Subgroups — C₂×C₆₇×#3556 #30742011

Order 17152.

6338 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×#3556	C₂	1	32	yes	maximal, direct product factor
2	C₂×#564×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂×#557×C₆₇	C₂	1	12	yes	maximal, complemented
4	C₂×#541×C₆₇	C₂	1	12	yes	maximal
5	C₂×#533×C₆₇	C₂	1	3	yes	maximal
6	#564×C₆₇	C₂²	1	64	yes	complemented
7	#557×C₆₇	C₂²	1	192	yes	complemented
8	#541×C₆₇	C₂²	1	192	yes	—
9	#533×C₆₇	C₂²	1	48	yes	—
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	6	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	12	yes	—
12	C₂×#126×C₆₇	C₂²	1	16	yes	—
13	C₂×#125×C₆₇	C₂²	1	6	yes	—
14	C₂×#125×C₆₇	C₂²	1	48	yes	—
15	C₂×#125×C₆₇	C₂²	1	12	yes	—
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	12	yes	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	12	yes	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	24	yes	—
19	C₂×#117×C₆₇	C₂²	1	6	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×Q₈×C₆₇	C₂³	1	12	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
23	C₄⋊Q₈×C₆₇	C₂³	1	48	yes	—
24	C₄⋊Q₈×C₆₇	C₂³	1	96	yes	—
25	#126×C₆₇	C₂³	1	128	yes	—
26	#125×C₆₇	C₂³	1	48	yes	—
27	#125×C₆₇	C₂³	1	96	yes	—
28	#125×C₆₇	C₂³	1	384	yes	—
29	C₂²⋊Q₈×C₆₇	C₂³	1	96	yes	—
30	C₄×Q₈×C₆₇	C₂³	1	96	yes	—
31	C₄×Q₈×C₆₇	C₂³	1	192	yes	—
32	#117×C₆₇	C₂³	1	48	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂³	1	24	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂³	1	48	yes	—
37	C₂×C₂²⋊C₄×C₆₇	C₂³	1	12	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	16	yes	—
43	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
44	C₂×Q₈×C₆₇	—	2	48	no	—
45	C₂×Q₈×C₆₇	C₂⁴	1	48	yes	—
46	C₂²×C₄×C₆₇	—	2	12	no	—
47	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
48	C₂²×C₄×C₆₇	C₂⁴	1	24	yes	—
49	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
50	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
51	C₄⋊C₄×C₆₇	C₂⁴	1	48	yes	—
52	C₄⋊C₄×C₆₇	C₂⁴	1	48	yes	—
53	C₄⋊C₄×C₆₇	C₂⁴	1	96	yes	—
54	C₄⋊C₄×C₆₇	C₂⁴	1	192	yes	—
55	C₂²⋊C₄×C₆₇	C₂⁴	1	48	yes	—
56	C₄²×C₆₇	C₂⁴	1	12	yes	—
57	C₄²×C₆₇	C₂⁴	1	4	yes	—
58	C₄²×C₆₇	C₂⁴	1	48	yes	—
59	C₄²×C₆₇	C₂⁴	1	64	yes	—
60	C₂³×C₆₇	—	2	6	no	—
61	C₂³×C₆₇	C₂⁵	1	2	yes	—
62	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
63	Q₈×C₆₇	—	2	96	no	—
64	C₂×C₄×C₆₇	—	2	6	no	—
65	C₂×C₄×C₆₇	—	2	48	no	—
66	C₂×C₄×C₆₇	—	2	24	no	—
67	C₂×C₄×C₆₇	2₋¹⁺⁴	1	12	yes	—
68	C₂×C₄×C₆₇	C₂⁵	1	48	yes	—
69	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
70	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
71	C₂²×C₆₇	—	2	6	no	—
72	C₂²×C₆₇	—	4	4	no	—
73	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	6	yes	—
74	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
75	C₄×C₆₇	—	2	6	no	—
76	C₄×C₆₇	—	2	48	no	—
77	C₄×C₆₇	C₂×2₋¹⁺⁴	1	12	yes	—
78	C₂×#3556	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×C₆₇	—	4	2	no	—
80	C₂×C₆₇	#3556	1	4	yes	direct product factor
81	C₂×C₆₇	C₂²×2₋¹⁺⁴	1	3	yes	—
82	#3556	C₂×C₆₇	1	32	yes	direct product factor
83	C₂×#564	C₂×C₆₇	1	4	yes	complemented
84	C₂×#557	C₂×C₆₇	1	12	yes	complemented
85	C₂×#541	C₂×C₆₇	1	12	yes	—
86	C₂×#533	C₂×C₆₇	1	3	yes	—
87	C₆₇	C₂×#3556	1	1	yes	characteristic, direct product factor, Sylow
88	#564	C₂²×C₆₇	1	64	yes	complemented
89	#557	C₂²×C₆₇	1	192	yes	complemented
90	#541	C₂²×C₆₇	1	192	yes	—
