Subgroups — #31915667

Order 18176.

614 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985800	C₂	1	4	yes	maximal, complemented
2	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#22588500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#493×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556062	—	2	1	no	—
7	C₂×#14556062	C₂²	1	2	yes	—
8	#17985869	—	2	1	no	—
9	#17985800	—	2	8	no	—
10	C₄×C₇₁⋊C₁₆	—	2	2	no	—
11	C₂²×D₈×C₇₁	—	2	1	no	—
12	C₂²×D₈×C₇₁	C₂²	1	2	yes	—
13	C₂×#127×C₇₁	—	2	1	no	—
14	#493×C₇₁	—	2	2	no	—
15	#493×C₇₁	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇₁	—	2	2	no	—
18	C₂×C₄⋊Dic₇₁	—	4	2	no	—
19	C₂×C₇₁⋊C₁₆	—	2	2	no	—
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	complemented
22	#14556062	—	2	4	no	—
23	#14556062	—	4	2	no	—
24	C₂²×D₄×C₇₁	—	2	2	no	—
25	C₂²×D₄×C₇₁	—	4	2	no	—
26	C₂×D₈×C₇₁	—	2	4	no	—
27	C₂×D₈×C₇₁	—	2	4	no	—
28	C₂×D₈×C₇₁	—	2	2	no	—
29	C₂×D₈×C₇₁	—	2	4	no	—
30	C₂×D₈×C₇₁	—	4	2	no	—
31	C₂×D₈×C₇₁	—	4	2	no	—
32	C₂×D₈×C₇₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	#127×C₇₁	—	2	4	no	—
36	#127×C₇₁	—	4	2	no	—
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₁	D₄	1	2	yes	—
40	C₂²×Dic₇₁	—	4	2	no	—
41	C₄⋊Dic₇₁	—	4	2	no	—
42	C₄⋊Dic₇₁	—	4	2	no	—
43	C₄⋊Dic₇₁	—	8	4	no	—
44	C₇₁⋊C₁₆	—	2	4	no	—
45	C₂⁴×C₇₁	—	4	2	no	—
46	C₂×D₄×C₇₁	—	2	4	no	—
47	C₂×D₄×C₇₁	—	4	8	no	—
48	C₂×D₄×C₇₁	—	4	2	no	—
49	C₂×D₄×C₇₁	—	4	2	no	—
50	C₂×D₄×C₇₁	—	4	2	no	—
51	C₂×D₄×C₇₁	—	4	2	no	—
52	C₂×D₄×C₇₁	—	4	2	no	—
53	C₂×D₄×C₇₁	—	8	4	no	—
54	C₂×D₄×C₇₁	—	8	4	no	—
55	C₂²×C₄×C₇₁	D₈	1	2	yes	—
56	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	D₈×C₇₁	—	2	8	no	—
58	D₈×C₇₁	—	4	8	no	—
59	D₈×C₇₁	—	4	4	no	—
60	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₁	D₈	1	2	yes	—
62	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
63	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	#41	1	2	yes	—
67	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₇₁	SD₁₆	1	2	yes	—
69	C₂³×C₇₁	—	4	4	no	—
70	C₂³×C₇₁	—	8	2	no	—
71	C₂³×C₇₁	—	8	2	no	—
72	C₂³×C₇₁	—	8	4	no	—
73	C₂³×C₇₁	—	8	4	no	—
74	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
75	D₄×C₇₁	—	4	4	no	—
76	D₄×C₇₁	—	4	4	no	—
77	D₄×C₇₁	—	8	16	no	—
78	D₄×C₇₁	—	8	2	no	—
79	D₄×C₇₁	—	8	2	no	—
80	C₂×C₄×C₇₁	#102	1	4	yes	—
81	C₂×C₄×C₇₁	D₁₆	1	4	yes	—
82	C₂×C₄×C₇₁	SD₃₂	1	4	yes	—
83	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×Dic₇₁	—	8	2	no	—
90	C₂×Dic₇₁	—	8	4	no	—
91	C₈×C₇₁	#102	1	4	yes	—
92	C₈×C₇₁	C₄×D₄	1	2	yes	—
93	C₈×C₇₁	#121	1	2	yes	—
94	C₂²×C₇₁	—	8	8	no	—
95	C₂²×C₇₁	—	8	4	no	—
96	C₂²×C₇₁	—	16	4	no	—
97	C₂²×C₇₁	—	16	2	no	—
98	C₂²×C₇₁	—	16	2	no	—
99	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
100	C₂²×C₇₁	#486	1	1	yes	characteristic
101	C₂²×C₇₁	#493	1	1	yes	characteristic
102	C₂²×C₇₁	C₂×D₁₆	1	2	yes	—
103	C₂²×C₇₁	C₂×SD₃₂	1	2	yes	—
104	C₄×C₇₁	#357	1	8	yes	complemented
105	C₄×C₇₁	#390	1	1	yes	characteristic
106	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
107	C₄×C₇₁	#486	1	1	yes	characteristic
108	C₄×C₇₁	#492	1	1	yes	characteristic
109	Dic₇₁	—	16	2	no	—
110	#13202	—	71	1	no	maximal, Sylow
111	C₂×C₇₁	—	16	4	no	—
112	C₂×C₇₁	#1957	1	1	yes	characteristic
113	C₂×C₇₁	C₂×#357	1	2	yes	—
114	C₂×C₇₁	#2240	1	2	yes	—
115	C₂×C₇₁	#2246	1	1	yes	characteristic
116	C₂×C₇₁	#2248	1	1	yes	characteristic
117	C₂×#493	D₇₁	1	1	yes	characteristic
118	C₂×#357	—	71	4	no	—
119	C₂×C₄×C₁₆	—	71	1	no	—
120	#1845	—	71	1	no	—
121	C₇₁	#13202	1	1	yes	characteristic, complemented, Sylow
122	C₂²×D₈	—	2	1	no	—
123	C₂²×D₈	C₂×D₇₁	1	2	yes	—
124	C₂×#127	—	2	1	no	—
125	C₂²×C₁₆	—	71	2	no	—
126	#493	—	2	2	no	—
127	#493	Dic₇₁	1	4	yes	complemented
128	C₂×#107	—	71	2	no	—
129	C₂×#107	—	142	1	no	—
