Subgroups — #31896007

Order 18176.

378 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589056	C₂	1	2	yes	maximal
2	C₄×#14556060	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
4	#22586179	C₂	1	2	yes	maximal
5	C₂×#334×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556060	C₄	1	2	yes	—
7	C₂×Dic₇₁⋊C₈	—	2	2	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₇₁	C₂²	1	2	yes	—
11	#17985762	C₄	1	1	yes	characteristic
12	#17985762	C₄	1	1	yes	characteristic
13	#17985753	—	2	2	no	—
14	#17985721	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#334×C₇₁	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	—	2	1	no	—
19	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₇₁	—	2	2	no	—
23	#14556060	—	2	4	no	—
24	Dic₇₁⋊C₈	—	4	4	no	—
25	C₄×C₇₁⋊C₈	—	2	1	no	—
26	C₄×C₇₁⋊C₈	—	2	1	no	—
27	C₄×C₇₁⋊C₈	—	2	1	no	—
28	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
29	C₇₁⋊#95	—	2	2	no	—
30	C₂²×C₈×C₇₁	—	2	2	no	—
31	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₁	—	2	2	no	—
35	C₄⋊C₈×C₇₁	—	2	2	no	—
36	C₄×C₈×C₇₁	D₄	1	2	yes	—
37	C₄×C₈×C₇₁	Q₈	1	2	yes	—
38	C₂²×Dic₇₁	—	2	2	no	—
39	C₂²×Dic₇₁	—	2	2	no	—
40	C₂×C₇₁⋊C₈	—	2	2	no	—
41	C₂×C₇₁⋊C₈	—	2	2	no	—
42	C₂×C₇₁⋊C₈	—	2	2	no	—
43	C₂×C₇₁⋊C₈	—	4	1	no	—
44	C₂×C₇₁⋊C₈	—	4	1	no	—
45	C₂×C₇₁⋊C₈	SD₁₆	1	4	yes	—
46	C₄⋊Dic₇₁	—	2	2	no	—
47	C₄⋊Dic₇₁	—	2	2	no	—
48	C₄⋊Dic₇₁	—	2	2	no	—
49	C₄⋊Dic₇₁	M₁₆	1	4	yes	—
50	C₄×Dic₇₁	—	4	2	no	—
51	C₄×Dic₇₁	—	4	2	no	—
52	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
55	C₂×C₈×C₇₁	—	2	2	no	—
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	4	2	no	—
58	C₂×C₈×C₇₁	—	4	2	no	—
59	C₂×C₈×C₇₁	—	4	2	no	—
60	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
62	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₇₁	#41	1	1	yes	characteristic
66	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	—	2	1	no	—
68	C₂×C₄×C₇₁	—	2	1	no	—
69	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₁	#125	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₇₁	#131	1	2	yes	—
77	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₇₁	#135	1	2	yes	—
79	C₂×Dic₇₁	—	2	4	no	—
80	C₂×Dic₇₁	—	4	2	no	—
81	C₂×Dic₇₁	—	4	2	no	—
82	C₂×Dic₇₁	—	4	2	no	—
83	C₂×Dic₇₁	—	4	2	no	—
84	C₂×Dic₇₁	—	4	2	no	—
85	C₈×C₇₁	—	2	2	no	—
86	C₈×C₇₁	—	2	2	no	—
87	C₈×C₇₁	—	8	2	no	—
88	C₇₁⋊C₈	—	2	4	no	—
89	C₇₁⋊C₈	—	4	2	no	—
90	C₂²×C₇₁	#389	1	1	yes	characteristic
91	C₂²×C₇₁	#435	1	2	yes	—
92	C₂²×C₇₁	C₄×SD₁₆	1	2	yes	—
93	C₂²×C₇₁	#444	1	1	yes	characteristic
94	C₂²×C₇₁	#444	1	1	yes	characteristic
95	C₄×C₇₁	—	2	1	no	—
96	C₄×C₇₁	—	2	1	no	—
97	C₄×C₇₁	#423	1	1	yes	characteristic
98	C₄×C₇₁	#424	1	1	yes	characteristic
99	C₄×C₇₁	#446	1	2	yes	—
100	C₄×C₇₁	#465	1	2	yes	—
101	C₄×C₇₁	#498	1	1	yes	characteristic
102	C₄×C₇₁	#499	1	1	yes	characteristic
103	Dic₇₁	—	4	4	no	—
104	Dic₇₁	—	8	2	no	—
105	#9597	—	71	1	no	maximal, Sylow
106	C₂×C₇₁	#1582	1	2	yes	—
107	C₂×C₇₁	#1583	1	2	yes	—
108	C₂×C₇₁	#1940	1	1	yes	characteristic
109	C₂×C₇₁	#1941	1	1	yes	characteristic
110	C₂×C₇₁	#1946	1	1	yes	characteristic
111	#1939	—	71	2	no	—
112	C₄×#106	—	71	1	no	—
113	C₂×#334	D₇₁	1	1	yes	characteristic
114	C₂×#322	—	71	1	no	—
115	#1279	—	71	2	no	—
116	C₇₁	#9597	1	1	yes	characteristic, complemented, Sylow
117	C₂×#106	—	71	2	no	—
118	C₂×C₄⋊C₈	—	142	2	no	—
119	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
120	C₂×C₄×C₈	—	71	1	no	—
121	C₂×C₄×C₈	—	71	1	no	—
122	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
