Subgroups — #31915558

Order 18176.

286 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985828	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#22588512	C₂	1	1	yes	maximal, characteristic
4	#22586433	C₂	1	2	yes	maximal
5	C₄×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14556064	—	2	1	no	—
8	C₂×#14556064	C₂²	1	2	yes	—
9	#17986023	—	2	1	no	—
10	#17985828	—	2	4	no	—
11	C₄⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	—
12	C₄⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	—
13	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₇₁	—	2	1	no	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#341×C₇₁	—	2	1	no	—
18	C₂×#11765849	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	—	2	2	no	—
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
22	#14556064	—	2	2	no	—
23	#14556064	—	2	2	no	—
24	#14556064	—	4	2	no	—
25	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
26	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
27	C₂×M₁₆×C₇₁	—	2	1	no	—
28	C₂×M₁₆×C₇₁	—	2	1	no	—
29	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
32	#108×C₇₁	—	2	1	no	—
33	#108×C₇₁	—	2	1	no	—
34	#108×C₇₁	D₄	1	2	yes	—
35	#108×C₇₁	Q₈	1	2	yes	—
36	#97×C₇₁	—	4	1	no	—
37	C₄×C₈×C₇₁	—	4	1	no	—
38	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
42	#11765849	—	4	2	no	—
43	#11765849	—	4	2	no	—
44	C₂×C₇₁⋊C₈	—	4	2	no	—
45	C₇₁⋊C₁₆	—	2	4	no	—
46	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₇₁	—	2	2	no	—
50	M₁₆×C₇₁	—	2	2	no	—
51	M₁₆×C₇₁	—	4	1	no	—
52	M₁₆×C₇₁	—	4	1	no	—
53	C₂×C₈×C₇₁	—	2	1	no	—
54	C₂×C₈×C₇₁	—	2	1	no	—
55	C₂×C₈×C₇₁	—	4	1	no	—
56	C₂×C₈×C₇₁	—	4	1	no	—
57	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
64	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
65	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
66	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₇₁	D₈	1	1	yes	characteristic
68	C₄²×C₇₁	D₈	1	1	yes	characteristic
69	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	—	4	1	no	—
72	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
73	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
82	C₈×C₇₁	—	2	1	no	—
83	C₈×C₇₁	—	2	1	no	—
84	C₈×C₇₁	—	4	2	no	—
85	C₈×C₇₁	—	4	2	no	—
86	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
87	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
88	C₈×C₇₁	#125	1	1	yes	characteristic
89	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
90	C₇₁⋊C₈	—	8	2	no	—
91	C₂²×C₇₁	—	2	1	no	—
92	C₂²×C₇₁	—	2	1	no	—
93	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
94	C₂²×C₇₁	#462	1	1	yes	characteristic
95	C₂²×C₇₁	#477	1	1	yes	characteristic
96	C₄×C₇₁	—	2	1	no	—
97	C₄×C₇₁	—	2	1	no	—
98	C₄×C₇₁	—	2	1	no	—
99	C₄×C₇₁	#368	1	2	yes	complemented
100	C₄×C₇₁	#389	1	1	yes	characteristic
101	C₄×C₇₁	#428	1	1	yes	characteristic
102	C₄×C₇₁	#459	1	1	yes	characteristic
103	C₄×C₇₁	#474	1	1	yes	characteristic
104	#13179	—	71	1	no	maximal, Sylow
105	C₂×C₇₁	—	2	1	no	—
106	C₂×C₇₁	—	2	1	no	—
107	C₂×C₇₁	#1947	1	1	yes	characteristic
108	C₂×C₇₁	C₂×#368	1	1	yes	characteristic
109	C₂×C₇₁	#2161	1	1	yes	characteristic
110	C₂×#368	—	71	2	no	—
111	C₂×C₄⋊C₁₆	—	71	1	no	—
112	#1848	—	71	1	no	—
113	C₄×#108	D₇₁	1	1	yes	characteristic
114	#1381	—	71	2	no	—
115	C₇₁	#13179	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₁₆	—	71	2	no	—
117	C₂×#108	—	71	2	no	—
118	C₂×#108	—	142	1	no	—
119	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
120	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
121	#423	—	142	1	no	—
122	C₄×M₁₆	—	2	1	no	—
123	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
124	#368	—	142	4	no	—
125	C₄⋊C₁₆	—	71	2	no	—
126	C₄⋊C₁₆	—	71	2	no	—
127	#341	—	2	1	no	—
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂×M₁₆	—	142	2	no	—
131	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
133	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
134	C₂×C₁₆	—	71	4	no	—
135	C₂×C₁₆	—	142	2	no	—
136	C₂×C₁₆	—	142	2	no	—
137	#108	—	2	1	no	—
138	#108	—	2	1	no	—
139	#108	—	284	2	no	—
140	#108	—	142	2	no	—
141	#108	—	142	2	no	—
142	#108	Dic₁₄₂	1	2	yes	—
143	#108	D₂₈₄	1	2	yes	—
144	C₄⋊C₈	—	284	1	no	—
145	C₄⋊C₈	—	284	1	no	—
146	#97	—	4	1	no	—
147	C₄×C₈	—	4	1	no	—
148	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
149	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
150	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
151	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
152	C₂²×C₄	#11765855	1	1	yes	characteristic
153	C₂²×C₄	#11765855	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
155	M₁₆	—	2	2	no	—
156	M₁₆	—	2	2	no	—
157	M₁₆	—	4	1	no	—
158	M₁₆	—	4	1	no	—
159	M₁₆	—	284	2	no	—
160	M₁₆	—	284	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	284	2	no	—
166	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
167	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
170	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
171	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
172	C₂×C₈	#11765865	1	1	yes	characteristic
173	C₂×C₈	#11765865	1	1	yes	characteristic
174	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
175	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
176	C₄²	#11765850	1	1	yes	characteristic
177	C₄²	#11765850	1	1	yes	characteristic
178	C₁₆	—	142	4	no	—
179	C₂³	C₂×#11765855	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	#14556061	1	1	yes	characteristic
183	C₂×C₄	#14556061	1	1	yes	characteristic
184	C₂×C₄	#14556061	1	1	yes	characteristic
185	C₂×C₄	#14556061	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
187	C₂×C₄	#14556102	1	1	yes	characteristic
188	C₂×C₄	#14556107	1	1	yes	characteristic
189	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
190	C₂×C₄	#14556153	1	1	yes	characteristic
191	C₂×C₄	#14556158	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	4	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	568	2	no	—
197	C₈	C₄×D₂₈₄	1	1	yes	characteristic
198	C₈	C₄×Dic₁₄₂	1	1	yes	characteristic
199	C₈	#14556120	1	1	yes	characteristic
200	C₈	#14556128	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×#14556061	1	1	yes	characteristic
204	C₂²	#17986209	1	1	yes	characteristic
205	C₂²	#17986294	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	#17985828	1	2	yes	complemented
210	C₄	#17985897	1	1	yes	characteristic
211	C₄	#17986042	1	1	yes	characteristic
212	C₄	#17986194	1	1	yes	characteristic
213	C₄	#17986279	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	#22589098	1	1	yes	characteristic
217	C₂	C₂×#17985828	1	1	yes	characteristic
218	C₂	#22590134	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₄×#108×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄×#108	D₇₁
hypocenter	C₇₁	#13179
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
99. C₄×C₇₁	#368	semidirect
115. C₇₁	#13179	semidirect
209. C₄	#17985828	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985828	2
C₂×C₄⋊(C₇₁⋊C₁₆)	2
#22588512	2
#22586433	2
C₄×#108×C₇₁	2
#13179	71