Subgroups — #30774048

Order 17152.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863199	C₂	1	4	yes	maximal
3	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆₈	—	2	1	no	—
6	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
7	C₂×#14524043	—	2	1	no	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#16863776	—	2	2	no	—
10	#16863776	C₄	1	4	yes	complemented
11	#16863199	—	2	8	no	—
12	C₄×C₆₇⋊C₁₆	—	2	2	no	—
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	#384×C₆₇	—	2	1	no	—
17	C₂²×Dic₁₃₄	—	2	2	no	—
18	C₂×Dic₂₆₈	—	2	4	no	—
19	C₂×Dic₂₆₈	—	2	4	no	—
20	C₂×Dic₂₆₈	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	4	no	—
22	C₂×Dic₂₆₈	—	4	2	no	—
23	C₂×Dic₂₆₈	—	4	2	no	—
24	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
25	#14524043	—	2	4	no	—
26	#14524043	—	4	2	no	—
27	C₂×C₄⋊Dic₆₇	—	4	2	no	—
28	C₂×C₆₇⋊C₁₆	—	2	2	no	—
29	C₂×C₆₇⋊C₁₆	—	2	2	no	—
30	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	#107×C₆₇	—	2	4	no	—
37	#107×C₆₇	—	4	2	no	—
38	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₇	D₄	1	2	yes	—
40	C₂×Dic₁₃₄	—	2	4	no	—
41	C₂×Dic₁₃₄	—	4	2	no	—
42	C₂×Dic₁₃₄	—	4	2	no	—
43	C₂×Dic₁₃₄	—	4	2	no	—
44	C₂×Dic₁₃₄	—	4	2	no	—
45	C₂×Dic₁₃₄	—	4	2	no	—
46	C₂²×Dic₆₇	—	4	2	no	—
47	Dic₂₆₈	—	2	8	no	—
48	Dic₂₆₈	—	4	8	no	—
49	Dic₂₆₈	—	4	4	no	—
50	C₄⋊Dic₆₇	—	4	8	no	—
51	C₆₇⋊C₁₆	—	2	4	no	—
52	C₂²×C₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	D₈	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₆₇	D₈	1	2	yes	—
57	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	#41	1	2	yes	—
62	C₄⋊C₄×C₆₇	—	4	2	no	—
63	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₄⋊C₄×C₆₇	—	8	4	no	—
65	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₆₇	SD₁₆	1	2	yes	—
67	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	—	8	2	no	—
69	C₂×C₄×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	#102	1	4	yes	—
71	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
72	C₂×C₄×C₆₇	Q₃₂	1	4	yes	—
73	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×Dic₆₇	—	4	4	no	—
80	C₂×Dic₆₇	—	8	2	no	—
81	C₂×Dic₆₇	—	8	2	no	—
82	Dic₁₃₄	—	4	4	no	—
83	Dic₁₃₄	—	4	4	no	—
84	Dic₁₃₄	—	8	2	no	—
85	Dic₁₃₄	—	8	2	no	—
86	C₈×C₆₇	#102	1	4	yes	—
87	C₈×C₆₇	C₄×D₄	1	2	yes	—
88	C₈×C₆₇	#121	1	2	yes	—
89	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
90	C₂²×C₆₇	#486	1	1	yes	characteristic
91	C₂²×C₆₇	#493	1	1	yes	characteristic
92	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
93	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
94	C₄×C₆₇	—	16	2	no	—
95	C₄×C₆₇	#358	1	8	yes	complemented
96	C₄×C₆₇	#390	1	1	yes	characteristic
97	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
98	C₄×C₆₇	#486	1	1	yes	characteristic
99	C₄×C₆₇	#492	1	1	yes	characteristic
100	Dic₆₇	—	8	4	no	—
101	#13203	—	67	1	no	maximal, Sylow
102	C₂×C₆₇	#1957	1	1	yes	characteristic
103	C₂×C₆₇	C₂×#358	1	2	yes	—
104	C₂×C₆₇	#2241	1	2	yes	—
105	C₂×C₆₇	#2247	1	1	yes	characteristic
106	C₂×C₆₇	#2248	1	1	yes	characteristic
107	C₂×#494	—	67	1	no	—
108	C₂×#358	—	67	4	no	—
109	C₂×C₄×C₁₆	—	67	1	no	—
110	#1845	D₆₇	1	1	yes	characteristic
111	C₆₇	#13203	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₁₆	—	67	2	no	—
113	C₂²×Q₁₆	—	134	1	no	—
114	C₂×C₄⋊Q₈	—	134	1	no	—
115	C₂²×C₁₆	—	67	2	no	—
116	#494	—	67	4	no	—
117	#494	—	134	2	no	—
118	C₂×#107	—	2	1	no	—
