Subgroups — #31011362

Order 17152.

482 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×#16863798	C₂	1	1	yes	maximal, characteristic
2	#22562492	C₂	1	4	yes	maximal, complemented
3	#22562485	C₂	1	4	yes	maximal, complemented
4	#22561998	C₂	1	4	yes	maximal
5	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524035	—	2	1	no	—
8	C₂×#14523983	C₂²	1	2	yes	—
9	#16863798	—	2	1	no	—
10	#16863798	—	2	1	no	—
11	#16863798	C₂²	1	2	yes	—
12	#16863798	C₂²	1	2	yes	—
13	C₂×#14523977	C₂²	1	2	yes	—
14	C₂×#14523975	—	2	1	no	—
15	#16863223	C₂²	1	8	yes	—
16	#16863221	C₂²	1	8	yes	—
17	#16863172	C₂²	1	2	yes	—
18	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
19	#533×C₆₇	—	2	1	no	—
20	#500×C₆₇	C₂²	1	2	yes	—
21	#499×C₆₇	—	2	1	no	—
22	#499×C₆₇	C₂²	1	2	yes	—
23	#498×C₆₇	—	2	1	no	—
24	#427×C₆₇	—	2	1	no	—
25	#427×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#14524035	—	2	2	no	—
28	#14524035	—	2	2	no	—
29	#14524035	—	4	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×Dic₆₇⋊C₄	—	4	2	no	—
32	#14523983	Q₈	1	8	yes	complemented
33	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
34	#14523977	—	2	2	no	—
35	#14523977	C₂³	1	4	yes	—
36	#14523975	—	2	2	no	—
37	#14523975	—	2	2	no	—
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₆₇	—	2	2	no	—
41	#125×C₆₇	—	2	2	no	—
42	C₂²⋊Q₈×C₆₇	—	4	2	no	—
43	C₄×Q₈×C₆₇	—	4	2	no	—
44	#117×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
46	#107×C₆₇	—	2	2	no	—
47	#107×C₆₇	C₂³	1	4	yes	—
48	#106×C₆₇	—	2	2	no	—
49	#106×C₆₇	—	2	2	no	—
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×Dic₆₇	—	4	2	no	—
55	C₄⋊Dic₆₇	—	2	4	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	Dic₆₇⋊C₄	—	4	4	no	—
58	Dic₆₇⋊C₄	—	4	4	no	—
59	C₆₇⋊C₁₆	C₂×Q₈	1	8	yes	—
60	C₂×Q₈×C₆₇	—	4	2	no	—
61	C₂²×C₄×C₆₇	D₈	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
64	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
68	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₄⋊C₄×C₆₇	—	4	4	no	—
71	C₄⋊C₄×C₆₇	—	4	2	no	—
72	C₂²⋊C₄×C₆₇	—	4	2	no	—
73	C₄²×C₆₇	—	4	2	no	—
74	C₄²×C₆₇	D₈	1	2	yes	—
75	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₆₇	—	8	4	no	—
79	C₂×C₄×C₆₇	—	4	2	no	—
80	C₂×C₄×C₆₇	—	4	4	no	—
81	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
87	C₂×Dic₆₇	—	4	4	no	—
88	C₂×Dic₆₇	—	8	4	no	—
89	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
90	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
91	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
92	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
93	C₂²×C₆₇	—	2	2	no	—
94	C₂²×C₆₇	#500	1	2	yes	—
95	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
96	C₄×C₆₇	—	2	2	no	—
97	C₄×C₆₇	—	2	2	no	—
98	C₄×C₆₇	—	8	4	no	—
99	C₄×C₆₇	#500	1	2	yes	—
100	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₆₇	—	8	4	no	—
103	#33732	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	—	2	2	no	—
105	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
106	C₂×C₆₇	#3414	1	2	yes	—
107	#3160	D₆₇	1	1	yes	characteristic
108	C₂×#499	—	67	1	no	—
109	#2255	—	67	4	no	—
