Subgroups — #31908750

Order 18176.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598854	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
3	#22586397	C₂	1	2	yes	maximal, complemented
4	#22586394	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2150	C₂	1	1	yes	maximal, characteristic
6	#17986556	—	2	1	no	—
7	#17986404	C₂²	1	1	yes	characteristic
8	#17986404	C₂²	1	1	yes	characteristic
9	#17986401	—	2	1	no	—
10	#17986401	C₄	1	2	yes	—
11	#17986397	—	2	1	no	—
12	#17986037	—	2	1	no	—
13	#17986037	C₄	1	2	yes	—
14	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
15	#17985752	—	2	1	no	—
16	#17985752	C₂²	1	2	yes	—
17	C₂×M₃₂×C₇₁	—	2	1	no	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₇₁	—	2	1	no	—
20	C₄⋊C₁₆×C₇₁	C₂²	1	2	yes	—
21	C₂²×C₇₁⋊C₈	—	4	1	no	—
22	#14556128	—	2	1	no	—
23	#14556128	—	2	1	no	—
24	#14556120	—	2	2	no	—
25	#14556107	—	4	1	no	—
26	#14556107	—	4	1	no	—
27	C₄×Dic₁₄₂	—	4	1	no	—
28	C₄×Dic₁₄₂	—	4	1	no	—
29	#14556087	—	2	2	no	—
30	#14556062	—	2	1	no	—
31	#14556062	—	2	1	no	—
32	#14556062	C₂×C₄	1	2	yes	—
33	#14556062	C₂×C₄	1	2	yes	—
34	#14556060	—	2	2	no	—
35	#14556060	—	2	2	no	—
36	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
38	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
41	M₃₂×C₇₁	—	4	2	no	—
42	C₂×C₁₆×C₇₁	—	2	2	no	—
43	C₄×C₈×C₇₁	C₂×C₄	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	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₄₂	—	4	1	no	—
48	C₂×Dic₁₄₂	—	4	1	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₂×C₇₁⋊C₈	—	8	2	no	—
51	C₄⋊Dic₇₁	—	2	2	no	—
52	C₄⋊Dic₇₁	—	2	2	no	—
53	C₄⋊Dic₇₁	—	4	1	no	—
54	C₄⋊Dic₇₁	—	4	1	no	—
55	Dic₇₁⋊C₄	—	4	2	no	—
56	Dic₇₁⋊C₄	—	4	2	no	—
57	C₂²⋊Dic₇₁	—	4	2	no	—
58	C₄×Dic₇₁	—	4	2	no	—
59	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
62	C₂×C₈×C₇₁	—	2	2	no	—
63	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
66	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
67	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₇₁	D₈	1	1	yes	characteristic
71	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
72	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₇₁	—	4	2	no	—
74	C₂³×C₇₁	#102	1	1	yes	characteristic
75	C₂×C₄×C₇₁	—	2	1	no	—
76	C₂×C₄×C₇₁	—	2	1	no	—
77	C₂×C₄×C₇₁	—	4	1	no	—
78	C₂×C₄×C₇₁	#101	1	2	yes	—
79	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
82	C₂×C₄×C₇₁	#103	1	2	yes	—
83	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₇₁	—	4	2	no	—
87	C₂×Dic₇₁	—	4	2	no	—
88	Dic₁₄₂	—	8	2	no	—
89	Dic₁₄₂	—	8	1	no	—
90	Dic₁₄₂	—	8	1	no	—
91	C₈×C₇₁	—	2	1	no	—
92	C₈×C₇₁	—	2	1	no	—
93	C₈×C₇₁	#103	1	2	yes	—
94	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
96	C₇₁⋊C₈	—	8	2	no	—
97	C₂²×C₇₁	—	2	1	no	—
98	C₂²×C₇₁	—	2	1	no	—
99	C₂²×C₇₁	#332	1	2	yes	—
100	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
101	C₄×C₇₁	—	2	1	no	—
102	C₄×C₇₁	—	2	1	no	—
103	C₄×C₇₁	—	2	1	no	—
104	C₄×C₇₁	#332	1	2	yes	—
105	C₄×C₇₁	C₂×#101	1	1	yes	characteristic
106	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
107	C₄×C₇₁	#509	1	1	yes	characteristic
108	C₄×C₇₁	#510	1	1	yes	characteristic
109	Dic₇₁	—	8	2	no	—
110	Dic₇₁	—	8	2	no	—
111	#11942	—	71	1	no	maximal, Sylow
112	C₂×C₇₁	—	2	2	no	—
113	C₂×C₇₁	C₂×#332	1	1	yes	characteristic
114	C₂×C₇₁	#2139	1	1	yes	characteristic
115	C₂×C₇₁	#2148	1	1	yes	characteristic
116	#3160	—	71	1	no	—
117	#2150	D₇₁	1	1	yes	characteristic
118	C₂×#335	—	71	1	no	—
119	#1365	—	71	2	no	—
120	#1364	—	71	2	no	—
121	C₇₁	#11942	1	1	yes	characteristic, complemented, Sylow
122	#533	—	142	1	no	—
123	C₂×M₃₂	—	2	1	no	—
