Subgroups — #30774359

Order 17152.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
3	#22558298	C₂	1	4	yes	maximal
4	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
9	#16863172	—	2	1	no	—
10	C₄×C₆₇⋊C₁₆	—	2	1	no	—
11	#533×C₆₇	C₄	1	1	yes	characteristic
12	#446×C₆₇	—	2	1	no	—
13	#446×C₆₇	—	2	1	no	—
14	C₈×Q₈×C₆₇	—	2	1	no	—
15	C₈×Q₈×C₆₇	—	2	1	no	—
16	#424×C₆₇	C₄	1	2	yes	—
17	#423×C₆₇	C₄	1	1	yes	characteristic
18	#405×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#14523983	—	2	2	no	—
21	#14523983	Q₈	1	4	yes	complemented
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	4	no	—
25	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₆₇	C₈	1	1	yes	characteristic
29	C₄⋊Q₈×C₆₇	C₈	1	1	yes	characteristic
30	#125×C₆₇	C₈	1	2	yes	—
31	C₂²⋊Q₈×C₆₇	—	2	1	no	—
32	C₂²⋊Q₈×C₆₇	—	2	1	no	—
33	C₄×Q₈×C₆₇	—	2	1	no	—
34	C₄×Q₈×C₆₇	—	2	1	no	—
35	#117×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₆₇	—	2	2	no	—
38	C₄⋊C₈×C₆₇	—	2	2	no	—
39	C₄⋊C₈×C₆₇	—	2	1	no	—
40	C₄⋊C₈×C₆₇	—	2	1	no	—
41	#97×C₆₇	—	2	2	no	—
42	C₄×C₈×C₆₇	—	2	2	no	—
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₆₇⋊C₁₆	—	2	4	no	—
48	C₆₇⋊C₁₆	—	4	2	no	—
49	C₂×Q₈×C₆₇	—	2	1	no	—
50	C₂×Q₈×C₆₇	—	2	1	no	—
51	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	M₁₆×C₆₇	—	2	4	no	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₆₇	#41	1	2	yes	—
63	C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
68	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
69	C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
75	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
76	Q₈×C₆₇	—	4	2	no	—
77	Q₈×C₆₇	—	4	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	4	1	no	—
82	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
88	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#131	1	2	yes	—
90	C₈×C₆₇	—	2	1	no	—
91	C₈×C₆₇	—	2	1	no	—
92	C₈×C₆₇	—	4	2	no	—
93	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
94	C₈×C₆₇	#124	1	1	yes	characteristic
95	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
96	C₂²×C₆₇	—	2	1	no	—
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
99	C₂²×C₆₇	#446	1	2	yes	—
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	—	2	1	no	—
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	4	2	no	—
104	C₄×C₆₇	—	4	2	no	—
105	C₄×C₆₇	#350	1	2	yes	—
106	C₄×C₆₇	#391	1	1	yes	characteristic
107	C₄×C₆₇	#408	1	1	yes	characteristic
108	C₄×C₆₇	C₈×Q₈	1	2	yes	—
109	#13265	—	67	1	no	maximal, Sylow
110	C₂×C₆₇	—	2	2	no	—
111	C₂×C₆₇	#1984	1	1	yes	characteristic
112	C₂×C₆₇	C₂×#350	1	1	yes	characteristic
113	C₂×C₆₇	#2117	1	1	yes	characteristic
114	#2960	D₆₇	1	1	yes	characteristic
115	C₂×C₄⋊C₁₆	—	67	1	no	—
116	C₄×M₃₂	—	67	1	no	—
117	#1379	—	67	4	no	—
118	C₆₇	#13265	1	1	yes	characteristic, complemented, Sylow
119	#533	Dic₆₇	1	1	yes	characteristic
120	C₂×M₃₂	—	67	2	no	—
121	C₂²×C₁₆	—	67	2	no	—
122	#446	—	2	1	no	—
123	#446	—	2	1	no	—
124	C₈×Q₈	—	2	1	no	—
125	C₈×Q₈	—	2	1	no	—
126	#424	Dic₆₇	1	2	yes	—
127	#423	Dic₆₇	1	1	yes	characteristic
128	#405	C₂×D₆₇	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
130	C₄⋊C₁₆	—	134	1	no	—
131	C₄⋊C₁₆	—	134	1	no	—
132	#346	—	134	1	no	—
133	C₄×C₁₆	—	134	1	no	—
134	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
135	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
137	C₄⋊Q₈	C₆₇⋊C₈	1	1	yes	characteristic
138	C₄⋊Q₈	C₆₇⋊C₈	1	1	yes	characteristic
139	#125	C₆₇⋊C₈	1	2	yes	—
140	C₂²⋊Q₈	—	2	1	no	—
141	C₂²⋊Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	—	2	1	no	—
144	#117	C₂×Dic₆₇	1	2	yes	—
145	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
146	M₃₂	—	67	4	no	—
147	M₃₂	—	134	2	no	—
148	C₂×C₁₆	—	134	2	no	—
149	C₂×C₁₆	—	134	4	no	—
150	C₂×C₁₆	—	134	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	1	no	—
155	#97	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂²×C₄	#11761612	1	1	yes	characteristic
164	C₂²×C₄	#11761612	1	1	yes	characteristic
165	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
166	M₁₆	—	2	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
173	C₂×C₈	Q₈×D₆₇	1	2	yes	—
174	C₂×C₈	#11761629	1	2	yes	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
180	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
184	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
185	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
186	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
187	C₁₆	—	268	2	no	—
188	C₁₆	—	134	4	no	—
189	C₂³	C₂×#11761612	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
198	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
199	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
201	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
202	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
203	C₂×C₄	#14524050	1	2	yes	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	4	2	no	—
207	C₈	#14524019	1	2	yes	—
208	C₈	#14524032	1	1	yes	characteristic
209	C₈	#14524041	1	1	yes	characteristic
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
213	C₂²	#16863527	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	2	no	—
219	C₄	#16863180	1	2	yes	—
220	C₄	#16863303	1	1	yes	characteristic
221	C₄	#16863363	1	1	yes	characteristic
222	C₄	Q₈×C₆₇⋊C₈	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	#22561146	1	1	yes	characteristic
225	C₂	C₂×#16863180	1	1	yes	characteristic
226	C₂	#22561837	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2960	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#13265
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
21. #14523983	Q₈	semidirect
118. C₆₇	#13265	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₄×#14523983	2
#22558298	2
C₆₇×#2960	2
#13265	67