130	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
131	#384	—	142	1	no	—
132	#357	—	142	8	no	—
133	C₄×C₁₆	—	142	2	no	—
134	C₂²×D₄	—	2	2	no	—
135	C₂²×D₄	—	4	2	no	—
136	C₂×D₈	—	2	4	no	—
137	C₂×D₈	—	2	4	no	—
138	C₂×D₈	—	2	2	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	4	2	no	—
141	C₂×D₈	—	4	2	no	—
142	C₂×D₈	C₂×Dic₇₁	1	4	yes	—
143	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
145	#127	—	2	4	no	—
146	#127	—	4	2	no	—
147	C₂×C₄⋊C₄	—	284	2	no	—
148	C₂×C₄⋊C₄	—	142	2	no	—
149	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
150	C₂×C₁₆	—	71	4	no	—
151	C₂×C₁₆	—	142	2	no	—
152	C₂×C₁₆	—	142	2	no	—
153	#107	—	284	2	no	—
154	#107	—	142	4	no	—
155	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
156	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
157	C₂⁴	—	4	2	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	4	8	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	8	4	no	—
166	C₂×D₄	—	8	4	no	—
167	C₂²×C₄	—	284	2	no	—
168	C₂²×C₄	#11765850	1	2	yes	—
169	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
170	D₈	—	2	8	no	—
171	D₈	—	4	8	no	—
172	D₈	—	4	4	no	—
173	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
174	C₂×C₈	#11765850	1	2	yes	—
175	C₂×C₈	#11765853	1	2	yes	—
176	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
177	C₂×C₈	D₄×D₇₁	1	2	yes	—
178	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
179	C₂×C₈	#11765864	1	2	yes	—
180	C₄⋊C₄	—	284	2	no	—
181	C₄⋊C₄	—	284	2	no	—
182	C₄⋊C₄	—	568	4	no	—
183	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
184	C₄²	#11765853	1	2	yes	—
185	C₁₆	—	142	4	no	—
186	C₂³	—	4	4	no	—
187	C₂³	—	8	2	no	—
188	C₂³	—	8	2	no	—
189	C₂³	—	8	4	no	—
190	C₂³	—	8	4	no	—
191	C₂³	C₂×#11765850	1	1	yes	characteristic
192	D₄	—	4	4	no	—
193	D₄	—	4	4	no	—
194	D₄	—	8	16	no	—
195	D₄	—	8	2	no	—
196	D₄	—	8	2	no	—
197	C₂×C₄	—	568	2	no	—
198	C₂×C₄	—	568	4	no	—
199	C₂×C₄	#14556049	1	4	yes	—
200	C₂×C₄	#14556069	1	4	yes	—
201	C₂×C₄	#14556072	1	4	yes	—
202	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
203	C₂×C₄	#14556115	1	1	yes	characteristic
204	C₂×C₄	#14556124	1	1	yes	characteristic
205	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
206	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
207	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
208	C₈	#14556049	1	4	yes	—
209	C₈	D₄×Dic₇₁	1	2	yes	—
210	C₈	#14556099	1	2	yes	—
211	C₂²	—	8	8	no	—
212	C₂²	—	8	4	no	—
213	C₂²	—	16	4	no	—
214	C₂²	—	16	2	no	—
215	C₂²	—	16	2	no	—
216	C₂²	C₂×#14556049	1	1	yes	characteristic
217	C₂²	#17986345	1	1	yes	characteristic
218	C₂²	#17986375	1	1	yes	characteristic
219	C₂²	C₂×#14556069	1	2	yes	—
220	C₂²	C₂×#14556072	1	2	yes	—
221	C₄	—	1136	2	no	—
222	C₄	#17985800	1	8	yes	complemented
223	C₄	#17985902	1	1	yes	characteristic
224	C₄	C₂×#14556049	1	1	yes	characteristic
225	C₄	#17986345	1	1	yes	characteristic
226	C₄	#17986371	1	1	yes	characteristic
227	C₂	—	16	4	no	—
228	C₂	#22589147	1	1	yes	characteristic
229	C₂	C₂×#17985800	1	2	yes	—
230	C₂	#22590563	1	2	yes	—
231	C₂	#22590589	1	1	yes	characteristic
232	C₂	#22590595	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting, prime radical	C₂×#493×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#493	D₇₁
hypocenter	C₇₁	#13202
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#17985800	C₂	semidirect
2. C₂×C₄×C₇₁⋊C₁₆	C₂	semidirect
3. #22588500	C₂	semidirect
15. #493×C₇₁	C₄	semidirect
21. C₂×C₇₁⋊C₁₆	D₄	semidirect
104. C₄×C₇₁	#357	semidirect
121. C₇₁	#13202	semidirect
127. #493	Dic₇₁	semidirect
222. C₄	#17985800	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985800	2
C₂×C₄×C₇₁⋊C₁₆	2
#22588500	2
C₂×#493×C₇₁	2
#13202	71