Subgroups — #30773866

Order 17152.

326 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863224	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863224	C₂	1	1	yes	maximal, characteristic
3	#22560942	C₂	1	2	yes	maximal
4	#22558301	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	C₂²	1	2	yes	—
7	C₂×#14523977	C₂²	1	2	yes	—
8	#16863294	—	2	2	no	—
9	#16863224	—	2	2	no	—
10	#16863224	C₄	1	4	yes	complemented
11	#16863159	—	2	2	no	—
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₆₇	—	2	2	no	—
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	C₂×Dic₆₇⋊C₄	—	4	2	no	—
19	#14523977	—	2	2	no	—
20	#14523977	—	2	2	no	—
21	#14523977	—	2	2	no	—
22	#14523977	C₂×C₄	1	4	yes	—
23	C₆₇⋊#95	—	4	2	no	—
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	—	2	1	no	—
28	C₂×C₁₆×C₆₇	—	2	1	no	—
29	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
31	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
32	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
33	C₄×C₈×C₆₇	—	2	1	no	—
34	C₄×C₈×C₆₇	—	2	1	no	—
35	C₂²×Dic₆₇	—	4	2	no	—
36	C₂²×Dic₆₇	—	4	2	no	—
37	C₄⋊Dic₆₇	—	2	4	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₄⋊Dic₆₇	—	4	2	no	—
41	Dic₆₇⋊C₄	—	8	4	no	—
42	C₂²×C₄×C₆₇	D₈	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	—	2	1	no	—
46	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
47	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
52	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
53	C₄²×C₆₇	—	2	1	no	—
54	C₄²×C₆₇	—	2	1	no	—
55	C₁₆×C₆₇	—	2	2	no	—
56	C₁₆×C₆₇	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
61	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
67	C₂×Dic₆₇	—	4	4	no	—
68	C₂×Dic₆₇	—	8	2	no	—
69	C₂×Dic₆₇	—	8	2	no	—
70	C₂×Dic₆₇	—	8	2	no	—
71	C₂×Dic₆₇	—	8	2	no	—
72	C₂×Dic₆₇	—	8	2	no	—
73	C₈×C₆₇	—	2	2	no	—
74	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
76	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
77	C₈×C₆₇	#125	1	1	yes	characteristic
78	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₆₇	#466	1	1	yes	characteristic
80	C₂²×C₆₇	#500	1	1	yes	characteristic
81	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
82	C₂²×C₆₇	#508	1	2	yes	—
83	C₄×C₆₇	—	2	4	no	—
84	C₄×C₆₇	#389	1	1	yes	characteristic
85	C₄×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
86	C₄×C₆₇	#467	1	1	yes	characteristic
87	C₄×C₆₇	#499	1	1	yes	characteristic
88	Dic₆₇	—	8	4	no	—
89	Dic₆₇	—	16	2	no	—
90	#13166	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	#1942	1	1	yes	characteristic
92	C₂×C₆₇	C₄×SD₃₂	1	2	yes	—
93	C₂×C₆₇	#2217	1	2	yes	—
94	C₂×C₆₇	#2253	1	1	yes	characteristic
95	C₂×C₆₇	#2254	1	1	yes	characteristic
96	C₂×#367	—	67	1	no	—
97	C₂×#367	—	67	1	no	—
98	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
99	#1942	—	67	2	no	—
100	#1380	—	67	2	no	—
101	C₆₇	#13166	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
103	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
104	C₂×#107	—	67	2	no	—
105	C₂×#107	—	67	2	no	—
106	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
107	#389	—	134	2	no	—
108	#367	—	67	4	no	—
109	#367	—	134	2	no	—
110	C₄×C₁₆	—	2	2	no	—
111	#340	—	134	2	no	—
112	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	268	2	no	—
115	C₂×C₄⋊C₄	—	134	2	no	—
116	C₂×C₄⋊C₄	—	134	2	no	—
117	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
121	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
122	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
124	#107	—	67	4	no	—
125	#107	—	134	2	no	—
126	#107	—	134	2	no	—
127	#107	—	134	2	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	#95	—	268	2	no	—
131	C₂²×C₄	—	268	2	no	—
132	C₂²×C₄	—	268	2	no	—
133	C₂²×C₄	D₅₃₆	1	2	yes	—
134	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	Dic₂₆₈	1	2	yes	—
138	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
142	C₂×C₈	#11761628	1	1	yes	characteristic
143	C₂×C₈	#11761628	1	1	yes	characteristic
144	C₄⋊C₄	—	268	2	no	—
145	C₄⋊C₄	—	268	2	no	—
146	C₄⋊C₄	—	268	2	no	—
147	C₄⋊C₄	—	536	4	no	—
148	C₄⋊C₄	—	134	4	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	Dic₆₇⋊C₄	1	4	yes	complemented
153	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	268	4	no	—
157	C₂×C₄	—	536	2	no	—
158	C₂×C₄	—	536	2	no	—
159	C₂×C₄	—	536	2	no	—
160	C₂×C₄	—	536	2	no	—
161	C₂×C₄	—	536	2	no	—
162	C₂×C₄	#14523988	1	4	yes	—
163	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
164	C₂×C₄	#14524017	1	1	yes	characteristic
165	C₂×C₄	#14524043	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
167	C₂×C₄	#14524065	1	1	yes	characteristic
168	C₂×C₄	#14524065	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
171	C₈	C₄×Dic₁₃₄	1	1	yes	characteristic
172	C₈	#14524022	1	1	yes	characteristic
173	C₈	#14524035	1	1	yes	characteristic
174	C₂²	C₄×D₅₃₆	1	1	yes	characteristic
175	C₂²	#16863632	1	1	yes	characteristic
176	C₂²	#16863801	1	1	yes	characteristic
177	C₂²	C₂×#14523988	1	2	yes	—
178	C₂²	#16863828	1	2	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	536	4	no	—
181	C₄	—	1072	2	no	—
182	C₄	#16863294	1	1	yes	characteristic
183	C₄	C₄×Dic₂₆₈	1	1	yes	characteristic
184	C₄	#16863637	1	1	yes	characteristic
185	C₄	#16863798	1	1	yes	characteristic
186	C₂	#22560942	1	1	yes	characteristic
187	C₂	C₄×#14523988	1	2	yes	—
188	C₂	#22562305	1	2	yes	—
189	C₂	#22562488	1	1	yes	characteristic
190	C₂	#22562491	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	#13166
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #16863224	C₄	semidirect
56. C₁₆×C₆₇	C₄⋊C₄	semidirect
101. C₆₇	#13166	semidirect
152. C₁₆	Dic₆₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863224	2
#22560942	2
#22558301	2
C₂×C₄×C₁₆×C₆₇	2
#13166	67