91	#533	C₂²×C₆₇	1	48	yes	—
92	C₂×C₄⋊Q₈	C₂²×C₆₇	1	6	yes	—
93	C₂×C₄⋊Q₈	C₂²×C₆₇	1	12	yes	—
94	C₂×#126	C₂²×C₆₇	1	16	yes	—
95	C₂×#125	C₂²×C₆₇	1	6	yes	—
96	C₂×#125	C₂²×C₆₇	1	12	yes	—
97	C₂×#125	C₂²×C₆₇	1	48	yes	—
98	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	12	yes	—
99	C₂×C₄×Q₈	C₂²×C₆₇	1	12	yes	—
100	C₂×C₄×Q₈	C₂²×C₆₇	1	24	yes	—
101	C₂×#117	C₂²×C₆₇	1	6	yes	—
102	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
103	C₂²×Q₈	C₂³×C₆₇	1	12	yes	—
104	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
105	C₄⋊Q₈	C₂³×C₆₇	1	48	yes	—
106	C₄⋊Q₈	C₂³×C₆₇	1	96	yes	—
107	#126	C₂³×C₆₇	1	128	yes	—
108	#125	C₂³×C₆₇	1	48	yes	—
109	#125	C₂³×C₆₇	1	96	yes	—
110	#125	C₂³×C₆₇	1	384	yes	—
111	C₂²⋊Q₈	C₂³×C₆₇	1	96	yes	—
112	C₄×Q₈	C₂³×C₆₇	1	96	yes	—
113	C₄×Q₈	C₂³×C₆₇	1	192	yes	—
114	#117	C₂³×C₆₇	1	48	yes	—
115	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
116	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
117	C₂×C₄⋊C₄	C₂³×C₆₇	1	24	yes	—
118	C₂×C₄⋊C₄	C₂³×C₆₇	1	48	yes	—
119	C₂×C₂²⋊C₄	C₂³×C₆₇	1	12	yes	—
120	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
121	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
122	C₂×C₄²	C₂³×C₆₇	1	16	yes	—
123	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
124	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
125	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
126	C₂×Q₈	—	2	48	no	—
127	C₂×Q₈	C₂⁴×C₆₇	1	48	yes	—
128	C₂²×C₄	—	2	12	no	—
129	C₂²×C₄	C₂⁴×C₆₇	1	24	yes	—
130	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
131	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
132	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
133	C₄⋊C₄	C₂⁴×C₆₇	1	48	yes	—
134	C₄⋊C₄	C₂⁴×C₆₇	1	48	yes	—
135	C₄⋊C₄	C₂⁴×C₆₇	1	96	yes	—
136	C₄⋊C₄	C₂⁴×C₆₇	1	192	yes	—
137	C₂²⋊C₄	C₂⁴×C₆₇	1	48	yes	—
138	C₄²	C₂⁴×C₆₇	1	12	yes	—
139	C₄²	C₂⁴×C₆₇	1	4	yes	—
140	C₄²	C₂⁴×C₆₇	1	48	yes	—
141	C₄²	C₂⁴×C₆₇	1	64	yes	—
142	C₂³	—	2	6	no	—
143	C₂³	C₂⁵×C₆₇	1	2	yes	—
144	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
145	Q₈	—	2	96	no	—
146	C₂×C₄	—	2	6	no	—
147	C₂×C₄	—	2	48	no	—
148	C₂×C₄	—	2	24	no	—
149	C₂×C₄	2₋¹⁺⁴×C₆₇	1	12	yes	—
150	C₂×C₄	C₂⁵×C₆₇	1	48	yes	—
151	C₂×C₄	C₂⁵×C₆₇	1	6	yes	—
152	C₂×C₄	C₂⁵×C₆₇	1	6	yes	—
153	C₂²	—	2	6	no	—
154	C₂²	—	4	4	no	—
155	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	6	yes	—
156	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
157	C₄	—	2	6	no	—
158	C₄	—	2	48	no	—
159	C₄	C₂×2₋¹⁺⁴×C₆₇	1	12	yes	—
160	C₂	—	4	2	no	—
161	C₂	C₆₇×#3556	1	4	yes	direct product factor
162	C₂	C₂²×2₋¹⁺⁴×C₆₇	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₇
prime radical	C₂⁴×C₆₇	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3556	C₂	direct (N × H)
2. C₂×#564×C₆₇	C₂	semidirect
3. C₂×#557×C₆₇	C₂	semidirect
6. #564×C₆₇	C₂²	semidirect
7. #557×C₆₇	C₂²	semidirect
78. C₂×#3556	C₆₇	direct (N × H)
80. C₂×C₆₇	#3556	direct (N × H)
82. #3556	C₂×C₆₇	direct (N × H)
83. C₂×#564	C₂×C₆₇	semidirect
84. C₂×#557	C₂×C₆₇	semidirect
87. C₆₇	C₂×#3556	direct (N × H)
88. #564	C₂²×C₆₇	semidirect
89. #557	C₂²×C₆₇	semidirect
161. C₂	C₆₇×#3556	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×#3556	2
C₂×#564×C₆₇	2
C₂×#557×C₆₇	2
C₂×#541×C₆₇	2
C₂×#533×C₆₇	2
C₂×#3556	67

Subgroups — C2×C67×#3556 #30742011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3556 #30742011