Subgroups — #32150945

Order 18176.

506 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597272	C₂	1	1	yes	maximal, characteristic
2	#22590247	C₂	1	4	yes	maximal, complemented
3	#22590242	C₂	1	4	yes	maximal, complemented
4	#22590098	C₂	1	1	yes	maximal, characteristic
5	#22590095	C₂	1	1	yes	maximal, characteristic
6	#22589939	C₂	1	1	yes	maximal, characteristic
7	#22589937	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
9	C₇₁×#2151	C₂	1	1	yes	maximal, characteristic
10	#17986552	C₄	1	1	yes	characteristic
11	C₂×#14556068	C₂²	1	1	yes	characteristic
12	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#17986127	C₄	1	4	yes	—
14	C₈×Dic₁₄₂	C₂²	1	4	yes	—
15	#17986027	C₄	1	1	yes	characteristic
16	#17986026	C₄	1	1	yes	characteristic
17	#17986022	C₄	1	1	yes	characteristic
18	#17985954	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
20	Dic₇₁⋊C₁₆	C₂²	1	4	yes	—
21	Dic₇₁⋊C₁₆	C₂²	1	4	yes	—
22	#17985775	C₂²	1	2	yes	—
23	#17985774	C₂²	1	2	yes	—
24	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
25	C₁₆×Dic₇₁	C₂²	1	2	yes	—
26	C₁₆×Dic₇₁	C₂²	1	2	yes	—
27	C₁₆×Dic₇₁	C₂²	1	2	yes	—
28	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
29	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
30	#405×C₇₁	C₂²	1	1	yes	characteristic
31	C₄⋊C₁₆×C₇₁	C₂²	1	4	yes	—
32	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
34	#14556127	C₈	1	2	yes	—
35	#14556119	C₈	1	2	yes	—
36	#14556105	C₈	1	4	yes	—
37	C₄×Dic₁₄₂	C₂×C₄	1	4	yes	—
38	#14556087	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
40	#14556068	—	2	2	no	—
41	C₂×C₇₁⋊C₁₆	Q₈	1	4	yes	complemented
42	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
43	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
45	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
46	Dic₇₁⋊C₈	C₂×C₄	1	4	yes	—
47	Dic₇₁⋊C₈	C₂×C₄	1	4	yes	—
48	#14556038	C₂×C₄	1	2	yes	—
49	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
50	C₈×Dic₇₁	C₂³	1	2	yes	—
51	C₈×Dic₇₁	C₂³	1	2	yes	—
52	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
54	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
55	M₃₂×C₇₁	—	2	2	no	—
56	C₂×C₁₆×C₇₁	—	2	2	no	—
57	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
58	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
59	#97×C₇₁	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₇₁	C₂³	1	2	yes	—
61	C₂×Dic₁₄₂	C₂×C₈	1	4	yes	—
62	C₂²×Dic₇₁	C₂×C₈	1	2	yes	—
63	#11765849	—	2	2	no	—
64	C₂×C₇₁⋊C₈	—	2	2	no	—
65	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
66	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
67	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
68	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
69	Dic₇₁⋊C₄	C₂×C₈	1	4	yes	—
70	Dic₇₁⋊C₄	C₂×C₈	1	4	yes	—
71	C₂²⋊Dic₇₁	C₂×C₈	1	2	yes	—
72	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
73	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
74	C₄×Dic₇₁	C₂²×C₄	1	2	yes	—
75	C₄×Dic₇₁	C₂²×C₄	1	2	yes	—
76	C₇₁⋊C₁₆	—	2	2	no	—
77	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	—
78	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
80	M₁₆×C₇₁	—	2	2	no	—
81	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₇₁	#41	1	2	yes	—
85	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
87	C₂²⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
88	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
89	C₁₆×C₇₁	—	2	2	no	—
90	C₁₆×C₇₁	—	2	2	no	—
91	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₇₁	#131	1	2	yes	—
96	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₇₁	—	2	4	no	—
98	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
99	C₂×Dic₇₁	C₂²×C₈	1	4	yes	—
100	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
101	Dic₁₄₂	—	2	8	no	—
102	C₈×C₇₁	—	2	2	no	—
103	C₈×C₇₁	C₄×Q₈	1	2	yes	—
104	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
105	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
106	C₇₁⋊C₈	—	2	2	no	—
107	C₇₁⋊C₈	—	2	2	no	—
108	C₂²×C₇₁	—	2	2	no	—
109	C₂²×C₇₁	C₈×Q₈	1	2	yes	—
110	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
111	C₄×C₇₁	—	2	4	no	—
112	C₄×C₇₁	C₈×Q₈	1	2	yes	—
113	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