119	C₂×#107	C₂×D₆₇	1	2	yes	—
120	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
121	#384	—	2	1	no	—
122	#358	—	134	8	no	—
123	C₄×C₁₆	—	134	2	no	—
124	C₂²×Q₈	—	134	2	no	—
125	C₂×Q₁₆	—	268	2	no	—
126	C₂×Q₁₆	—	268	2	no	—
127	C₂×Q₁₆	—	67	4	no	—
128	C₂×Q₁₆	—	134	4	no	—
129	C₂×Q₁₆	—	134	4	no	—
130	C₂×Q₁₆	—	134	2	no	—
131	C₂×Q₁₆	—	134	4	no	—
132	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
133	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
134	C₄⋊Q₈	—	268	2	no	—
135	C₄⋊Q₈	—	134	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	4	2	no	—
138	C₂×C₄⋊C₄	—	268	2	no	—
139	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
140	C₂×C₁₆	—	67	4	no	—
141	C₂×C₁₆	—	134	2	no	—
142	C₂×C₁₆	—	134	2	no	—
143	#107	—	2	4	no	—
144	#107	—	4	2	no	—
145	C₄×C₈	C₄×D₆₇	1	2	yes	—
146	C₄×C₈	D₂₆₈	1	2	yes	—
147	C₂×Q₈	—	268	2	no	—
148	C₂×Q₈	—	268	2	no	—
149	C₂×Q₈	—	268	2	no	—
150	C₂×Q₈	—	268	2	no	—
151	C₂×Q₈	—	268	2	no	—
152	C₂×Q₈	—	134	4	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	—	268	2	no	—
155	C₂²×C₄	#11761613	1	2	yes	—
156	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
157	Q₁₆	—	268	8	no	—
158	Q₁₆	—	268	4	no	—
159	Q₁₆	—	134	8	no	—
160	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
161	C₂×C₈	#11761613	1	2	yes	—
162	C₂×C₈	#11761615	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
164	C₂×C₈	D₄×D₆₇	1	2	yes	—
165	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
166	C₂×C₈	#11761629	1	2	yes	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	8	4	no	—
170	C₄⋊C₄	—	268	8	no	—
171	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
172	C₄²	#11761615	1	2	yes	—
173	C₁₆	—	134	4	no	—
174	C₂³	C₂×#11761613	1	1	yes	characteristic
175	Q₈	—	268	4	no	—
176	Q₈	—	268	4	no	—
177	Q₈	—	536	2	no	—
178	Q₈	—	536	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	—	268	4	no	—
182	C₂×C₄	—	536	2	no	—
183	C₂×C₄	—	536	2	no	—
184	C₂×C₄	#14523963	1	4	yes	—
185	C₂×C₄	#14523987	1	4	yes	—
186	C₂×C₄	#14523989	1	4	yes	—
187	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
188	C₂×C₄	#14524032	1	1	yes	characteristic
189	C₂×C₄	#14524039	1	1	yes	characteristic
190	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
191	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
192	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
193	C₈	#14523963	1	4	yes	—
194	C₈	#14524005	1	2	yes	—
195	C₈	#14524018	1	2	yes	—
196	C₂²	C₂×#14523963	1	1	yes	characteristic
197	C₂²	#16863744	1	1	yes	characteristic
198	C₂²	#16863772	1	1	yes	characteristic
199	C₂²	C₂×#14523987	1	2	yes	—
200	C₂²	C₂×#14523989	1	2	yes	—
201	C₄	—	16	2	no	—
202	C₄	—	536	4	no	—
203	C₄	#16863199	1	8	yes	complemented
204	C₄	#16863297	1	1	yes	characteristic
205	C₄	C₂×#14523963	1	1	yes	characteristic
206	C₄	#16863744	1	1	yes	characteristic
207	C₄	#16863770	1	1	yes	characteristic
208	C₂	#22561014	1	1	yes	characteristic
209	C₂	C₂×#16863199	1	2	yes	—
210	C₂	#22562438	1	2	yes	—
211	C₂	#22562461	1	1	yes	characteristic
212	C₂	#22562464	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#13203
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
10. #16863776	C₄	semidirect
95. C₄×C₆₇	#358	semidirect
111. C₆₇	#13203	semidirect
203. C₄	#16863199	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863776	2
C₂×#16863199	2
C₂×C₄×C₆₇⋊C₁₆	2
C₆₇×#1845	2
#13203	67