Subgroups — #31915599

Order 18176.

294 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	C₂×#17985822	C₂	1	2	yes	maximal
2	#22588504	C₂	1	1	yes	maximal, characteristic
3	#22588219	C₂	1	1	yes	maximal, characteristic
4	#22586446	C₂	1	2	yes	maximal
5	C₂×#364×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556064	C₂²	1	1	yes	characteristic
7	#17986037	C₂²	1	1	yes	characteristic
8	C₂×#14556062	—	2	1	no	—
9	C₂×#14556060	C₂²	1	2	yes	—
10	#17985954	—	2	1	no	—
11	#17985865	—	2	1	no	—
12	#17985822	—	2	4	no	—
13	#17985759	—	2	1	no	—
14	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#364×C₇₁	C₄	1	4	yes	complemented
17	C₂×#11765849	—	2	1	no	—
18	#14556087	—	2	1	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	#14556064	—	2	1	no	—
21	#14556064	D₄	1	2	yes	—
22	#14556062	—	4	2	no	—
23	#14556062	Q₈	1	2	yes	—
24	#14556060	—	2	2	no	—
25	#14556060	—	2	1	no	—
26	#14556060	—	2	2	no	—
27	#14556038	—	4	1	no	—
28	C₈×Dic₇₁	—	4	1	no	—
29	C₇₁⋊#95	—	4	2	no	—
30	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
33	M₃₂×C₇₁	—	2	2	no	—
34	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
35	C₂×C₁₆×C₇₁	—	2	2	no	—
36	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₇₁	Q₈	1	2	yes	—
38	C₂²×Dic₇₁	—	4	2	no	—
39	#11765849	—	2	2	no	—
40	#11765849	—	4	1	no	—
41	C₂×C₇₁⋊C₈	—	4	1	no	—
42	C₄⋊Dic₇₁	—	2	2	no	—
43	C₄⋊Dic₇₁	—	4	2	no	—
44	C₄⋊Dic₇₁	—	4	2	no	—
45	C₂²⋊Dic₇₁	—	4	1	no	—
46	C₄×Dic₇₁	—	4	1	no	—
47	C₂²×C₄×C₇₁	—	2	1	no	—
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
53	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
58	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
59	C₄²×C₇₁	—	2	1	no	—
60	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
61	C₁₆×C₇₁	—	2	4	no	—
62	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₇₁	—	2	2	no	—
64	C₂×C₄×C₇₁	—	4	2	no	—
65	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
71	C₂×Dic₇₁	—	4	2	no	—
72	C₂×Dic₇₁	—	8	4	no	—
73	C₂×Dic₇₁	—	8	2	no	—
74	C₈×C₇₁	#106	1	2	yes	—
75	C₈×C₇₁	#106	1	2	yes	—
76	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
77	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
78	C₈×C₇₁	#125	1	1	yes	characteristic
79	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₇₁⋊C₈	—	4	2	no	—
81	C₂²×C₇₁	—	2	2	no	—
82	C₂²×C₇₁	#428	1	1	yes	characteristic
83	C₂²×C₇₁	#460	1	1	yes	characteristic
84	C₂²×C₇₁	#476	1	1	yes	characteristic
85	C₄×C₇₁	—	4	2	no	—
86	C₄×C₇₁	#389	1	1	yes	characteristic
87	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
88	C₄×C₇₁	#461	1	1	yes	characteristic
89	C₄×C₇₁	#475	1	1	yes	characteristic
90	Dic₇₁	—	8	2	no	—
91	Dic₇₁	—	16	2	no	—
92	#13187	—	71	1	no	maximal, Sylow
93	C₂×C₇₁	—	2	1	no	—
94	C₂×C₇₁	—	2	1	no	—
95	C₂×C₇₁	#1948	1	1	yes	characteristic
96	C₂×C₇₁	#2231	1	2	yes	—
97	C₂×#365	—	71	2	no	—
98	C₂×#364	D₇₁	1	1	yes	characteristic
99	#1846	—	71	1	no	—
100	#1772	—	71	1	no	—
101	#1386	—	71	2	no	—
102	C₇₁	#13187	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
104	C₂×#108	—	71	1	no	—
105	#427	—	71	1	no	—
106	C₂×#107	—	142	1	no	—
107	C₂×#106	—	71	2	no	—
108	#405	—	142	1	no	—
109	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	#382	—	142	1	no	—
