Subgroups — #31888591

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#17985721	C₂	1	2	yes	maximal
3	C₇₁⋊#1273	C₂	1	1	yes	maximal, characteristic
4	C₇₁⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₄×#97×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₄	1	4	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
9	#17985754	C₄	1	4	yes	—
10	#17985721	—	2	4	no	—
11	C₂×#97×C₇₁	C₄	1	2	yes	—
12	C₂×#97×C₇₁	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄³×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₄⋊(C₇₁⋊C₈)	—	2	8	no	—
19	C₄×C₇₁⋊C₈	—	2	2	no	—
20	C₄×C₇₁⋊C₈	—	2	2	no	—
21	C₄×C₇₁⋊C₈	—	2	2	no	—
22	C₄×C₇₁⋊C₈	—	2	2	no	—
23	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
30	#97×C₇₁	—	2	4	no	—
31	#97×C₇₁	—	2	4	no	—
32	C₄×C₈×C₇₁	—	2	2	no	—
33	C₄×C₈×C₇₁	D₄	1	2	yes	—
34	C₄×C₈×C₇₁	Q₈	1	2	yes	—
35	C₂×C₇₁⋊C₈	—	2	4	no	—
36	C₂×C₇₁⋊C₈	—	2	4	no	—
37	C₂×C₇₁⋊C₈	—	2	4	no	—
38	C₂×C₇₁⋊C₈	—	2	4	no	—
39	C₂×C₇₁⋊C₈	—	2	4	no	—
40	C₂×C₇₁⋊C₈	M₁₆	1	8	yes	—
41	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
44	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	—	2	2	no	—
47	C₂×C₈×C₇₁	—	2	2	no	—
48	C₂×C₈×C₇₁	—	2	2	no	—
49	C₂×C₈×C₇₁	—	2	2	no	—
50	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
51	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
52	C₄²×C₇₁	—	2	2	no	—
53	C₄²×C₇₁	—	2	4	no	—
54	C₄²×C₇₁	—	2	4	no	—
55	C₄²×C₇₁	M₁₆	1	4	yes	—
56	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₇₁	#41	1	2	yes	—
59	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	—	2	2	no	—
63	C₂×C₄×C₇₁	—	2	1	no	—
64	C₂×C₄×C₇₁	—	2	1	no	—
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	#97	1	4	yes	—
67	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
72	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₇₁	#131	1	4	yes	—
75	C₈×C₇₁	—	2	4	no	—
76	C₈×C₇₁	—	2	4	no	—
77	C₇₁⋊C₈	—	2	8	no	—
78	C₇₁⋊C₈	—	4	4	no	—
79	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
81	C₂²×C₇₁	C₄×M₁₆	1	2	yes	—
82	C₂²×C₇₁	C₄×M₁₆	1	1	yes	characteristic
83	C₂²×C₇₁	#405	1	2	yes	—
84	C₄×C₇₁	—	2	4	no	—
85	C₄×C₇₁	—	4	1	no	—
86	C₄×C₇₁	—	4	1	no	—
87	C₄×C₇₁	#423	1	2	yes	—
88	C₄×C₇₁	#432	1	2	yes	—
89	C₄×C₇₁	#435	1	4	yes	—
90	C₄×C₇₁	#446	1	4	yes	—
91	#8263	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	#1451	1	2	yes	—
93	C₂×C₇₁	#1453	1	1	yes	characteristic
94	C₂×C₇₁	#1455	1	1	yes	characteristic
95	C₂×C₇₁	#1767	1	1	yes	characteristic
96	C₂×C₇₁	#1771	1	2	yes	—
97	C₄×C₄⋊C₈	—	71	2	no	—
98	C₄×#97	D₇₁	1	1	yes	characteristic
99	C₂×#322	—	71	2	no	—
100	#1273	—	71	1	no	—
101	#1273	—	71	1	no	—
102	C₇₁	#8263	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊C₈	—	71	4	no	—
104	C₂×#97	Dic₇₁	1	2	yes	—
105	C₂×#97	Dic₇₁	1	2	yes	—
106	C₂×C₄×C₈	—	71	2	no	—
107	C₂×C₄×C₈	—	71	2	no	—
108	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	C₄³	C₂×D₇₁	1	1	yes	characteristic
111	#336	—	71	4	no	—
112	#322	—	142	4	no	—
113	C₂²×C₈	—	71	4	no	—
114	C₂²×C₈	—	71	4	no	—
115	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
116	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
117	C₂×C₄²	C₄×D₇₁	1	2	yes	—
118	C₂×C₄²	C₄×D₇₁	1	2	yes	—
119	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
120	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
121	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
122	C₄⋊C₈	—	142	8	no	—
123	#97	—	2	4	no	—
124	#97	—	2	4	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	142	2	no	—
127	C₄×C₈	—	142	2	no	—
128	C₄×C₈	—	142	2	no	—
129	C₄×C₈	—	142	2	no	—
130	C₄×C₈	Dic₁₄₂	1	2	yes	—
131	C₄×C₈	D₂₈₄	1	2	yes	—
132	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
133	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
134	C₂²×C₄	C₄×Dic₇₁	1	2	yes	—
135	C₂²×C₄	C₂×C₄×D₇₁	1	2	yes	—
136	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	71	8	no	—
142	C₂×C₈	—	142	4	no	—
143	C₂×C₈	—	142	4	no	—
144	C₂×C₈	—	142	4	no	—
145	C₂×C₈	—	142	4	no	—
146	C₂×C₈	—	142	4	no	—
147	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
148	C₂×C₈	#11765849	1	4	yes	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	4	no	—
152	C₄²	#11765849	1	4	yes	—
153	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
154	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
155	C₄²	#11765865	1	2	yes	—
156	C₂³	C₂×C₄×Dic₇₁	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	#14556039	1	4	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
165	C₂×C₄	#14556087	1	1	yes	characteristic
166	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
167	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
168	C₂×C₄	M₁₆×D₇₁	1	4	yes	—
169	C₂×C₄	C₂×#11765849	1	2	yes	—
170	C₂×C₄	C₂×#11765849	1	2	yes	—
171	C₂×C₄	#14556136	1	4	yes	—
172	C₈	—	2	4	no	—
173	C₈	—	2	4	no	—
174	C₈	—	284	4	no	—
175	C₈	—	142	8	no	—
176	C₂²	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
177	C₂²	C₂×#14556039	1	1	yes	characteristic
178	C₂²	M₁₆×Dic₇₁	1	2	yes	—
179	C₂²	C₄×#11765849	1	1	yes	characteristic
180	C₂²	#17985957	1	2	yes	—
181	C₄	—	2	4	no	—
182	C₄	—	4	1	no	—
183	C₄	—	4	1	no	—
184	C₄	#17986023	1	2	yes	—
185	C₄	#17986056	1	2	yes	—
186	C₄	#17986072	1	4	yes	—
187	C₄	#17986127	1	4	yes	—
188	C₂	#22586627	1	2	yes	—
189	C₂	#22586637	1	1	yes	characteristic
190	C₂	#22586643	1	1	yes	characteristic
191	C₂	#22588193	1	1	yes	characteristic
192	C₂	#22588212	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator, socle, prime radical	C₂³×C₇₁	C₂×C₄²
Fitting	C₄×#97×C₇₁	C₂
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#97	D₇₁
hypocenter	C₇₁	#8263
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
102. C₇₁	#8263	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₇₁⋊C₈)	2
C₂×#17985721	2
C₇₁⋊#1273	2
C₄×#97×C₇₁	2
#8263	71