Subgroups — #31011323

Order 17152.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570723	C₂	1	2	yes	maximal
2	#22562491	C₂	1	2	yes	maximal, complemented
3	#22562488	C₂	1	4	yes	maximal, complemented
4	#22562484	C₂	1	2	yes	maximal, complemented
5	#22561993	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863953	—	2	2	no	—
8	#16863801	C₂²	1	4	yes	—
9	#16863798	—	2	2	no	—
10	#16863798	C₂²	1	4	yes	—
11	#16863794	—	2	2	no	—
12	#16863434	—	2	2	no	—
13	#16863434	C₂²	1	4	yes	—
14	#16863224	C₂²	1	8	yes	—
15	#16863222	C₂²	1	8	yes	—
16	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
20	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	#14524043	—	2	4	no	—
22	#14524035	—	2	4	no	—
23	#14524022	—	4	4	no	—
24	C₄×Dic₁₃₄	—	4	4	no	—
25	#14524002	—	2	4	no	—
26	#14523977	—	2	4	no	—
27	#14523977	C₂³	1	8	yes	—
28	#14523975	—	2	4	no	—
29	#14523975	—	2	4	no	—
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₆₇	Q₈	1	8	yes	complemented
34	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
35	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
36	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₃₄	—	4	4	no	—
41	C₄⋊Dic₆₇	—	2	8	no	—
42	C₄⋊Dic₆₇	—	4	4	no	—
43	Dic₆₇⋊C₄	—	4	8	no	—
44	C₂²⋊Dic₆₇	—	4	4	no	—
45	C₄×Dic₆₇	—	4	4	no	—
46	C₂²×C₄×C₆₇	D₈	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
49	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
53	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
54	C₄²×C₆₇	D₈	1	2	yes	—
55	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₁₆×C₆₇	C₂×Q₈	1	4	yes	—
58	C₁₆×C₆₇	C₂×Q₈	1	4	yes	—
59	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	2	4	no	—
61	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
66	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×Dic₆₇	—	4	8	no	—
68	Dic₁₃₄	—	8	8	no	—
69	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
70	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
71	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
72	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
73	C₂²×C₆₇	—	2	2	no	—
74	C₂²×C₆₇	#500	1	2	yes	—
75	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
76	C₄×C₆₇	—	2	2	no	—
77	C₄×C₆₇	#500	1	2	yes	—
78	C₄×C₆₇	#508	1	4	yes	—
79	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
80	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
81	Dic₆₇	—	8	8	no	—
82	#33726	—	67	1	no	maximal, Sylow
83	C₂×C₆₇	—	2	2	no	—
84	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
85	C₂×C₆₇	C₂×#508	1	2	yes	—
86	#3160	—	67	2	no	—
87	#2254	—	67	2	no	—
88	#2253	—	67	4	no	—
89	#2252	—	67	2	no	—
90	#2158	—	67	4	no	—
91	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
92	C₆₇	#33726	1	1	yes	characteristic, complemented, Sylow
93	#533	—	134	2	no	—
94	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
95	#500	—	67	4	no	—
96	#499	—	67	4	no	—
97	#499	—	134	2	no	—
98	#498	—	134	2	no	—
99	#427	—	67	4	no	—
100	#427	—	134	2	no	—
101	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
102	#367	—	67	8	no	—
103	#366	—	67	8	no	—
104	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
105	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
106	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
107	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
109	C₄⋊Q₈	—	134	4	no	—
110	#125	—	134	4	no	—
111	C₂²⋊Q₈	—	268	4	no	—
112	C₄×Q₈	—	268	4	no	—
113	#117	—	134	4	no	—
114	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
115	C₂×C₁₆	Dic₁₃₄	1	8	yes	complemented
116	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
117	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
118	#107	—	67	8	no	—
119	#107	—	134	4	no	—
120	#106	—	134	4	no	—
121	#106	—	134	4	no	—
122	C₄×C₈	D₂₆₈	1	1	yes	characteristic
123	C₄×C₈	D₂₆₈	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
125	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
126	C₂×Q₈	—	268	4	no	—
127	C₂²×C₄	D₅₃₆	1	2	yes	—
128	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
129	C₂×C₈	Dic₂₆₈	1	4	yes	—
130	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
131	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
132	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
133	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
134	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
135	C₄⋊C₄	—	268	8	no	—
136	C₄⋊C₄	—	268	4	no	—
137	C₄⋊C₄	—	134	8	no	—
138	C₂²⋊C₄	—	268	4	no	—
139	C₄²	—	268	4	no	—
140	C₄²	D₅₃₆	1	2	yes	—
141	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
142	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
143	C₁₆	C₂×Dic₁₃₄	1	4	yes	—
144	C₁₆	C₂×Dic₁₃₄	1	4	yes	—
145	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
146	Q₈	—	536	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	268	8	no	—
149	C₂×C₄	#14524043	1	2	yes	—
150	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
151	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
152	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
153	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
154	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
155	C₈	#14524043	1	2	yes	—
156	C₈	C₂×Dic₂₆₈	1	2	yes	—
157	C₈	C₂×Dic₂₆₈	1	2	yes	—
158	C₈	C₂²×Dic₁₃₄	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#16863801	1	2	yes	—
161	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	536	8	no	—
164	C₄	#16863801	1	2	yes	—
165	C₄	#16863828	1	4	yes	—
166	C₄	C₂×#14524043	1	1	yes	characteristic
167	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#16863801	1	1	yes	characteristic
170	C₂	C₂×#16863828	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	#33726
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22562491	C₂	semidirect
3. #22562488	C₂	semidirect
4. #22562484	C₂	semidirect
33. C₂×C₁₆×C₆₇	Q₈	semidirect
92. C₆₇	#33726	semidirect
115. C₂×C₁₆	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22562491	2
#22562488	2
#22562484	2
#22561993	2
C₂×C₄×C₁₆×C₆₇	2
#33726	67