Subgroups — #30765641

Order 17152.

334 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569143	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22558187	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
5	#16863953	C₄	1	1	yes	characteristic
6	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
7	#16863526	—	2	1	no	—
8	#16863526	C₄	1	2	yes	—
9	C₈×Dic₁₃₄	—	2	1	no	—
10	C₈×Dic₁₃₄	C₂²	1	2	yes	—
11	#16863422	—	2	1	no	—
12	#16863420	C₄	1	1	yes	characteristic
13	#16863351	—	2	1	no	—
14	C₄×C₆₇⋊C₁₆	—	2	1	no	—
15	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
16	C₂×M₃₂×C₆₇	—	2	1	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₆₇	—	2	1	no	—
19	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂×#11761612	—	2	1	no	—
21	#14524043	C₈	1	1	yes	characteristic
22	#14524043	C₈	1	1	yes	characteristic
23	#14524035	—	2	1	no	—
24	#14524022	—	2	1	no	—
25	#14524022	C₈	1	2	yes	—
26	C₄×Dic₁₃₄	—	2	1	no	—
27	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
28	#14524002	—	2	1	no	—
29	C₂×C₆₇⋊C₁₆	—	2	4	no	—
30	C₂×C₆₇⋊C₁₆	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
33	Dic₆₇⋊C₈	—	2	2	no	—
34	Dic₆₇⋊C₈	—	2	2	no	—
35	#14523953	—	2	2	no	—
36	C₈×Dic₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₆₇	—	4	2	no	—
41	C₂×C₁₆×C₆₇	—	2	2	no	—
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₃₄	—	2	1	no	—
47	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
48	#11761612	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	2	2	no	—
50	C₄⋊Dic₆₇	—	2	1	no	—
51	C₄⋊Dic₆₇	—	2	1	no	—
52	C₄⋊Dic₆₇	—	2	1	no	—
53	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
54	Dic₆₇⋊C₄	—	2	2	no	—
55	Dic₆₇⋊C₄	—	2	2	no	—
56	C₂²⋊Dic₆₇	—	2	2	no	—
57	C₄×Dic₆₇	—	2	2	no	—
58	C₆₇⋊C₁₆	—	2	4	no	—
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
66	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
72	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₁₆×C₆₇	—	4	2	no	—
74	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#103	1	2	yes	—
81	C₂×C₄×C₆₇	#104	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₆₇	—	2	2	no	—
86	C₂×Dic₆₇	—	2	2	no	—
87	Dic₁₃₄	—	2	4	no	—
88	Dic₁₃₄	—	4	2	no	—
89	C₈×C₆₇	—	2	1	no	—
90	C₈×C₆₇	—	2	1	no	—
91	C₈×C₆₇	#103	1	2	yes	—
92	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₆₇⋊C₈	—	4	2	no	—
95	C₂²×C₆₇	—	2	1	no	—
96	C₂²×C₆₇	—	2	1	no	—
97	C₂²×C₆₇	Q₈⋊C₈	1	2	yes	—
98	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	Q₈⋊C₈	1	2	yes	—
102	C₄×C₆₇	#350	1	2	yes	—
103	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
104	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
105	C₄×C₆₇	#504	1	2	yes	—
106	Dic₆₇	—	4	2	no	—
107	Dic₆₇	—	4	2	no	—
108	#11608	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	—	2	2	no	—
110	C₂×C₆₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
111	C₂×C₆₇	#2113	1	1	yes	characteristic
112	C₂×C₆₇	C₂×#350	1	1	yes	characteristic
113	#2960	—	67	1	no	—
114	#2150	D₆₇	1	1	yes	characteristic
115	C₂×C₄×C₁₆	—	67	1	no	—
116	Q₈⋊C₁₆	—	67	4	no	—
117	C₆₇	#11608	1	1	yes	characteristic, complemented, Sylow
118	#533	—	67	1	no	—
119	C₂×M₃₂	—	2	1	no	—
120	C₂²×C₁₆	—	134	1	no	—
121	#446	—	67	2	no	—
122	#446	—	134	1	no	—
123	C₈×Q₈	—	67	2	no	—
124	C₈×Q₈	—	134	1	no	—
125	#424	—	134	1	no	—
126	#423	—	67	1	no	—
127	#405	—	134	1	no	—
128	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
