C₄⋊C₄	—	2	8	no	—
120	C₄⋊C₄	—	4	8	no	—
121	C₄²	—	2	8	no	—
122	C₄²	C₂×D₄×C₆₇	1	2	yes	—
123	C₄²	C₂×D₄×C₆₇	1	2	yes	—
124	C₄²	C₂×Q₈×C₆₇	1	8	yes	—
125	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
126	Q₈	—	8	16	no	—
127	C₂×C₄	—	2	8	no	—
128	C₂×C₄	—	4	8	no	—
129	C₂×C₄	—	4	16	no	—
130	C₂×C₄	#127×C₆₇	1	4	yes	—
131	C₂×C₄	C₄⋊Q₈×C₆₇	1	8	yes	—
132	C₂×C₄	#135×C₆₇	1	8	yes	—
133	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
134	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
135	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
136	C₈	C₄⋊Q₈×C₆₇	1	8	yes	—
137	C₈	C₄⋊Q₈×C₆₇	1	8	yes	—
138	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
139	C₂²	C₂×C₄⋊Q₈×C₆₇	1	2	yes	—
140	C₂²	C₂×#135×C₆₇	1	4	yes	—
141	C₄	—	2	8	no	—
142	C₄	—	8	8	no	—
143	C₄	#495×C₆₇	1	8	yes	—
144	C₄	C₂×C₄⋊Q₈×C₆₇	1	2	yes	—
145	C₄	C₂×C₄⋊Q₈×C₆₇	1	2	yes	—
146	C₂	C₆₇×#2868	1	1	yes	characteristic
147	C₂	C₂×#495×C₆₇	1	2	yes	—
148	C₂	C₆₇×#3160	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
31. C₄×C₈×C₆₇	Q₈	semidirect
71. #33313	C₆₇	direct (N × H)
80. C₆₇	#33313	direct (N × H)
106. C₄×C₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₆₇	2
C₂×#498×C₆₇	2
C₆₇×#2862	2
C₆₇×#1846	2
C₄²×C₈×C₆₇	2
#33313	67

Subgroups — C67×#33313 #30712564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33313 #30712564