Subgroups — #31916011

Order 18176.

366 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986397	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985820	C₂	1	1	yes	maximal, characteristic
3	#22586445	C₂	1	2	yes	maximal
4	#22586440	C₂	1	2	yes	maximal
5	C₇₁×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	—	2	1	no	—
7	C₂×#14556068	C₂²	1	2	yes	—
8	#17986397	—	2	2	no	—
9	#17986397	C₄	1	4	yes	complemented
10	C₂×#14556060	—	2	1	no	—
11	C₂×#14556060	C₂²	1	2	yes	—
12	#17985820	—	2	2	no	—
13	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
14	#427×C₇₁	C₂²	1	1	yes	characteristic
15	#405×C₇₁	—	2	1	no	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#339×C₇₁	—	2	1	no	—
18	C₂²×Dic₁₄₂	—	4	1	no	—
19	#14556128	—	2	4	no	—
20	#14556128	—	4	2	no	—
21	C₂×C₄⋊Dic₇₁	—	2	2	no	—
22	C₂×C₄⋊Dic₇₁	—	4	1	no	—
23	#14556068	—	2	2	no	—
24	#14556068	—	2	2	no	—
25	C₂×C₇₁⋊C₁₆	—	2	2	no	—
26	#14556060	—	2	4	no	—
27	#14556060	—	2	2	no	—
28	#14556060	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₇₁	—	2	1	no	—
30	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	#117×C₇₁	—	2	1	no	—
33	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
34	#108×C₇₁	—	2	1	no	—
35	#108×C₇₁	Q₈	1	2	yes	complemented
36	#107×C₇₁	Q₈	1	2	yes	complemented
37	#106×C₇₁	—	2	1	no	—
38	#97×C₇₁	—	4	1	no	—
39	C₄×C₈×C₇₁	—	4	1	no	—
40	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₇₁	D₄	1	2	yes	—
42	C₂×Dic₁₄₂	—	4	4	no	—
43	C₂×Dic₁₄₂	—	8	4	no	—
44	C₂²×Dic₇₁	—	4	2	no	—
45	C₄⋊Dic₇₁	—	2	4	no	—
46	C₄⋊Dic₇₁	—	4	4	no	—
47	C₄⋊Dic₇₁	—	4	2	no	—
48	C₇₁⋊C₁₆	—	4	2	no	—
49	C₂²×C₄×C₇₁	—	2	1	no	—
50	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₇₁	—	2	2	no	—
52	M₁₆×C₇₁	—	4	1	no	—
53	C₂×C₈×C₇₁	—	4	1	no	—
54	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
58	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
63	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
64	C₄⋊C₄×C₇₁	—	2	2	no	—
65	C₂²⋊C₄×C₇₁	—	4	1	no	—
66	C₄²×C₇₁	—	2	1	no	—
67	C₄²×C₇₁	—	4	1	no	—
68	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
69	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂×C₄×C₇₁	—	2	4	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	—	4	2	no	—
73	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×Dic₇₁	—	4	4	no	—
80	C₂×Dic₇₁	—	8	4	no	—
81	Dic₁₄₂	—	8	8	no	—
82	C₈×C₇₁	—	4	2	no	—
83	C₈×C₇₁	#103	1	2	yes	—
84	C₈×C₇₁	#103	1	2	yes	—
85	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₇₁	#124	1	1	yes	characteristic
88	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
89	C₂²×C₇₁	—	2	2	no	—
90	C₂²×C₇₁	#418	1	1	yes	characteristic
91	C₂²×C₇₁	#476	1	1	yes	characteristic
92	C₂²×C₇₁	#477	1	1	yes	characteristic
93	C₄×C₇₁	—	2	4	no	—
94	C₄×C₇₁	—	8	2	no	—
95	C₄×C₇₁	#391	1	1	yes	characteristic
96	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
97	C₄×C₇₁	#476	1	1	yes	characteristic
98	C₄×C₇₁	#477	1	1	yes	characteristic
99	Dic₇₁	—	8	4	no	—
100	#13272	—	71	1	no	maximal, Sylow
101	C₂×C₇₁	—	2	1	no	—
102	C₂×C₇₁	—	2	1	no	—
103	C₂×C₇₁	#1987	1	1	yes	characteristic
104	C₂×C₇₁	#2230	1	2	yes	—
105	C₂×#498	—	71	1	no	—
106	C₂×#364	—	71	1	no	—
107	#1772	D₇₁	1	1	yes	characteristic
108	#1386	—	71	2	no	—
109	#1383	—	71	2	no	—
110	C₇₁	#13272	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄⋊Q₈	—	142	1	no	—
112	C₂×M₃₂	—	71	2	no	—
113	#498	—	71	4	no	—
114	#498	—	142	2	no	—