124	#500	—	71	1	no	—
125	#500	—	71	1	no	—
126	#499	—	71	2	no	—
127	#499	—	142	1	no	—
128	#498	—	142	1	no	—
129	#427	—	71	2	no	—
130	#427	—	142	1	no	—
131	C₂×C₄⋊C₈	—	142	1	no	—
132	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
133	C₄⋊C₁₆	—	2	1	no	—
134	C₄⋊C₁₆	C₂×D₇₁	1	2	yes	—
135	#335	—	71	2	no	—
136	#335	—	142	1	no	—
137	C₂²×C₈	—	284	1	no	—
138	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
140	C₄⋊Q₈	—	142	1	no	—
141	C₄⋊Q₈	—	142	1	no	—
142	#125	—	142	2	no	—
143	C₂²⋊Q₈	—	284	1	no	—
144	C₂²⋊Q₈	—	284	1	no	—
145	C₄×Q₈	—	284	1	no	—
146	C₄×Q₈	—	284	1	no	—
147	#117	—	142	2	no	—
148	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
149	M₃₂	—	4	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#107	—	71	2	no	—
152	#107	—	71	2	no	—
153	#107	—	142	1	no	—
154	#107	—	142	1	no	—
155	#106	—	142	2	no	—
156	#106	—	142	2	no	—
157	C₄⋊C₈	—	284	1	no	—
158	C₄⋊C₈	—	142	2	no	—
159	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
160	C₄×C₈	D₂₈₄	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
163	C₂×Q₈	—	284	1	no	—
164	C₂×Q₈	—	284	1	no	—
165	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
167	C₂²×C₄	#11765854	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	284	2	no	—
170	C₂×C₈	—	568	2	no	—
171	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
172	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	#11765853	1	2	yes	—
174	C₂×C₈	#11765855	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
177	C₄⋊C₄	—	284	2	no	—
178	C₄⋊C₄	—	284	2	no	—
179	C₄⋊C₄	—	284	1	no	—
180	C₄⋊C₄	—	284	1	no	—
181	C₄⋊C₄	—	142	2	no	—
182	C₄⋊C₄	—	142	2	no	—
183	C₂²⋊C₄	—	284	2	no	—
184	C₄²	—	284	2	no	—
185	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
186	C₄²	D₅₆₈	1	1	yes	characteristic
187	C₄²	#11765854	1	1	yes	characteristic
188	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
189	C₁₆	—	4	2	no	—
190	C₂³	#14556050	1	1	yes	characteristic
191	Q₈	—	568	2	no	—
192	Q₈	—	568	1	no	—
193	Q₈	—	568	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	284	2	no	—
198	C₂×C₄	—	284	2	no	—
199	C₂×C₄	#14556046	1	2	yes	—
200	C₂×C₄	#14556050	1	1	yes	characteristic
201	C₂×C₄	#14556050	1	1	yes	characteristic
202	C₂×C₄	#14556050	1	1	yes	characteristic
203	C₂×C₄	#14556051	1	2	yes	—
204	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
206	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	568	2	no	—
210	C₈	#14556051	1	2	yes	—
211	C₈	C₂×#11765853	1	1	yes	characteristic
212	C₈	C₂×#11765855	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#17985744	1	2	yes	—
216	C₂²	C₂×#14556050	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	568	2	no	—
221	C₄	—	568	2	no	—
222	C₄	#17985744	1	2	yes	—
223	C₄	C₂×#14556046	1	1	yes	characteristic
224	C₄	C₂×#14556051	1	1	yes	characteristic
225	C₄	#17986437	1	1	yes	characteristic
226	C₄	#17986442	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#17985744	1	1	yes	characteristic
229	C₂	#22590053	1	1	yes	characteristic
230	C₂	#22590090	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556050
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2150	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	#102
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#2150	D₇₁
hypocenter	C₇₁	#11942
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #22586397	C₂	semidirect
4. #22586394	C₂	semidirect
121. C₇₁	#11942	semidirect

Maximal subgroups

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

subgroup	index
#22598854	2
C₂×#17985752	2
#22586397	2
#22586394	2
C₇₁×#2150	2
#11942	71