Subgroups — C₇₁×#13854 #31834726

Order 18176.

1422 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₇₁	C₂	1	2	yes	maximal
2	C₂×#376×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1442	C₂	1	8	yes	maximal
4	C₄×#95×C₇₁	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₇₁	C₂²	1	4	yes	—
6	C₂²×C₄²×C₇₁	C₂²	1	4	yes	—
7	C₂²×C₄²×C₇₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₇₁	—	2	8	no	—
10	#376×C₇₁	C₄	1	8	yes	complemented
11	C₂×#95×C₇₁	C₂²	1	16	yes	—
12	C₂×#95×C₇₁	C₂²	1	4	yes	—
13	C₂×#95×C₇₁	C₂²	1	4	yes	—
14	C₂³×C₄×C₇₁	C₂³	1	4	yes	—
15	C₂³×C₄×C₇₁	C₂³	1	8	yes	—
16	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
17	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₁	—	2	8	no	—
19	C₂×C₄⋊C₄×C₇₁	—	2	16	no	—
20	C₂×C₄²×C₇₁	—	2	8	no	—
21	C₂×C₄²×C₇₁	—	2	8	no	—
22	C₂×C₄²×C₇₁	—	2	16	no	—
23	C₂×C₄²×C₇₁	C₂×C₄	1	8	yes	—
24	C₂×C₄²×C₇₁	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₇₁	D₄	1	4	yes	—
26	C₂×C₄²×C₇₁	Q₈	1	4	yes	—
27	#95×C₇₁	—	2	8	no	—
28	#95×C₇₁	—	2	32	no	—
29	#95×C₇₁	C₂×C₄	1	16	yes	—
30	C₂⁴×C₇₁	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	—	2	8	no	—
32	C₂²×C₄×C₇₁	—	2	8	no	—
33	C₂²×C₄×C₇₁	—	2	16	no	—
34	C₂²×C₄×C₇₁	—	2	16	no	—
35	C₂²×C₄×C₇₁	—	2	8	no	—
36	C₂²×C₄×C₇₁	—	2	16	no	—
37	C₂²×C₄×C₇₁	—	2	8	no	—
38	C₂²×C₄×C₇₁	C₂²×C₄	1	8	yes	—
39	C₂²×C₄×C₇₁	C₂²×C₄	1	4	yes	—
40	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
42	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₇₁	C₂×Q₈	1	4	yes	—
44	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₇₁	#41	1	16	yes	—
46	C₂²×C₄×C₇₁	#41	1	8	yes	—
47	C₄⋊C₄×C₇₁	—	2	32	no	—
48	C₄²×C₇₁	—	2	16	no	—
49	C₄²×C₇₁	—	4	16	no	—
50	C₄²×C₇₁	C₄⋊C₄	1	16	yes	—
51	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
52	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
53	C₂³×C₇₁	C₂²×Q₈	1	1	yes	characteristic
54	C₂³×C₇₁	C₂×#41	1	4	yes	—
55	C₂³×C₇₁	C₂×#41	1	8	yes	—
56	C₂×C₄×C₇₁	—	2	8	no	—
57	C₂×C₄×C₇₁	—	2	8	no	—
58	C₂×C₄×C₇₁	—	2	8	no	—
59	C₂×C₄×C₇₁	—	2	8	no	—
60	C₂×C₄×C₇₁	—	2	16	no	—
61	C₂×C₄×C₇₁	—	2	16	no	—
62	C₂×C₄×C₇₁	—	2	32	no	—
63	C₂×C₄×C₇₁	—	4	8	no	—
64	C₂×C₄×C₇₁	—	4	8	no	—
65	C₂×C₄×C₇₁	—	4	16	no	—
66	C₂×C₄×C₇₁	—	4	16	no	—
67	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	8	yes	—
68	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	4	yes	—
69	C₂×C₄×C₇₁	#117	1	8	yes	—
70	C₂×C₄×C₇₁	#124	1	8	yes	—
71	C₂×C₄×C₇₁	#125	1	8	yes	—
72	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₇₁	C₂×#117	1	2	yes	—
74	C₂²×C₇₁	C₄×#41	1	4	yes	—
75	C₂²×C₇₁	#525	1	4	yes	—
76	C₂²×C₇₁	C₂×#124	1	2	yes	—
77	C₂²×C₇₁	C₂×#125	1	2	yes	—
78	C₂²×C₇₁	#529	1	16	yes	—
79	C₂²×C₇₁	#533	1	4	yes	—
80	C₄×C₇₁	—	2	8	no	—
81	C₄×C₇₁	—	4	16	no	—
82	C₄×C₇₁	—	4	8	no	—
83	C₄×C₇₁	#382	1	8	yes	—
84	#13854	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×C₇₁	C₂×#382	1	1	yes	characteristic
86	C₂×C₇₁	#2296	1	2	yes	—
87	C₂×C₇₁	#2298	1	4	yes	—
88	C₂×C₇₁	#2398	1	4	yes	—
89	C₂×C₇₁	#2402	1	4	yes	—
90	C₂×C₄×C₄⋊C₄	C₂×C₇₁	1	2	yes	—
91	C₂×#376	C₂×C₇₁	1	1	yes	characteristic
92	#1442	C₂×C₇₁	1	8	yes	—
93	C₄×#95	C₂×C₇₁	1	4	yes	—