115	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
116	#427	C₂×D₇₁	1	1	yes	characteristic
117	C₂×#106	—	71	2	no	—
118	C₂×#106	—	142	1	no	—
119	#405	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
121	#364	—	142	2	no	—
122	#339	—	2	1	no	—
123	C₂²×Q₈	—	284	1	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
127	C₄⋊Q₈	—	284	2	no	—
128	C₄⋊Q₈	—	142	4	no	—
129	#117	—	2	1	no	—
130	C₂×C₄⋊C₄	—	284	1	no	—
131	C₂×C₄⋊C₄	—	142	2	no	—
132	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
133	M₃₂	—	142	2	no	—
134	M₃₂	—	142	2	no	—
135	C₂×C₁₆	—	142	2	no	—
136	#108	—	2	1	no	—
137	#108	Dic₁₄₂	1	2	yes	complemented
138	#107	Dic₁₄₂	1	2	yes	complemented
139	#106	—	2	1	no	—
140	#106	—	71	4	no	—
141	#106	—	142	4	no	—
142	#106	—	142	2	no	—
143	#97	—	4	1	no	—
144	C₄×C₈	—	4	1	no	—
145	C₄×C₈	C₄×D₇₁	1	2	yes	—
146	C₄×C₈	D₂₈₄	1	2	yes	—
147	C₂×Q₈	—	284	4	no	—
148	C₂×Q₈	—	568	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	284	2	no	—
151	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	4	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
156	C₂×C₈	#11765853	1	1	yes	characteristic
157	C₂×C₈	#11765853	1	1	yes	characteristic
158	C₂×C₈	#11765855	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
160	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
161	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
162	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
163	C₂×C₈	#11765865	1	1	yes	characteristic
164	C₂×C₈	#11765865	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	284	4	no	—
167	C₄⋊C₄	—	284	2	no	—
168	C₄⋊C₄	—	142	4	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
173	C₁₆	—	284	2	no	—
174	C₂³	#14556153	1	1	yes	characteristic
175	Q₈	—	568	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	284	4	no	—
180	C₂×C₄	—	568	4	no	—
181	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
182	C₂×C₄	#14556107	1	1	yes	characteristic
183	C₂×C₄	#14556107	1	1	yes	characteristic
184	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
185	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
186	C₂×C₄	#14556153	1	1	yes	characteristic
187	C₈	—	4	2	no	—
188	C₈	#14556051	1	2	yes	—
189	C₈	#14556051	1	2	yes	—
190	C₈	C₄×Dic₁₄₂	1	1	yes	characteristic
191	C₈	C₄×Dic₁₄₂	1	1	yes	characteristic
192	C₈	#14556116	1	1	yes	characteristic
193	C₈	#14556128	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	#17986003	1	1	yes	characteristic
196	C₂²	#17986289	1	1	yes	characteristic
197	C₂²	#17986294	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	2	no	—
200	C₄	—	568	4	no	—
201	C₄	#17985905	1	1	yes	characteristic
202	C₄	C₂×#14556051	1	1	yes	characteristic
203	C₄	#17986289	1	1	yes	characteristic
204	C₄	#17986294	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#22589288	1	1	yes	characteristic
208	C₂	#22590511	1	2	yes	—

	subgroup	G/H
center	C₂²	#17986289
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1772	C₂
socle	C₂²×C₇₁	#476
prime radical	C₂³×C₇₁	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1772	D₇₁
hypocenter	C₇₁	#13272
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
9. #17986397	C₄	semidirect
35. #108×C₇₁	Q₈	semidirect
36. #107×C₇₁	Q₈	semidirect
110. C₇₁	#13272	semidirect
137. #108	Dic₁₄₂	semidirect
138. #107	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986397	2
C₂×#17985820	2
#22586445	2
#22586440	2
C₇₁×#1772	2
#13272	71