111	#365	—	142	4	no	—
112	#364	Dic₇₁	1	4	yes	complemented
113	#339	—	142	1	no	—
114	C₂×M₁₆	—	142	1	no	—
115	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	#117	—	142	1	no	—
118	C₂×C₄⋊C₄	—	142	2	no	—
119	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
120	M₃₂	—	2	2	no	—
121	M₃₂	C₂×Dic₇₁	1	4	yes	—
122	C₂×C₁₆	—	2	2	no	—
123	#108	—	71	2	no	—
124	#108	—	142	1	no	—
125	#107	—	284	2	no	—
126	#107	—	71	2	no	—
127	#106	—	142	2	no	—
128	#106	—	142	2	no	—
129	#106	—	142	1	no	—
130	#97	—	284	1	no	—
131	C₄×C₈	—	284	1	no	—
132	C₄×C₈	Dic₁₄₂	1	2	yes	—
133	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
134	#95	—	284	2	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	284	2	no	—
137	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
138	M₁₆	—	284	1	no	—
139	M₁₆	—	142	2	no	—
140	C₂×C₈	—	284	1	no	—
141	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
142	C₂×C₈	#11765854	1	1	yes	characteristic
143	C₂×C₈	#11765854	1	1	yes	characteristic
144	C₂×C₈	#11765854	1	2	yes	—
145	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
146	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
148	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
149	C₂×C₈	#11765864	1	1	yes	characteristic
150	C₂×C₈	#11765866	1	1	yes	characteristic
151	C₄⋊C₄	—	284	2	no	—
152	C₄⋊C₄	—	284	2	no	—
153	C₄⋊C₄	—	142	2	no	—
154	C₂²⋊C₄	—	284	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	284	1	no	—
157	C₄²	C₄⋊Dic₇₁	1	2	yes	—
158	C₁₆	—	2	4	no	—
159	C₂³	#14556156	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	284	2	no	—
163	C₂×C₄	—	568	4	no	—
164	C₂×C₄	—	568	2	no	—
165	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
166	C₂×C₄	#14556103	1	1	yes	characteristic
167	C₂×C₄	#14556108	1	1	yes	characteristic
168	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
169	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
170	C₂×C₄	#14556161	1	1	yes	characteristic
171	C₈	—	284	2	no	—
172	C₈	#14556060	1	2	yes	—
173	C₈	#14556060	1	2	yes	—
174	C₈	D₄×Dic₇₁	1	1	yes	characteristic
175	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
176	C₈	#14556119	1	1	yes	characteristic
177	C₈	#14556127	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#17986041	1	1	yes	characteristic
180	C₂²	#17986200	1	1	yes	characteristic
181	C₂²	#17986290	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	568	2	no	—
184	C₄	—	1136	2	no	—
185	C₄	#17985898	1	1	yes	characteristic
186	C₄	C₂×#14556060	1	1	yes	characteristic
187	C₄	#17986205	1	1	yes	characteristic
188	C₄	#17986285	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#22589104	1	1	yes	characteristic
192	C₂	#22590517	1	2	yes	—

	subgroup	G/H
center	C₂²	#17986290
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#364×C₇₁	C₂
socle	C₂²×C₇₁	#476
prime radical	C₂³×C₇₁	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#364	D₇₁
hypocenter	C₇₁	#13187
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
16. #364×C₇₁	C₄	semidirect
102. C₇₁	#13187	semidirect
112. #364	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985822	2
#22588504	2
#22588219	2
#22586446	2
C₂×#364×C₇₁	2
#13187	71