110	#2253	—	67	4	no	—
111	#2159	—	67	4	no	—
112	C₄×M₃₂	—	67	1	no	—
113	C₆₇	#33732	1	1	yes	characteristic, complemented, Sylow
114	#533	—	2	1	no	—
115	C₂×#125	—	134	1	no	—
116	C₂×M₃₂	—	67	2	no	—
117	#500	C₂×D₆₇	1	2	yes	—
118	#499	—	2	1	no	—
119	#499	—	67	2	no	—
120	#499	—	67	2	no	—
121	#499	—	134	1	no	—
122	#499	—	134	1	no	—
123	#499	C₂×D₆₇	1	2	yes	—
124	#498	—	2	1	no	—
125	#427	—	2	1	no	—
126	#427	C₂×D₆₇	1	2	yes	—
127	C₂×#107	—	67	2	no	—
128	C₂×#106	—	134	1	no	—
129	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
130	#367	—	67	8	no	—
131	#366	—	67	8	no	—
132	#346	—	67	2	no	—
133	C₄×C₁₆	—	67	2	no	—
134	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	2	no	—
137	#125	—	2	2	no	—
138	#125	—	268	2	no	—
139	#125	—	134	2	no	—
140	#125	—	134	2	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₄×Q₈	—	4	2	no	—
143	#117	—	2	2	no	—
144	C₂×C₄⋊C₄	—	268	2	no	—
145	C₂×C₄⋊C₄	—	134	2	no	—
146	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
147	M₃₂	—	67	8	no	—
148	C₂×C₁₆	—	67	4	no	—
149	#107	—	2	2	no	—
150	#107	—	67	4	no	—
151	#107	—	134	2	no	—
152	#107	C₂²×D₆₇	1	4	yes	—
153	#106	—	2	2	no	—
154	#106	—	2	2	no	—
155	#106	—	134	2	no	—
156	#106	—	134	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	—	4	2	no	—
162	C₂²×C₄	—	268	2	no	—
163	C₂²×C₄	#11761613	1	2	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂×C₈	#11761618	1	4	yes	—
166	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₆₇	1	4	yes	—
170	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	268	4	no	—
175	C₄⋊C₄	—	268	4	no	—
176	C₄⋊C₄	—	268	2	no	—
177	C₄⋊C₄	—	134	4	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	#11761613	1	2	yes	—
181	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
183	C₁₆	—	67	8	no	—
184	C₂³	C₂×#11761613	1	1	yes	characteristic
185	Q₈	—	8	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	268	4	no	—
189	C₂×C₄	—	536	4	no	—
190	C₂×C₄	#14524041	1	2	yes	—
191	C₂×C₄	C₂×#11761613	1	2	yes	—
192	C₂×C₄	C₂×#11761613	1	2	yes	—
193	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
194	C₂×C₄	C₂×#11761618	1	2	yes	—
195	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
196	C₈	#14524041	1	2	yes	—
197	C₈	C₂×#11761618	1	2	yes	—
198	C₈	C₂×#11761618	1	2	yes	—
199	C₈	C₂×Q₈×D₆₇	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	#16863799	1	2	yes	—
202	C₂²	C₂²×#11761613	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	—	536	4	no	—
207	C₄	#16863799	1	2	yes	—
208	C₄	C₂×#14524041	1	1	yes	characteristic
209	C₄	C₂²×#11761618	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#16863799	1	1	yes	characteristic
212	C₂	#22573343	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3160	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#3160	D₆₇
hypocenter	C₆₇	#33732
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #22562492	C₂	semidirect
3. #22562485	C₂	semidirect
32. #14523983	Q₈	semidirect
113. C₆₇	#33732	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863798	2
#22562492	2
#22562485	2
#22561998	2
C₄×#14523983	2
C₆₇×#3160	2
#33732	67