Subgroups — #31915992

Order 18176.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590096	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556068	C₂	1	1	yes	maximal, characteristic
3	#22586424	C₂	1	4	yes	maximal
4	C₂×#446×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	C₂²	1	2	yes	—
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	#17985775	—	2	1	no	—
10	C₄×C₇₁⋊C₁₆	—	2	1	no	—
11	C₂×C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
12	#446×C₇₁	—	2	1	no	—
13	#446×C₇₁	—	2	1	no	—
14	#446×C₇₁	—	2	1	no	—
15	#446×C₇₁	—	2	1	no	—
16	C₂×C₄⋊C₈×C₇₁	C₄	1	2	yes	—
17	C₂×C₄⋊C₈×C₇₁	C₄	1	1	yes	characteristic
18	C₂×#97×C₇₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	#14556068	—	2	2	no	—
21	#14556068	—	2	4	no	—
22	#14556068	Q₈	1	4	yes	complemented
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	C₂²×Q₈×C₇₁	C₈	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₇₁	—	2	1	no	—
30	C₄×Q₈×C₇₁	—	2	1	no	—
31	C₄×Q₈×C₇₁	—	2	1	no	—
32	C₄×Q₈×C₇₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₁	C₈	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₇₁	C₈	1	2	yes	—
35	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₇₁	—	2	2	no	—
38	C₄⋊C₈×C₇₁	—	2	2	no	—
39	C₄⋊C₈×C₇₁	—	2	1	no	—
40	C₄⋊C₈×C₇₁	—	2	1	no	—
41	#97×C₇₁	—	2	2	no	—
42	#97×C₇₁	—	2	2	no	—
43	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₇₁⋊C₁₆	—	2	4	no	—
48	C₇₁⋊C₁₆	—	4	2	no	—
49	C₂×Q₈×C₇₁	—	2	1	no	—
50	C₂×Q₈×C₇₁	—	2	1	no	—
51	C₂×Q₈×C₇₁	—	4	2	no	—
52	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	—	2	2	no	—
59	C₂×C₈×C₇₁	—	2	4	no	—
60	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
65	C₂×C₈×C₇₁	#41	1	2	yes	—
66	C₄⋊C₄×C₇₁	—	2	1	no	—
67	C₄⋊C₄×C₇₁	—	2	1	no	—
68	C₄⋊C₄×C₇₁	—	2	2	no	—
69	C₄⋊C₄×C₇₁	—	2	2	no	—
70	C₄²×C₇₁	—	2	2	no	—
71	C₄²×C₇₁	—	2	2	no	—
72	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
75	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
76	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
77	Q₈×C₇₁	—	4	2	no	—
78	Q₈×C₇₁	—	4	2	no	—
79	C₂×C₄×C₇₁	—	2	2	no	—
80	C₂×C₄×C₇₁	—	2	2	no	—
81	C₂×C₄×C₇₁	—	2	2	no	—
82	C₂×C₄×C₇₁	—	4	1	no	—
83	C₂×C₄×C₇₁	—	4	2	no	—
84	C₂×C₄×C₇₁	—	4	2	no	—
85	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
91	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#131	1	2	yes	—
93	C₈×C₇₁	—	2	1	no	—
94	C₈×C₇₁	—	2	1	no	—
95	C₈×C₇₁	—	4	2	no	—
96	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
97	C₈×C₇₁	#124	1	1	yes	characteristic
98	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
99	C₂²×C₇₁	—	2	1	no	—
100	C₂²×C₇₁	—	2	1	no	—
101	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×C₇₁	C₈×Q₈	1	2	yes	—
103	C₄×C₇₁	—	2	1	no	—
104	C₄×C₇₁	—	2	1	no	—
105	C₄×C₇₁	—	2	1	no	—
106	C₄×C₇₁	—	4	2	no	—
107	C₄×C₇₁	—	4	2	no	—
108	C₄×C₇₁	#349	1	2	yes	—
109	C₄×C₇₁	#391	1	1	yes	characteristic
110	C₄×C₇₁	#408	1	1	yes	characteristic
111	C₄×C₇₁	#446	1	2	yes	—
112	#13268	—	71	1	no	maximal, Sylow
113	C₂×C₇₁	—	2	2	no	—
114	C₂×C₇₁	#1984	1	1	yes	characteristic
115	C₂×C₇₁	C₂×#349	1	1	yes	characteristic
116	C₂×C₇₁	#2115	1	1	yes	characteristic
117	C₂×#446	D₇₁	1	1	yes	characteristic
118	#2150	—	71	1	no	—
119	C₄×M₃₂	—	71	1	no	—
120	#1377	—	71	4	no	—
121	C₇₁	#13268	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄×Q₈	Dic₇₁	1	1	yes	characteristic
123	C₂×M₃₂	—	71	2	no	—
124	C₂×M₃₂	—	71	2	no	—
125	#446	—	2	1	no	—
126	#446	—	2	1	no	—
127	#446	—	2	1	no	—
128	#446	—	2	1	no	—
129	C₂×C₄⋊C₈	Dic₇₁	1	2	yes	—
130	C₂×C₄⋊C₈	Dic₇₁	1	1	yes	characteristic
131	C₂×#97	C₂×D₇₁	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
133	C₄⋊C₁₆	—	142	1	no	—
134	C₄⋊C₁₆	—	142	1	no	—
135	#346	—	142	1	no	—
136	C₄×C₁₆	—	142	1	no	—
137	C₂²×Q₈	C₇₁⋊C₈	1	1	yes	characteristic
138	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
139	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	—	2	1	no	—
144	C₄×Q₈	—	2	1	no	—
145	C₂×C₄⋊C₄	C₇₁⋊C₈	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₇₁⋊C₈	1	2	yes	—
147	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
148	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
149	M₃₂	—	71	4	no	—
150	M₃₂	—	142	4	no	—
151	M₃₂	—	142	2	no	—
152	C₂×C₁₆	—	142	2	no	—
153	C₂×C₁₆	—	142	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	2	1	no	—
158	#97	—	2	2	no	—
159	#97	—	2	2	no	—
160	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
169	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
170	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
179	C₂×C₈	Q₈×D₇₁	1	2	yes	—
180	C₂×C₈	#11765866	1	2	yes	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
189	C₄²	#11765849	1	1	yes	characteristic
190	C₄²	#11765849	1	1	yes	characteristic
191	C₁₆	—	284	2	no	—
192	C₁₆	—	142	4	no	—
193	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
194	Q₈	—	4	2	no	—
195	Q₈	—	4	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
204	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
205	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
206	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
207	C₂×C₄	Q₈×Dic₇₁	1	2	yes	—
208	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
209	C₂×C₄	#14556135	1	2	yes	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	#14556104	1	2	yes	—
214	C₈	#14556117	1	1	yes	characteristic
215	C₈	#14556126	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
219	C₂²	Q₈×C₇₁⋊C₈	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	#17985781	1	2	yes	—
226	C₄	#17985906	1	1	yes	characteristic
227	C₄	#17985966	1	1	yes	characteristic
228	C₄	#17986130	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	#22589277	1	1	yes	characteristic
231	C₂	C₂×#17985781	1	1	yes	characteristic
232	C₂	#22589961	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇₁
commutator, prime radical	C₂³×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#446×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#446	D₇₁
hypocenter	C₇₁	#13268
LCS	G ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
22. #14556068	Q₈	semidirect
121. C₇₁	#13268	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
C₄×#14556068	2
#22586424	2
C₂×#446×C₇₁	2
#13268	71