115	Dic₇₁	—	2	4	no	—
116	Dic₇₁	—	2	2	no	—
117	Dic₇₁	#504	1	4	yes	—
118	#33474	—	71	1	no	maximal, Sylow
119	C₂×C₇₁	—	2	2	no	—
120	C₂×C₇₁	C₂×C₈×Q₈	1	1	yes	characteristic
121	C₂×C₇₁	C₂×#504	1	2	yes	—
122	#2960	—	71	1	no	—
123	#2183	—	71	4	no	—
124	Q₈×C₁₆	—	71	4	no	—
125	#2151	—	71	1	no	—
126	#2151	D₇₁	1	1	yes	characteristic
127	#2150	—	71	1	no	—
128	#2108	—	71	1	no	—
129	#2108	—	71	1	no	—
130	C₂×C₄×C₁₆	—	71	1	no	—
131	C₇₁	#33474	1	1	yes	characteristic, complemented, Sylow
132	#533	—	71	1	no	—
133	C₂×M₃₂	—	71	1	no	—
134	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
135	C₂²×C₁₆	—	71	1	no	—
136	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
137	#446	—	71	4	no	—
138	C₈×Q₈	—	71	4	no	—
139	#424	—	71	1	no	—
140	#424	—	71	1	no	—
141	#423	—	71	1	no	—
142	#405	—	71	1	no	—
143	#405	C₂×D₇₁	1	1	yes	characteristic
144	C₂×C₄×C₈	—	71	1	no	—
145	C₄⋊C₁₆	—	71	4	no	—
146	C₄⋊C₁₆	—	71	4	no	—
147	C₄⋊C₁₆	C₂×D₇₁	1	4	yes	—
148	#346	—	71	2	no	—
149	#346	—	71	2	no	—
150	C₄×C₁₆	—	71	2	no	—
151	C₄×C₁₆	—	71	2	no	—
152	C₄×C₁₆	—	71	2	no	—
153	C₄×C₁₆	—	71	2	no	—
154	C₂×M₁₆	—	71	1	no	—
155	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
156	C₂²×C₈	—	71	1	no	—
157	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
158	C₄⋊Q₈	—	71	2	no	—
159	#125	—	71	2	no	—
160	C₂²⋊Q₈	—	71	4	no	—
161	C₄×Q₈	—	71	4	no	—
162	#117	—	71	1	no	—
163	#117	C₄×D₇₁	1	1	yes	characteristic
164	C₂×C₄²	—	71	1	no	—
165	M₃₂	—	2	2	no	—
166	M₃₂	—	142	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	71	4	no	—
169	C₂×C₁₆	—	71	1	no	—
170	C₂×C₁₆	—	71	1	no	—
171	C₂×C₁₆	—	71	2	no	—
172	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
173	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
174	C₄⋊C₈	—	71	4	no	—
175	C₄⋊C₈	—	71	4	no	—
176	C₄⋊C₈	—	71	4	no	—
177	#97	—	71	2	no	—
178	#97	C₄×D₇₁	1	2	yes	—
179	C₄×C₈	—	71	2	no	—
180	C₄×C₈	—	71	2	no	—
181	C₄×C₈	—	71	2	no	—
182	C₄×C₈	C₂²×D₇₁	1	2	yes	—
183	C₂×Q₈	—	71	4	no	—
184	C₂²×C₄	—	71	2	no	—
185	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	142	2	no	—
188	C₂×C₈	—	71	2	no	—
189	C₂×C₈	—	71	2	no	—
190	C₂×C₈	—	142	2	no	—
191	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
192	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
193	C₂×C₈	Q₈×D₇₁	1	2	yes	—
194	C₂×C₈	#11765864	1	2	yes	—
195	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
196	C₄⋊C₄	—	71	4	no	—
197	C₄⋊C₄	—	71	4	no	—
198	C₄⋊C₄	—	71	2	no	—
199	C₄⋊C₄	—	71	4	no	—
200	C₄⋊C₄	C₈×D₇₁	1	2	yes	—
201	C₂²⋊C₄	—	71	2	no	—
202	C₂²⋊C₄	C₈×D₇₁	1	2	yes	—
203	C₄²	—	71	2	no	—
204	C₄²	—	71	2	no	—
205	C₄²	—	71	1	no	—
206	C₄²	—	71	1	no	—
207	C₄²	C₂×C₄×D₇₁	1	2	yes	—
208	C₁₆	—	2	2	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	—	71	2	no	—
211	C₁₆	—	71	2	no	—
212	C₁₆	—	142	2	no	—
213	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
214	Q₈	—	142	8	no	—
215	C₂×C₄	—	71	2	no	—
216	C₂×C₄	—	71	2	no	—
217	C₂×C₄	—	71	4	no	—
218	C₂×C₄	—	142	4	no	—
219	C₂×C₄	#14556093	1	2	yes	—
220	C₂×C₄	C₂×C₈×D₇₁	1	4	yes	—
221	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
222	C₂×C₄	#14556136	1	2	yes	—
223	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	142	2	no	—
226	C₈	—	142	2	no	—
227	C₈	#14556093	1	2	yes	—
228	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
229	C₈	C₂×#11765864	1	1	yes	characteristic
230	C₂²	—	2	2	no	—
231	C₂²	#17986123	1	2	yes	—
232	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	71	4	no	—
235	C₄	—	142	2	no	—
236	C₄	—	142	4	no	—
237	C₄	#17986123	1	2	yes	—
238	C₄	C₂×#14556093	1	1	yes	characteristic
239	C₄	C₂×#14556136	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	C₂×#17986123	1	1	yes	characteristic
242	C₂	#504×D₇₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₇₁×#2151	C₂
prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2151	D₇₁
hypocenter	C₇₁	#33474
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
2. #22590247	C₂	semidirect
3. #22590242	C₂	semidirect
41. C₂×C₇₁⋊C₁₆	Q₈	semidirect
131. C₇₁	#33474	semidirect

Maximal subgroups

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

subgroup	index
#22597272	2
#22590247	2
#22590242	2
#22590098	2
#22590095	2
#22589939	2
#22589937	2
C₂×C₁₆×Dic₇₁	2
C₇₁×#2151	2
#33474	71