Subgroups — C₂²×#368×C₆₇ #30712663

Order 17152.

662 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	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₆₇	C₂	1	2	yes	maximal
3	C₂×#368×C₆₇	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₁₆×C₆₇	—	2	2	no	—
5	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₆₇	C₄	1	2	yes	—
7	C₂²×C₁₆×C₆₇	C₄	1	6	yes	—
8	C₂²×C₁₆×C₆₇	C₂²	1	6	yes	—
9	C₂×#108×C₆₇	—	2	6	no	—
10	C₂×#108×C₆₇	C₂²	1	12	yes	—
11	#368×C₆₇	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
13	C₂×M₁₆×C₆₇	—	2	12	no	—
14	C₂×M₁₆×C₆₇	—	4	6	no	—
15	C₂²×C₈×C₆₇	—	4	2	no	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
18	C₂²×C₈×C₆₇	D₄	1	3	yes	—
19	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	Q₈	1	3	yes	—
21	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
22	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	—
23	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	—
24	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
25	#108×C₆₇	—	2	12	no	—
26	#108×C₆₇	C₂³	1	8	yes	—
27	C₂⁴×C₆₇	Q₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	C₄⋊C₄	1	3	yes	—
30	C₂²×C₄×C₆₇	D₈	1	3	yes	—
31	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	Q₁₆	1	3	yes	—
33	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
34	M₁₆×C₆₇	—	2	8	no	—
35	M₁₆×C₆₇	—	4	12	no	—
36	C₂×C₈×C₆₇	—	4	12	no	—
37	C₂×C₈×C₆₇	C₄⋊C₄	1	6	yes	—
38	C₂×C₈×C₆₇	C₄⋊C₄	1	3	yes	—
39	C₂×C₈×C₆₇	C₄⋊C₄	1	3	yes	—
40	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
41	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
42	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
43	C₂×C₈×C₆₇	C₂×Q₈	1	3	yes	—
44	C₂×C₈×C₆₇	C₂×Q₈	1	3	yes	—
45	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₁₆×C₆₇	C₂²×C₄	1	8	yes	—
47	C₂³×C₆₇	—	2	4	no	—
48	C₂³×C₆₇	#107	1	3	yes	—
49	C₂³×C₆₇	#107	1	1	yes	characteristic
50	C₂³×C₆₇	C₂×Q₁₆	1	3	yes	—
51	C₂×C₄×C₆₇	#107	1	6	yes	—
52	C₂×C₄×C₆₇	#107	1	3	yes	—
53	C₂×C₄×C₆₇	#107	1	3	yes	—
54	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
57	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
58	C₂×C₄×C₆₇	C₂×Q₁₆	1	3	yes	—
59	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
60	C₈×C₆₇	—	4	8	no	—
61	C₈×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
62	C₈×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
63	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
64	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂²×C₆₇	—	2	6	no	—
66	C₂²×C₆₇	—	2	6	no	—
67	C₂²×C₆₇	#368	1	4	yes	direct product factor
68	C₂²×C₆₇	C₂×#107	1	3	yes	—
69	C₂²×C₆₇	C₂×#107	1	3	yes	—
70	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
71	C₄×C₆₇	C₂×#107	1	3	yes	—
72	C₄×C₆₇	C₂×#107	1	3	yes	—
73	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
75	C₂²×#368	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×C₆₇	—	2	1	no	—
77	C₂×C₆₇	—	2	3	no	—
78	C₂×C₆₇	C₂×#368	1	6	yes	direct product factor
79	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
80	C₂³×C₁₆	C₂×C₆₇	1	1	yes	characteristic
81	C₂²×#108	C₂×C₆₇	1	2	yes	—
82	C₂×#368	C₂×C₆₇	1	12	yes	direct product factor
83	C₆₇	C₂²×#368	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²×M₁₆	—	2	2	no	—
85	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
86	C₂²×C₁₆	C₄×C₆₇	1	6	yes	—
87	C₂²×C₁₆	C₄×C₆₇	1	2	yes	—
88	C₂²×C₁₆	C₂²×C₆₇	1	6	yes	—
89	C₂×#108	—	2	6	no	—
