Subgroups — #31906505

Order 18176.

246 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590143	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985720	C₂	1	1	yes	maximal, characteristic
3	#22586262	C₂	1	2	yes	maximal, complemented
4	#22586260	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2163	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	—
9	C₂²⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	—
10	#17985775	C₂²	1	2	yes	—
11	#17985720	C₄	1	1	yes	characteristic
12	#17985720	C₄	1	1	yes	characteristic
13	#17985720	C₂²	1	1	yes	characteristic
14	#17985720	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₇₁	C₄	1	2	yes	—
17	C₂²⋊C₁₆×C₇₁	C₄	1	2	yes	—
18	#346×C₇₁	C₂²	1	2	yes	—
19	C₂²×C₇₁⋊C₈	—	2	1	no	—
20	C₂²×C₇₁⋊C₈	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₇₁⋊C₈	—	2	1	no	—
23	C₄×C₇₁⋊C₈	—	2	1	no	—
24	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
36	C₂×C₇₁⋊C₈	—	2	1	no	—
37	C₂×C₇₁⋊C₈	—	2	1	no	—
38	C₂×C₇₁⋊C₈	—	2	1	no	—
39	C₂×C₇₁⋊C₈	—	2	1	no	—
40	C₂×C₇₁⋊C₈	—	4	2	no	—
41	C₂×C₇₁⋊C₈	—	4	2	no	—
42	C₇₁⋊C₁₆	—	2	4	no	—
43	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₁	—	2	2	no	—
47	C₂×C₈×C₇₁	—	2	2	no	—
48	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₇₁	C₄²	1	1	yes	characteristic
53	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
54	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
55	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₁₆×C₇₁	—	2	4	no	—
57	C₂³×C₇₁	#97	1	1	yes	characteristic
58	C₂×C₄×C₇₁	—	2	2	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	1	no	—
61	C₂×C₄×C₇₁	—	2	1	no	—
62	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₇₁	—	2	1	no	—
69	C₈×C₇₁	—	2	1	no	—
70	C₈×C₇₁	#131	1	2	yes	—
71	C₈×C₇₁	#131	1	2	yes	—
72	C₇₁⋊C₈	—	4	1	no	—
73	C₇₁⋊C₈	—	4	1	no	—
74	C₇₁⋊C₈	—	4	1	no	—
75	C₇₁⋊C₈	—	4	1	no	—
76	C₂²×C₇₁	—	2	1	no	—
77	C₂²×C₇₁	—	2	1	no	—
78	C₂²×C₇₁	#347	1	2	yes	—
79	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
80	C₄×C₇₁	—	2	1	no	—
81	C₄×C₇₁	—	2	1	no	—
82	C₄×C₇₁	—	2	1	no	—
83	C₄×C₇₁	—	2	1	no	—
84	C₄×C₇₁	#347	1	2	yes	—
85	C₄×C₇₁	#405	1	1	yes	characteristic
86	C₄×C₇₁	#405	1	1	yes	characteristic
87	#11415	—	71	1	no	maximal, Sylow
88	C₂×C₇₁	—	2	2	no	—
89	C₂×C₇₁	#1452	1	1	yes	characteristic
90	C₂×C₇₁	C₂×#347	1	1	yes	characteristic
91	C₂×C₇₁	C₂×#347	1	1	yes	characteristic
92	#2163	—	71	1	no	—
93	#2163	D₇₁	1	1	yes	characteristic
94	C₂×#322	—	71	1	no	—
95	#1313	—	71	2	no	—
96	#1313	—	71	2	no	—
97	C₇₁	#11415	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄×C₈	—	71	1	no	—
99	C₂×C₄×C₈	—	71	1	no	—
100	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
101	C₄⋊C₁₆	—	71	2	no	—
102	C₄⋊C₁₆	Dic₇₁	1	2	yes	—
103	C₂²⋊C₁₆	—	71	2	no	—
104	C₂²⋊C₁₆	Dic₇₁	1	2	yes	—
105	#346	—	71	2	no	—
106	#346	C₂×D₇₁	1	2	yes	—
107	#322	—	71	1	no	—
108	#322	—	71	1	no	—
109	#322	—	71	1	no	—
110	#322	—	71	1	no	—
111	C₂²×C₈	—	142	1	no	—
112	C₂²×C₈	—	142	1	no	—
113	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
116	C₂×C₁₆	—	71	4	no	—
117	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
118	C₄×C₈	—	71	1	no	—
119	C₄×C₈	—	71	1	no	—
120	C₄×C₈	—	71	1	no	—
121	C₄×C₈	—	71	1	no	—
122	C₄×C₈	—	142	1	no	—
123	C₄×C₈	—	142	1	no	—
124	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
125	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
127	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
128	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
129	C₂²×C₄	#11765848	1	1	yes	characteristic
130	C₂²×C₄	#11765848	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	284	2	no	—
134	C₂×C₈	—	284	2	no	—
135	C₂×C₈	—	142	1	no	—
136	C₂×C₈	—	142	1	no	—
137	C₂×C₈	—	142	1	no	—
138	C₂×C₈	—	142	1	no	—
139	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
140	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
143	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
144	C₄²	#11765848	1	1	yes	characteristic
145	C₄²	#11765848	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
147	C₁₆	—	2	4	no	—
148	C₁₆	—	142	4	no	—
149	C₂³	#14556038	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	#14556038	1	1	yes	characteristic
155	C₂×C₄	#14556038	1	1	yes	characteristic
156	C₂×C₄	#14556038	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
158	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
159	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	284	1	no	—
163	C₈	—	284	1	no	—
164	C₈	—	284	1	no	—
165	C₈	—	284	1	no	—
166	C₈	#14556135	1	2	yes	—
167	C₈	#14556136	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#17985776	1	2	yes	—
171	C₂²	C₂×#14556038	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#17985776	1	2	yes	—
177	C₄	#17985955	1	1	yes	characteristic
178	C₄	#17985957	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	#22586633	1	1	yes	characteristic
181	C₂	C₂×#17985776	1	1	yes	characteristic
182	C₂	C₂×#17985776	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2163	C₂
socle	C₂²×C₇₁	C₂×#97
prime radical	C₂³×C₇₁	#97
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#2163	D₇₁
hypocenter	C₇₁	#11415
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #22586262	C₂	semidirect
4. #22586260	C₂	semidirect
97. C₇₁	#11415	semidirect

Maximal subgroups

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

subgroup	index
#22590143	2
C₂×#17985720	2
#22586262	2
#22586260	2
C₇₁×#2163	2
#11415	71