Subgroups — #31896019

Order 18176.

378 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589058	C₂	1	2	yes	maximal
2	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985750	C₂	1	1	yes	maximal, characteristic
4	#22586180	C₂	1	2	yes	maximal
5	C₂×#322×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	C₄	1	2	yes	—
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
8	C₂×Dic₇₁⋊C₈	—	2	2	no	—
9	C₄×C₄⋊Dic₇₁	C₂²	1	2	yes	—
10	#17985762	C₄	1	1	yes	characteristic
11	#17985762	C₄	1	1	yes	characteristic
12	#17985753	—	2	2	no	—
13	#17985750	—	2	2	no	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#322×C₇₁	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₇₁	—	2	2	no	—
22	#14556062	—	2	4	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	Dic₇₁⋊C₈	—	4	4	no	—
26	C₇₁⋊#95	—	2	2	no	—
27	C₂²×C₈×C₇₁	—	2	1	no	—
28	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	D₄	1	2	yes	—
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₇₁	—	2	1	no	—
33	C₄×C₈×C₇₁	—	2	1	no	—
34	C₄×C₈×C₇₁	—	2	1	no	—
35	C₄×C₈×C₇₁	D₄	1	2	yes	—
36	C₄×C₈×C₇₁	Q₈	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	—	4	2	no	—
41	C₂×C₇₁⋊C₈	—	4	2	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₄⋊Dic₇₁	—	2	2	no	—
44	C₄⋊Dic₇₁	—	2	2	no	—
45	C₄⋊Dic₇₁	—	2	2	no	—
46	C₄⋊Dic₇₁	M₁₆	1	4	yes	—
47	C₄×Dic₇₁	—	4	2	no	—
48	C₄×Dic₇₁	—	4	2	no	—
49	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
52	C₂×C₈×C₇₁	—	2	4	no	—
53	C₂×C₈×C₇₁	—	2	2	no	—
54	C₂×C₈×C₇₁	—	2	2	no	—
55	C₂×C₈×C₇₁	—	2	2	no	—
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	—	2	2	no	—
59	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
61	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₇₁	#41	1	1	yes	characteristic
65	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#125	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₇₁	#131	1	2	yes	—
75	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₇₁	#137	1	2	yes	—
77	C₂×Dic₇₁	—	2	4	no	—
78	C₂×Dic₇₁	—	4	2	no	—
79	C₂×Dic₇₁	—	4	2	no	—
80	C₂×Dic₇₁	—	4	2	no	—
81	C₂×Dic₇₁	—	4	2	no	—
82	C₂×Dic₇₁	—	4	2	no	—
83	C₈×C₇₁	—	2	4	no	—
84	C₈×C₇₁	—	2	2	no	—
85	C₈×C₇₁	—	2	2	no	—
86	C₈×C₇₁	—	4	2	no	—
87	C₇₁⋊C₈	—	8	2	no	—
88	C₂²×C₇₁	#389	1	1	yes	characteristic
89	C₂²×C₇₁	#435	1	2	yes	—
90	C₂²×C₇₁	#441	1	1	yes	characteristic
91	C₂²×C₇₁	#442	1	1	yes	characteristic
92	C₂²×C₇₁	#443	1	2	yes	—
93	C₄×C₇₁	—	2	1	no	—
94	C₄×C₇₁	—	2	1	no	—
95	C₄×C₇₁	#423	1	1	yes	characteristic
96	C₄×C₇₁	#424	1	1	yes	characteristic
97	C₄×C₇₁	#446	1	2	yes	—
98	C₄×C₇₁	#468	1	2	yes	—
99	C₄×C₇₁	#501	1	2	yes	—
100	Dic₇₁	—	4	4	no	—
101	Dic₇₁	—	8	2	no	—
102	#9599	—	71	1	no	maximal, Sylow
103	C₂×C₇₁	#1586	1	2	yes	—
104	C₂×C₇₁	#1587	1	2	yes	—
105	C₂×C₇₁	#1940	1	1	yes	characteristic
106	C₂×C₇₁	#1943	1	1	yes	characteristic
107	C₂×C₇₁	#1945	1	1	yes	characteristic
108	#1939	—	71	2	no	—
109	C₄×#107	—	71	1	no	—
110	C₂×#334	—	71	1	no	—
111	C₂×#322	D₇₁	1	1	yes	characteristic
112	#1279	—	71	2	no	—
113	C₇₁	#9599	1	1	yes	characteristic, complemented, Sylow
114	C₂×#107	—	71	2	no	—
115	C₂×C₄⋊C₈	—	71	2	no	—
116	C₂×C₄⋊C₈	—	142	2	no	—
117	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	71	2	no	—
121	#340	—	71	1	no	—
122	#340	—	71	1	no	—