90	C₂×#108	C₂²×C₆₇	1	12	yes	—
91	#368	C₂²×C₆₇	1	16	yes	direct product factor
92	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
93	C₂×M₁₆	—	2	12	no	—
94	C₂×M₁₆	—	4	6	no	—
95	C₂²×C₈	—	4	2	no	—
96	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
97	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
98	C₂²×C₈	D₄×C₆₇	1	3	yes	—
99	C₂²×C₈	Q₈×C₆₇	1	3	yes	—
100	C₂²×C₈	Q₈×C₆₇	1	1	yes	characteristic
101	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
102	C₂×C₁₆	C₂×C₄×C₆₇	1	12	yes	—
103	C₂×C₁₆	C₂×C₄×C₆₇	1	12	yes	—
104	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
105	#108	—	2	12	no	—
106	#108	C₂³×C₆₇	1	8	yes	—
107	C₂⁴	Q₁₆×C₆₇	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊C₄×C₆₇	1	3	yes	—
109	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
110	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
111	C₂²×C₄	D₈×C₆₇	1	3	yes	—
112	C₂²×C₄	Q₁₆×C₆₇	1	3	yes	—
113	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
114	M₁₆	—	2	8	no	—
115	M₁₆	—	4	12	no	—
116	C₂×C₈	—	4	12	no	—
117	C₂×C₈	C₄⋊C₄×C₆₇	1	6	yes	—
118	C₂×C₈	C₄⋊C₄×C₆₇	1	3	yes	—
119	C₂×C₈	C₄⋊C₄×C₆₇	1	3	yes	—
120	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
121	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
122	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
123	C₂×C₈	C₂×Q₈×C₆₇	1	3	yes	—
124	C₂×C₈	C₂×Q₈×C₆₇	1	3	yes	—
125	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
126	C₁₆	C₂²×C₄×C₆₇	1	8	yes	—
127	C₂³	—	2	4	no	—
128	C₂³	#107×C₆₇	1	3	yes	—
129	C₂³	#107×C₆₇	1	1	yes	characteristic
130	C₂³	C₂×Q₁₆×C₆₇	1	3	yes	—
131	C₂×C₄	#107×C₆₇	1	6	yes	—
132	C₂×C₄	#107×C₆₇	1	3	yes	—
133	C₂×C₄	#107×C₆₇	1	3	yes	—
134	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	3	yes	—
135	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	3	yes	—
136	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
137	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
138	C₂×C₄	C₂×Q₁₆×C₆₇	1	3	yes	—
139	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
140	C₈	—	4	8	no	—
141	C₈	C₂×C₄⋊C₄×C₆₇	1	3	yes	—
142	C₈	C₂×C₄⋊C₄×C₆₇	1	3	yes	—
143	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
144	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
145	C₂²	—	2	6	no	—
146	C₂²	—	2	6	no	—
147	C₂²	#368×C₆₇	1	4	yes	direct product factor
148	C₂²	C₂×#107×C₆₇	1	3	yes	—
149	C₂²	C₂×#107×C₆₇	1	3	yes	—
150	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
151	C₄	C₂×#107×C₆₇	1	3	yes	—
152	C₄	C₂×#107×C₆₇	1	3	yes	—
153	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
154	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	3	no	—
157	C₂	C₂×#368×C₆₇	1	6	yes	direct product factor
158	C₂	C₂²×#107×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂³×C₆₇	#107
prime radical	C₂⁴×C₆₇	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#368×C₆₇	C₂	direct (N × H)
11. #368×C₆₇	C₂²	direct (N × H)
67. C₂²×C₆₇	#368	direct (N × H)
75. C₂²×#368	C₆₇	direct (N × H)
78. C₂×C₆₇	C₂×#368	direct (N × H)
82. C₂×#368	C₂×C₆₇	direct (N × H)
83. C₆₇	C₂²×#368	direct (N × H)
91. #368	C₂²×C₆₇	direct (N × H)
147. C₂²	#368×C₆₇	direct (N × H)
157. C₂	C₂×#368×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₆₇	2
C₂²×#108×C₆₇	2
C₂×#368×C₆₇	2
C₂²×#368	67

Subgroups — C22×#368×C67 #30712663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#368×C₆₇ #30712663