Subgroups — #31908787

Order 18176.

310 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590139	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
3	#22586398	C₂	1	2	yes	maximal, complemented
4	#22586389	C₂	1	2	yes	maximal, complemented
5	C₇₁×#3170	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
8	C₂²⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	C₄×C₇₁⋊C₁₆	—	2	1	no	—
10	#17985752	—	2	1	no	—
11	#17985752	C₂²	1	2	yes	—
12	#533×C₇₁	—	2	1	no	—
13	#500×C₇₁	C₂²	1	1	yes	characteristic
14	#494×C₇₁	C₂²	1	1	yes	characteristic
15	#487×C₇₁	—	2	1	no	—
16	#467×C₇₁	—	2	1	no	—
17	#467×C₇₁	C₄	1	2	yes	—
18	C₄×Q₁₆×C₇₁	—	2	1	no	—
19	C₄×Q₁₆×C₇₁	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂²×C₇₁⋊C₈	—	4	1	no	—
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
25	C₂×Q₁₆×C₇₁	—	2	1	no	—
26	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₇₁	—	2	1	no	—
30	C₄⋊Q₈×C₇₁	—	2	1	no	—
31	#125×C₇₁	—	4	1	no	—
32	C₂²⋊Q₈×C₇₁	—	2	2	no	—
33	C₂²⋊Q₈×C₇₁	—	4	1	no	—
34	C₄×Q₈×C₇₁	—	2	2	no	—
35	C₄×Q₈×C₇₁	—	4	1	no	—
36	#117×C₇₁	—	4	1	no	—
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	#107×C₇₁	—	2	1	no	—
39	#107×C₇₁	C₂×C₄	1	2	yes	—
40	#103×C₇₁	—	2	2	no	—
41	#103×C₇₁	—	2	2	no	—
42	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂×C₇₁⋊C₈	—	4	2	no	—
47	C₂×C₇₁⋊C₈	—	8	2	no	—
48	C₇₁⋊C₁₆	—	4	2	no	—
49	C₂×Q₈×C₇₁	—	2	2	no	—
50	C₂×Q₈×C₇₁	—	4	1	no	—
51	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
54	Q₁₆×C₇₁	—	4	2	no	—
55	Q₁₆×C₇₁	—	4	2	no	—
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	4	1	no	—
58	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₇₁	—	2	2	no	—
63	C₄⋊C₄×C₇₁	—	4	2	no	—
64	C₄⋊C₄×C₇₁	—	4	2	no	—
65	C₄⋊C₄×C₇₁	—	4	1	no	—
66	C₄⋊C₄×C₇₁	—	4	1	no	—
67	C₄⋊C₄×C₇₁	—	4	1	no	—
68	C₂²⋊C₄×C₇₁	—	4	2	no	—
69	C₄²×C₇₁	—	4	2	no	—
70	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₇₁	D₈	1	1	yes	characteristic
72	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
73	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₇₁	#102	1	1	yes	characteristic
75	Q₈×C₇₁	—	4	2	no	—
76	Q₈×C₇₁	—	4	2	no	—
77	Q₈×C₇₁	—	8	2	no	—
78	C₂×C₄×C₇₁	—	2	1	no	—
79	C₂×C₄×C₇₁	—	2	1	no	—
80	C₂×C₄×C₇₁	—	4	1	no	—
81	C₂×C₄×C₇₁	—	4	2	no	—
82	C₂×C₄×C₇₁	—	4	2	no	—
83	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
87	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
88	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₇₁	—	2	1	no	—
92	C₈×C₇₁	—	2	1	no	—
93	C₈×C₇₁	—	2	1	no	—
94	C₈×C₇₁	#135	1	2	yes	—
95	C₇₁⋊C₈	—	8	2	no	—
96	C₂²×C₇₁	—	2	1	no	—
97	C₂²×C₇₁	—	2	1	no	—
98	C₂²×C₇₁	#358	1	2	yes	—
99	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
100	C₄×C₇₁	—	2	1	no	—
101	C₄×C₇₁	—	2	1	no	—
102	C₄×C₇₁	—	2	1	no	—
103	C₄×C₇₁	—	8	2	no	—
104	C₄×C₇₁	—	8	2	no	—
105	C₄×C₇₁	#358	1	2	yes	—
106	C₄×C₇₁	#413	1	1	yes	characteristic
107	C₄×C₇₁	#416	1	1	yes	characteristic
108	C₄×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
109	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
110	#11949	—	71	1	no	maximal, Sylow
111	C₂×C₇₁	—	2	2	no	—
112	C₂×C₇₁	#1540	1	1	yes	characteristic
113	C₂×C₇₁	C₂×#358	1	1	yes	characteristic
114	C₂×C₇₁	#2148	1	1	yes	characteristic