123	#336	—	142	2	no	—
124	#334	—	142	2	no	—
125	#322	—	2	2	no	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	142	2	no	—
128	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
129	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
130	C₂²×C₈	D₂₈₄	1	2	yes	—
131	C₂×C₄⋊C₄	—	71	2	no	—
132	C₂×C₄⋊C₄	—	71	2	no	—
133	C₂×C₄²	—	142	2	no	—
134	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
135	#107	—	142	4	no	—
136	C₄⋊C₈	—	284	4	no	—
137	C₄⋊C₈	—	142	2	no	—
138	C₄⋊C₈	—	142	2	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	—	2	1	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	Dic₁₄₂	1	2	yes	—
143	C₄×C₈	Dic₁₄₂	1	2	yes	—
144	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
145	#95	—	142	2	no	—
146	C₂²×C₄	—	142	2	no	—
147	C₂²×C₄	—	142	2	no	—
148	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
150	C₂²×C₄	#11765865	1	1	yes	characteristic
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	284	2	no	—
159	C₂×C₈	—	284	2	no	—
160	C₂×C₈	—	284	2	no	—
161	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
162	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
163	C₄⋊C₄	—	71	4	no	—
164	C₄⋊C₄	—	142	2	no	—
165	C₄⋊C₄	—	142	2	no	—
166	C₄⋊C₄	—	142	2	no	—
167	C₄²	—	284	2	no	—
168	C₄²	—	284	2	no	—
169	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
170	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
171	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
172	C₄²	#11765865	1	1	yes	characteristic
173	C₂³	C₄×D₂₈₄	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	284	2	no	—
176	C₂×C₄	—	284	2	no	—
177	C₂×C₄	—	284	2	no	—
178	C₂×C₄	—	284	2	no	—
179	C₂×C₄	—	284	2	no	—
180	C₂×C₄	—	142	4	no	—
181	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
182	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
183	C₂×C₄	#14556102	1	1	yes	characteristic
184	C₂×C₄	#14556107	1	1	yes	characteristic
185	C₂×C₄	#14556120	1	1	yes	characteristic
186	C₂×C₄	#14556128	1	1	yes	characteristic
187	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
188	C₂×C₄	#14556136	1	2	yes	—
189	C₂×C₄	#14556156	1	2	yes	—
190	C₂×C₄	#14556161	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	568	2	no	—
196	C₂²	#17985897	1	1	yes	characteristic
197	C₂²	#17986072	1	2	yes	—
198	C₂²	#17986107	1	1	yes	characteristic
199	C₂²	#17986112	1	1	yes	characteristic
200	C₂²	#17986117	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	284	4	no	—
204	C₄	—	568	2	no	—
205	C₄	#17986022	1	1	yes	characteristic
206	C₄	#17986027	1	1	yes	characteristic
207	C₄	#17986127	1	2	yes	—
208	C₄	#17986245	1	2	yes	—
209	C₄	#17986408	1	2	yes	—
210	C₂	#22587206	1	2	yes	—
211	C₂	#22587213	1	2	yes	—
212	C₂	#22589063	1	1	yes	characteristic
213	C₂	#22589078	1	1	yes	characteristic
214	C₂	#22589088	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈₄
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#322×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#322	D₇₁
hypocenter	C₇₁	#9599
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
113. C₇₁	#9599	semidirect

Maximal subgroups

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

subgroup	index
#22589058	2
C₄×#14556062	2
C₂×#17985750	2
#22586180	2
C₂×#322×C₇₁	2
#9599	71