Subgroups — #31011361

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	#22570723	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863798	C₂	1	1	yes	maximal, characteristic
3	#22562495	C₂	1	4	yes	maximal, complemented
4	#22562488	C₂	1	4	yes	maximal, complemented
5	#22561997	C₂	1	4	yes	maximal
6	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863953	—	2	1	no	—
8	C₂×#14524035	—	2	1	no	—
9	#16863801	C₂²	1	2	yes	—
10	#16863798	—	2	1	no	—
11	#16863798	—	2	1	no	—
12	#16863798	—	2	1	no	—
13	#16863798	C₂²	1	2	yes	—
14	#16863798	C₂²	1	2	yes	—
15	#16863798	C₂²	1	2	yes	—
16	#16863794	—	2	1	no	—
17	#16863434	—	2	1	no	—
18	#16863434	C₂²	1	2	yes	—
19	C₂×#14523977	C₂²	1	2	yes	—
20	C₂×#14523975	—	2	1	no	—
21	#16863224	C₂²	1	8	yes	—
22	#16863222	C₂²	1	8	yes	—
23	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#346×C₆₇	C₂²	1	2	yes	—
26	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
27	#14524043	—	2	2	no	—
28	#14524035	—	2	2	no	—
29	#14524035	—	2	2	no	—
30	#14524035	—	2	2	no	—
31	#14524035	—	4	2	no	—
32	#14524022	—	4	2	no	—
33	C₄×Dic₁₃₄	—	4	2	no	—
34	#14524002	—	2	2	no	—
35	C₂×C₄⋊Dic₆₇	—	2	2	no	—
36	C₂×Dic₆₇⋊C₄	—	4	2	no	—
37	#14523977	—	2	2	no	—
38	#14523977	—	2	2	no	—
39	#14523977	C₂³	1	4	yes	—
40	#14523977	C₂³	1	4	yes	—
41	#14523975	—	2	2	no	—
42	#14523975	—	2	2	no	—
43	#14523975	—	2	2	no	—
44	#14523975	—	2	2	no	—
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	M₃₂×C₆₇	Q₈	1	8	yes	complemented
49	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
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	C₂×Dic₁₃₄	—	4	2	no	—
55	C₂²×Dic₆₇	—	4	2	no	—
56	C₄⋊Dic₆₇	—	2	4	no	—
57	C₄⋊Dic₆₇	—	2	4	no	—
58	C₄⋊Dic₆₇	—	4	2	no	—
59	C₄⋊Dic₆₇	—	4	2	no	—
60	Dic₆₇⋊C₄	—	4	4	no	—
61	Dic₆₇⋊C₄	—	4	4	no	—
62	Dic₆₇⋊C₄	—	4	4	no	—
63	C₂²⋊Dic₆₇	—	4	2	no	—
64	C₄×Dic₆₇	—	4	2	no	—
65	C₂²×C₄×C₆₇	D₈	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
68	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
72	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₄²×C₆₇	D₈	1	2	yes	—
74	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₆₇	C₂×Q₈	1	8	yes	—
77	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₆₇	—	4	4	no	—
86	C₂×Dic₆₇	—	4	4	no	—
87	C₂×Dic₆₇	—	8	4	no	—
88	Dic₁₃₄	—	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	#500	1	2	yes	—
99	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
100	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
101	Dic₆₇	—	8	4	no	—
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	—	67	1	no	—
108	C₂×#499	—	67	1	no	—
109	#2255	—	67	4	no	—
110	#2253	—	67	4	no	—
111	#2159	—	67	4	no	—
112	C₄×M₃₂	D₆₇	1	1	yes	characteristic
113	C₆₇	#33732	1	1	yes	characteristic, complemented, Sylow
114	#533	—	134	1	no	—
115	C₂×#125	—	134	1	no	—
116	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
117	#500	—	67	2	no	—
118	#499	—	67	2	no	—
119	#499	—	67	2	no	—
120	#499	—	67	2	no	—
121	#499	—	134	1	no	—
122	#499	—	134	1	no	—
123	#499	—	134	1	no	—
124	#498	—	134	1	no	—
125	#427	—	67	2	no	—
126	#427	—	134	1	no	—
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	C₂×D₆₇	1	2	yes	—
133	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
134	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	134	2	no	—
137	#125	—	268	2	no	—
138	#125	—	134	2	no	—
139	#125	—	134	2	no	—
140	#125	—	134	2	no	—
141	C₂²⋊Q₈	—	268	2	no	—
142	C₄×Q₈	—	268	2	no	—
143	#117	—	134	2	no	—
144	C₂×C₄⋊C₄	—	268	2	no	—
145	C₂×C₄⋊C₄	—	134	2	no	—
146	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
147	M₃₂	Dic₁₃₄	1	8	yes	complemented
148	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
149	#107	—	67	4	no	—
150	#107	—	67	4	no	—
151	#107	—	134	2	no	—
152	#107	—	134	2	no	—
153	#106	—	134	2	no	—
154	#106	—	134	2	no	—
155	#106	—	134	2	no	—
156	#106	—	134	2	no	—
157	C₄×C₈	D₂₆₈	1	1	yes	characteristic
158	C₄×C₈	D₂₆₈	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	—	268	2	no	—
162	C₂²×C₄	—	268	2	no	—
163	C₂²×C₄	D₅₃₆	1	2	yes	—
164	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
165	C₂×C₈	Dic₂₆₈	1	4	yes	—
166	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
167	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
168	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
169	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
170	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
171	C₄⋊C₄	—	268	4	no	—
172	C₄⋊C₄	—	268	4	no	—
173	C₄⋊C₄	—	268	2	no	—
174	C₄⋊C₄	—	268	4	no	—
175	C₄⋊C₄	—	268	2	no	—
176	C₄⋊C₄	—	134	4	no	—
177	C₄⋊C₄	—	134	4	no	—
178	C₂²⋊C₄	—	268	2	no	—
179	C₄²	—	268	2	no	—
180	C₄²	D₅₃₆	1	2	yes	—
181	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
182	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
183	C₁₆	C₂×Dic₁₃₄	1	8	yes	—
184	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
185	Q₈	—	536	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	268	4	no	—
188	C₂×C₄	—	268	4	no	—
189	C₂×C₄	—	536	4	no	—
190	C₂×C₄	#14524043	1	2	yes	—
191	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
192	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
193	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
194	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
195	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
196	C₈	#14524043	1	2	yes	—
197	C₈	C₂×Dic₂₆₈	1	2	yes	—
198	C₈	C₂×Dic₂₆₈	1	2	yes	—
199	C₈	C₂²×Dic₁₃₄	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	#16863801	1	2	yes	—
202	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	536	4	no	—
206	C₄	—	536	4	no	—
207	C₄	#16863801	1	2	yes	—
208	C₄	C₂×#14524043	1	1	yes	characteristic
209	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#16863801	1	1	yes	characteristic
212	C₂	#22573345	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#33732
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22562495	C₂	semidirect
4. #22562488	C₂	semidirect
48. M₃₂×C₆₇	Q₈	semidirect
113. C₆₇	#33732	semidirect
147. M₃₂	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
C₂×#16863798	2
#22562495	2
#22562488	2
#22561997	2
C₄×M₃₂×C₆₇	2
#33732	67