129	C₄⋊C₁₆	—	2	1	no	—
130	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
131	C₄×C₁₆	—	67	2	no	—
132	C₄×C₁₆	—	134	1	no	—
133	C₂×M₁₆	—	134	1	no	—
134	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	67	1	no	—
137	C₄⋊Q₈	—	67	1	no	—
138	#125	—	134	1	no	—
139	C₂²⋊Q₈	—	67	2	no	—
140	C₂²⋊Q₈	—	134	1	no	—
141	C₄×Q₈	—	67	2	no	—
142	C₄×Q₈	—	134	1	no	—
143	#117	—	134	1	no	—
144	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
145	M₃₂	—	4	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	134	4	no	—
148	C₂×C₁₆	—	134	2	no	—
149	C₄⋊C₈	—	67	2	no	—
150	C₄⋊C₈	—	134	2	no	—
151	C₄⋊C₈	—	134	2	no	—
152	C₄⋊C₈	—	134	1	no	—
153	#97	—	134	2	no	—
154	C₄×C₈	—	134	2	no	—
155	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
156	C₄×C₈	D₂₆₈	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
159	C₂×Q₈	—	67	2	no	—
160	C₂×Q₈	—	134	1	no	—
161	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
163	C₂²×C₄	#11761611	1	1	yes	characteristic
164	M₁₆	—	268	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	134	2	no	—
167	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
168	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
169	C₂×C₈	#11761616	1	2	yes	—
170	C₂×C₈	#11761618	1	2	yes	—
171	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
172	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
173	C₄⋊C₄	—	67	2	no	—
174	C₄⋊C₄	—	134	1	no	—
175	C₄⋊C₄	—	134	1	no	—
176	C₄⋊C₄	—	134	2	no	—
177	C₄⋊C₄	—	134	1	no	—
178	C₄⋊C₄	—	134	2	no	—
179	C₂²⋊C₄	—	134	2	no	—
180	C₄²	—	134	2	no	—
181	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
182	C₄²	C₈×D₆₇	1	1	yes	characteristic
183	C₄²	#11761611	1	1	yes	characteristic
184	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
185	C₁₆	—	4	2	no	—
186	C₁₆	—	134	4	no	—
187	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
188	Q₈	—	268	2	no	—
189	Q₈	—	134	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	134	2	no	—
193	C₂×C₄	—	134	2	no	—
194	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
195	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
196	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
197	C₂×C₄	#14523966	1	2	yes	—
198	C₂×C₄	#14523969	1	2	yes	—
199	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
200	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
201	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	268	2	no	—
205	C₈	#14523966	1	2	yes	—
206	C₈	C₂×#11761616	1	1	yes	characteristic
207	C₈	C₂×#11761618	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#16863126	1	2	yes	—
211	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	268	2	no	—
215	C₄	—	268	2	no	—
216	C₄	#16863126	1	2	yes	—
217	C₄	#16863181	1	2	yes	—
218	C₄	C₂×#14523966	1	1	yes	characteristic
219	C₄	C₂×#14523969	1	1	yes	characteristic
220	C₄	#16863814	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	C₂×#16863126	1	1	yes	characteristic
223	C₂	#22561824	1	1	yes	characteristic
224	C₂	C₂×#16863181	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₇	C₂²⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#11608
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #22558187	C₂	semidirect
117. C₆₇	#11608	semidirect

Maximal subgroups

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

subgroup	index
#22569143	2
C₂×C₄×C₆₇⋊C₁₆	2
#22558187	2
C₆₇×#2150	2
#11608	67