115	#3170	D₇₁	1	1	yes	characteristic
116	#2162	—	71	1	no	—
117	C₂×#335	—	71	1	no	—
118	#1365	—	71	2	no	—
119	#1362	—	71	2	no	—
120	C₇₁	#11949	1	1	yes	characteristic, complemented, Sylow
121	#533	—	2	1	no	—
122	#500	C₂×D₇₁	1	1	yes	characteristic
123	#494	C₂×D₇₁	1	1	yes	characteristic
124	#487	—	2	1	no	—
125	#467	—	2	1	no	—
126	#467	Dic₇₁	1	2	yes	—
127	C₄×Q₁₆	—	2	1	no	—
128	C₄×Q₁₆	Dic₇₁	1	2	yes	—
129	C₂×C₄⋊C₈	—	142	1	no	—
130	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
131	C₄⋊C₁₆	—	71	2	no	—
132	C₂²⋊C₁₆	—	142	1	no	—
133	C₄×C₁₆	—	142	1	no	—
134	#335	—	71	2	no	—
135	#335	—	142	1	no	—
136	C₂×Q₁₆	—	2	1	no	—
137	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	—
138	C₂²×C₈	—	284	1	no	—
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	#125	—	4	1	no	—
144	C₂²⋊Q₈	—	2	2	no	—
145	C₂²⋊Q₈	—	4	1	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	4	1	no	—
148	#117	—	4	1	no	—
149	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
150	C₂×C₁₆	—	142	2	no	—
151	#107	—	2	1	no	—
152	#107	C₂×Dic₇₁	1	2	yes	—
153	C₄⋊C₈	—	284	1	no	—
154	C₄⋊C₈	—	142	2	no	—
155	#103	—	2	2	no	—
156	#103	—	2	2	no	—
157	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
164	C₂²×C₄	#11765850	1	1	yes	characteristic
165	C₂²×C₄	#11765853	1	1	yes	characteristic
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	4	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	284	2	no	—
171	C₂×C₈	—	568	2	no	—
172	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
174	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
185	C₄²	#11765850	1	1	yes	characteristic
186	C₄²	#11765853	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
188	C₁₆	—	284	2	no	—
189	C₂³	#14556049	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#14556049	1	1	yes	characteristic
199	C₂×C₄	#14556049	1	1	yes	characteristic
200	C₂×C₄	#14556049	1	1	yes	characteristic
201	C₂×C₄	#14556071	1	2	yes	—
202	C₂×C₄	#14556074	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
204	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
205	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	568	2	no	—
210	C₈	#14556148	1	2	yes	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#17985803	1	2	yes	—
214	C₂²	C₂×#14556049	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	2	no	—
220	C₄	#17985803	1	2	yes	—
221	C₄	#17985983	1	1	yes	characteristic
222	C₄	#17985995	1	1	yes	characteristic
223	C₄	C₂×#14556071	1	1	yes	characteristic
224	C₄	C₂×#14556074	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	#22586989	1	1	yes	characteristic
227	C₂	C₂×#17985803	1	1	yes	characteristic
228	C₂	#22590089	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556049
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#3170	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	#102
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#3170	D₇₁
hypocenter	C₇₁	#11949
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #22586398	C₂	semidirect
4. #22586389	C₂	semidirect
120. C₇₁	#11949	semidirect

Maximal subgroups

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

subgroup	index
#22590139	2
C₂×#17985752	2
#22586398	2
#22586389	2
C₇₁×#3170	2
#11949	71