123	C₄×C₄⋊C₄	—	71	2	no	—
124	#340	—	71	1	no	—
125	#340	—	71	1	no	—
126	#336	—	142	2	no	—
127	#334	—	2	2	no	—
128	#322	—	142	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	71	2	no	—
131	C₂²×C₈	—	142	1	no	—
132	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
133	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
134	C₂×C₄⋊C₄	—	71	2	no	—
135	C₂×C₄⋊C₄	—	71	2	no	—
136	C₂×C₄²	—	142	2	no	—
137	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
138	#106	—	142	4	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	284	4	no	—
142	C₄×C₈	—	71	2	no	—
143	C₄×C₈	—	142	1	no	—
144	C₄×C₈	—	142	1	no	—
145	C₄×C₈	—	142	1	no	—
146	C₄×C₈	Dic₁₄₂	1	2	yes	—
147	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
148	#95	—	142	2	no	—
149	C₂²×C₄	—	142	2	no	—
150	C₂²×C₄	—	142	2	no	—
151	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
152	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
153	C₂²×C₄	#11765866	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	284	1	no	—
160	C₂×C₈	—	284	1	no	—
161	C₂×C₈	—	71	4	no	—
162	C₂×C₈	—	142	2	no	—
163	C₂×C₈	—	142	2	no	—
164	C₂×C₈	—	142	2	no	—
165	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
166	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
167	C₄⋊C₄	—	71	4	no	—
168	C₄⋊C₄	—	142	2	no	—
169	C₄⋊C₄	—	142	2	no	—
170	C₄⋊C₄	—	142	2	no	—
171	C₄²	—	284	2	no	—
172	C₄²	—	284	2	no	—
173	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
174	C₄²	Q₈×D₇₁	1	1	yes	characteristic
175	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
176	C₄²	#11765864	1	1	yes	characteristic
177	C₂³	#14556090	1	1	yes	characteristic
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	284	2	no	—
181	C₂×C₄	—	284	2	no	—
182	C₂×C₄	—	284	2	no	—
183	C₂×C₄	—	284	2	no	—
184	C₂×C₄	—	284	2	no	—
185	C₂×C₄	—	142	4	no	—
186	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
187	C₂×C₄	#14556093	1	1	yes	characteristic
188	C₂×C₄	#14556099	1	1	yes	characteristic
189	C₂×C₄	#14556104	1	1	yes	characteristic
190	C₂×C₄	#14556119	1	1	yes	characteristic
191	C₂×C₄	#14556127	1	1	yes	characteristic
192	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
193	C₂×C₄	#14556134	1	2	yes	—
194	C₂×C₄	SD₁₆×D₇₁	1	2	yes	—
195	C₂×C₄	#14556149	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	2	2	no	—
198	C₈	—	8	2	no	—
199	C₈	—	284	2	no	—
200	C₈	—	142	4	no	—
201	C₂²	#17985895	1	1	yes	characteristic
202	C₂²	#17986069	1	2	yes	—
203	C₂²	#17986086	1	2	yes	—
204	C₂²	#17986120	1	1	yes	characteristic
205	C₂²	#17986120	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	284	4	no	—
209	C₄	—	568	2	no	—
210	C₄	#17986022	1	1	yes	characteristic
211	C₄	#17986025	1	1	yes	characteristic
212	C₄	#17986128	1	2	yes	—
213	C₄	#17986225	1	2	yes	—
214	C₄	#17986396	1	1	yes	characteristic
215	C₄	#17986399	1	1	yes	characteristic
216	C₂	#22587175	1	2	yes	—
217	C₂	#22587182	1	2	yes	—
218	C₂	#22589061	1	1	yes	characteristic
219	C₂	#22589066	1	1	yes	characteristic
220	C₂	#22589091	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₁
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#334×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#334	D₇₁
hypocenter	C₇₁	#9597
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
116. C₇₁	#9597	semidirect

Maximal subgroups

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

subgroup	index
#22589056	2
C₄×#14556060	2
C₂×#17985721	2
#22586179	2
C₂×#334×C₇₁	2
#9597	71