Subgroups — #30774437

Order 17152.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
2	#22560373	C₂	1	1	yes	maximal, characteristic
3	#22558300	C₂	1	4	yes	maximal
4	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14523977	C₂²	1	2	yes	—
7	C₂×#14523975	—	2	1	no	—
8	#16863262	—	2	1	no	—
9	#16863172	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
11	#500×C₆₇	—	2	2	no	—
12	#500×C₆₇	C₄	1	4	yes	complemented
13	C₂×#107×C₆₇	—	2	1	no	—
14	C₂×#107×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	C₂×C₆₇⋊C₁₆	—	2	2	no	—
18	C₂×C₆₇⋊C₁₆	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
20	#14523977	—	2	4	no	—
21	#14523975	—	4	2	no	—
22	C₆₇⋊#95	—	4	1	no	—
23	C₆₇⋊#95	—	4	1	no	—
24	C₂²×Q₈×C₆₇	—	4	1	no	—
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₆₇	—	2	4	no	—
28	C₄⋊Q₈×C₆₇	—	4	2	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
31	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
32	#107×C₆₇	—	2	4	no	—
33	#107×C₆₇	—	2	2	no	—
34	#107×C₆₇	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₂²×Dic₆₇	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₆₇⋊C₁₆	—	2	4	no	—
41	C₂×Q₈×C₆₇	—	4	4	no	—
42	C₂×Q₈×C₆₇	—	8	4	no	—
43	C₂²×C₄×C₆₇	—	4	2	no	—
44	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
49	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₆₇	#41	1	2	yes	—
54	C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₄⋊C₄×C₆₇	—	4	4	no	—
56	C₄⋊C₄×C₆₇	—	4	2	no	—
57	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₆₇	SD₁₆	1	2	yes	—
59	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₆₇	—	8	8	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	4	4	no	—
64	C₂×C₄×C₆₇	—	8	4	no	—
65	C₂×C₄×C₆₇	#102	1	2	yes	—
66	C₂×C₄×C₆₇	#102	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×Dic₆₇	—	8	2	no	—
74	C₂×Dic₆₇	—	8	2	no	—
75	C₂×Dic₆₇	—	8	2	no	—
76	C₈×C₆₇	#103	1	4	yes	—
77	C₈×C₆₇	C₄×Q₈	1	2	yes	—
78	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
80	C₂²×C₆₇	#486	1	1	yes	characteristic
81	C₂²×C₆₇	#500	1	1	yes	characteristic
82	C₂²×C₆₇	#509	1	2	yes	—
83	C₂²×C₆₇	#510	1	2	yes	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	2	2	no	—
86	C₄×C₆₇	—	8	4	no	—
87	C₄×C₆₇	#391	1	1	yes	characteristic
88	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
89	C₄×C₆₇	#487	1	1	yes	characteristic
90	C₄×C₆₇	#498	1	1	yes	characteristic
91	Dic₆₇	—	16	2	no	—
92	#13283	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	#1990	1	1	yes	characteristic
94	C₂×C₆₇	#2140	1	1	yes	characteristic
95	C₂×C₆₇	#2141	1	1	yes	characteristic
96	C₂×C₆₇	#2243	1	2	yes	—
97	C₂×C₆₇	#2255	1	2	yes	—
98	C₂×#500	D₆₇	1	1	yes	characteristic
99	C₂×#346	—	67	1	no	—
100	#1846	—	67	1	no	—
101	#1380	—	67	4	no	—
102	C₆₇	#13283	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	2	1	no	—
104	C₂²×C₁₆	—	67	2	no	—
105	#500	—	2	2	no	—
106	#500	Dic₆₇	1	4	yes	complemented
107	C₂×#107	—	2	1	no	—
108	C₂×#107	—	67	2	no	—
109	C₂×#107	C₂×D₆₇	1	2	yes	—
110	C₂×#106	—	134	1	no	—
111	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
112	#382	—	134	1	no	—
113	#346	—	134	2	no	—
114	C₂²×Q₈	—	4	1	no	—
115	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	4	no	—
118	C₄⋊Q₈	—	4	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄⋊C₄	—	134	2	no	—
122	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₂×C₁₆	—	67	4	no	—
124	C₂×C₁₆	—	134	2	no	—
125	C₂×C₁₆	—	134	2	no	—
126	#107	—	2	4	no	—
127	#107	—	2	2	no	—
128	#107	—	134	4	no	—
129	#107	C₂×Dic₆₇	1	4	yes	—
130	#106	—	268	2	no	—
131	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
132	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
133	#95	—	268	1	no	—
134	#95	—	268	1	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	8	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	268	2	no	—
139	C₂²×C₄	#11761613	1	1	yes	characteristic
140	C₂²×C₄	#11761613	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
142	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
143	C₂×C₈	#11761615	1	2	yes	—
144	C₂×C₈	#11761618	1	2	yes	—
145	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
146	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₆₇	1	2	yes	—
148	C₂×C₈	#11761629	1	2	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	268	2	no	—
153	C₄⋊C₄	—	268	2	no	—
154	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
155	C₄²	#11761616	1	2	yes	—
156	C₁₆	—	134	4	no	—
157	C₂³	C₂×#11761613	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	536	2	no	—
164	C₂×C₄	—	536	2	no	—
165	C₂×C₄	—	536	2	no	—
166	C₂×C₄	#14523964	1	2	yes	—
167	C₂×C₄	#14523964	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
169	C₂×C₄	#14524032	1	1	yes	characteristic
170	C₂×C₄	#14524041	1	1	yes	characteristic
171	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
172	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
173	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
174	C₈	#14523967	1	4	yes	—
175	C₈	Q₈×Dic₆₇	1	2	yes	—
176	C₈	#14524019	1	2	yes	—
177	C₂²	C₂×#14523964	1	1	yes	characteristic
178	C₂²	#16863744	1	1	yes	characteristic
179	C₂²	#16863799	1	1	yes	characteristic
180	C₂²	#16863832	1	2	yes	—
181	C₂²	#16863837	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	1072	2	no	—
186	C₄	#16863303	1	1	yes	characteristic
187	C₄	C₂×#14523967	1	1	yes	characteristic
188	C₄	#16863748	1	1	yes	characteristic
189	C₄	#16863792	1	1	yes	characteristic
190	C₂	#22561171	1	1	yes	characteristic
191	C₂	#22561925	1	1	yes	characteristic
192	C₂	#22561929	1	1	yes	characteristic
193	C₂	#22562448	1	2	yes	—
194	C₂	#22562492	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#13283
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. #500×C₆₇	C₄	semidirect
19. C₂×C₆₇⋊C₁₆	Q₈	semidirect
102. C₆₇	#13283	semidirect
106. #500	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863172	2
#22560373	2
#22558300	2
C₂×#500×C₆₇	2
#13283	67