94	C₇₁	#13854	1	1	yes	characteristic, direct product factor, Sylow
95	C₂²×C₄⋊C₄	C₂²×C₇₁	1	4	yes	—
96	C₂²×C₄²	C₂²×C₇₁	1	4	yes	—
97	C₂²×C₄²	C₂²×C₇₁	1	2	yes	—
98	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	2	8	no	—
100	#376	C₄×C₇₁	1	8	yes	complemented
101	C₂×#95	C₂²×C₇₁	1	16	yes	—
102	C₂×#95	C₂²×C₇₁	1	4	yes	—
103	C₂×#95	C₂²×C₇₁	1	4	yes	—
104	C₂³×C₄	C₂³×C₇₁	1	4	yes	—
105	C₂³×C₄	C₂³×C₇₁	1	8	yes	—
106	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
107	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	2	8	no	—
109	C₂×C₄⋊C₄	—	2	16	no	—
110	C₂×C₄²	—	2	8	no	—
111	C₂×C₄²	—	2	8	no	—
112	C₂×C₄²	—	2	16	no	—
113	C₂×C₄²	C₂×C₄×C₇₁	1	8	yes	—
114	C₂×C₄²	C₂×C₄×C₇₁	1	4	yes	—
115	C₂×C₄²	D₄×C₇₁	1	4	yes	—
116	C₂×C₄²	Q₈×C₇₁	1	4	yes	—
117	#95	—	2	8	no	—
118	#95	—	2	32	no	—
119	#95	C₂×C₄×C₇₁	1	16	yes	—
120	C₂⁴	C₂⁴×C₇₁	1	1	yes	characteristic
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	—	2	8	no	—
123	C₂²×C₄	—	2	16	no	—
124	C₂²×C₄	—	2	16	no	—
125	C₂²×C₄	—	2	8	no	—
126	C₂²×C₄	—	2	16	no	—
127	C₂²×C₄	—	2	8	no	—
128	C₂²×C₄	C₂²×C₄×C₇₁	1	8	yes	—
129	C₂²×C₄	C₂²×C₄×C₇₁	1	4	yes	—
130	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
131	C₂²×C₄	C₂×D₄×C₇₁	1	4	yes	—
132	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
133	C₂²×C₄	C₂×Q₈×C₇₁	1	4	yes	—
134	C₂²×C₄	C₂×Q₈×C₇₁	1	2	yes	—
135	C₂²×C₄	#41×C₇₁	1	16	yes	—
136	C₂²×C₄	#41×C₇₁	1	8	yes	—
137	C₄⋊C₄	—	2	32	no	—
138	C₄²	—	2	16	no	—
139	C₄²	—	4	16	no	—
140	C₄²	C₄⋊C₄×C₇₁	1	16	yes	—
141	C₂³	C₂³×C₄×C₇₁	1	1	yes	characteristic
142	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
143	C₂³	C₂²×Q₈×C₇₁	1	1	yes	characteristic
144	C₂³	C₂×#41×C₇₁	1	4	yes	—
145	C₂³	C₂×#41×C₇₁	1	8	yes	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	2	8	no	—
150	C₂×C₄	—	2	16	no	—
151	C₂×C₄	—	2	16	no	—
152	C₂×C₄	—	2	32	no	—
153	C₂×C₄	—	4	8	no	—
154	C₂×C₄	—	4	8	no	—
155	C₂×C₄	—	4	16	no	—
156	C₂×C₄	—	4	16	no	—
157	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	8	yes	—
158	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	4	yes	—
159	C₂×C₄	#117×C₇₁	1	8	yes	—
160	C₂×C₄	#124×C₇₁	1	8	yes	—
161	C₂×C₄	#125×C₇₁	1	8	yes	—
162	C₂²	C₂²×C₄⋊C₄×C₇₁	1	1	yes	characteristic
163	C₂²	C₂×#117×C₇₁	1	2	yes	—
164	C₂²	C₄×#41×C₇₁	1	4	yes	—
165	C₂²	#525×C₇₁	1	4	yes	—
166	C₂²	C₂×#124×C₇₁	1	2	yes	—
167	C₂²	C₂×#125×C₇₁	1	2	yes	—
168	C₂²	#529×C₇₁	1	16	yes	—
169	C₂²	#533×C₇₁	1	4	yes	—
170	C₄	—	2	8	no	—
171	C₄	—	4	8	no	—
172	C₄	—	4	16	no	—
173	C₄	#382×C₇₁	1	8	yes	—
174	C₂	C₂×#382×C₇₁	1	1	yes	characteristic
175	C₂	C₇₁×#2296	1	2	yes	—
176	C₂	C₇₁×#2298	1	4	yes	—
177	C₂	C₇₁×#2398	1	4	yes	—
178	C₂	C₇₁×#2402	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇₁	C₂³
commutator	C₂³	C₂³×C₄×C₇₁
Frattini, cosocle	C₂⁴	C₂⁴×C₇₁
socle, prime radical	C₂⁴×C₇₁	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. #376×C₇₁	C₄	semidirect
84. #13854	C₇₁	direct (N × H)
94. C₇₁	#13854	direct (N × H)
100. #376	C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊C₄×C₇₁	2
C₂×#376×C₇₁	2
C₇₁×#1442	2
C₄×#95×C₇₁	2
#13854	71

Subgroups — C71×#13854 #31834726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#13854 #31834726