Subgroups — C₂×#22588500 #31997854

Order 18176.

1358 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556062	C₂	1	4	yes	maximal
2	C₂×#17985869	C₂	1	2	yes	maximal
3	#22588500	C₂	1	8	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₁	—	2	4	no	—
6	C₂²×C₄⋊Dic₇₁	C₂²	1	4	yes	—
7	C₂×#14556062	—	2	16	no	—
8	C₂×#14556062	C₂²	1	16	yes	—
9	#17985869	—	2	4	no	—
10	#17985869	C₂²	1	8	yes	—
11	C₂³×C₈×C₇₁	C₂²	1	2	yes	—
12	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₁	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₇₁	C₂²	1	4	yes	—
15	C₂³×Dic₇₁	—	2	4	no	—
16	C₂×C₄⋊Dic₇₁	—	2	16	no	—
17	C₂×C₄⋊Dic₇₁	—	2	4	no	—
18	C₂×C₄⋊Dic₇₁	—	2	16	no	—
19	C₂×C₄⋊Dic₇₁	—	4	16	no	—
20	C₂×C₄⋊Dic₇₁	C₂³	1	8	yes	—
21	#14556062	—	2	32	no	—
22	C₂³×C₄×C₇₁	D₄	1	2	yes	—
23	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₇₁	D₄	1	8	yes	—
26	C₂²×C₈×C₇₁	Q₈	1	8	yes	—
27	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
28	C₂×C₄²×C₇₁	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₇₁	D₄	1	2	yes	—
30	C₂×C₄²×C₇₁	Q₈	1	4	yes	—
31	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
32	C₄×C₈×C₇₁	C₂×C₄	1	16	yes	complemented
33	C₂²×Dic₇₁	—	2	8	no	—
34	C₂²×Dic₇₁	—	4	8	no	—
35	C₂²×Dic₇₁	—	4	16	no	—
36	C₄⋊Dic₇₁	—	2	16	no	—
37	C₄⋊Dic₇₁	—	4	8	no	—
38	C₄⋊Dic₇₁	—	4	32	no	—
39	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₄⋊C₄	1	8	yes	—
41	C₂²×C₄×C₇₁	D₈	1	8	yes	—
42	C₂²×C₄×C₇₁	Q₁₆	1	8	yes	—
43	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₇₁	C₂×Q₈	1	4	yes	—
48	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₄⋊C₄	1	32	yes	complemented
51	C₂×C₈×C₇₁	C₂²×C₄	1	8	yes	—
52	C₂×C₈×C₇₁	C₂×D₄	1	8	yes	—
53	C₂×C₈×C₇₁	C₂×Q₈	1	8	yes	—
54	C₄²×C₇₁	C₄⋊C₄	1	8	yes	—
55	C₄²×C₇₁	C₂²×C₄	1	4	yes	—
56	C₄²×C₇₁	C₂×Q₈	1	4	yes	—
57	C₂³×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
58	C₂³×C₇₁	#127	1	2	yes	—
59	C₂³×C₇₁	C₄⋊Q₈	1	2	yes	—
60	C₂³×C₇₁	C₂×D₈	1	4	yes	—
61	C₂³×C₇₁	C₂×Q₁₆	1	4	yes	—
62	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#107	1	32	yes	complemented
64	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	8	yes	—
65	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
66	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	8	yes	—
67	C₂×C₄×C₇₁	#127	1	2	yes	—
68	C₂×C₄×C₇₁	C₄⋊Q₈	1	8	yes	—
69	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
70	C₂×C₄×C₇₁	C₂×D₈	1	8	yes	—
71	C₂×C₄×C₇₁	C₂×Q₁₆	1	8	yes	—
72	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
74	C₂×C₄×C₇₁	C₂²×Q₈	1	2	yes	—
75	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
76	C₂×Dic₇₁	—	4	8	no	—
77	C₂×Dic₇₁	—	4	16	no	—
78	C₂×Dic₇₁	—	8	16	no	—
79	C₈×C₇₁	C₂×C₄⋊C₄	1	16	yes	complemented
80	C₂²×C₇₁	#384	1	4	yes	—
81	C₂²×C₇₁	C₂×#107	1	8	yes	—
82	C₂²×C₇₁	#493	1	4	yes	—
83	C₂²×C₇₁	#494	1	4	yes	—
84	C₂²×C₇₁	#500	1	8	yes	—
85	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₇₁	C₂×#127	1	1	yes	characteristic
87	C₂²×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₇₁	C₂²×D₈	1	2	yes	—
89	C₂²×C₇₁	C₂²×Q₁₆	1	2	yes	—
90	C₄×C₇₁	#384	1	4	yes	—
91	C₄×C₇₁	C₂×#107	1	16	yes	complemented
92	C₄×C₇₁	C₂²×C₄⋊C₄	1	2	yes	—
93	C₄×C₇₁	C₂×C₄⋊Q₈	1	2	yes	—
94	Dic₇₁	—	8	8	no	—
95	C₂×#1845	—	71	1	no	maximal, Sylow
96	C₂×C₇₁	#1845	1	8	yes	complemented
97	C₂×C₇₁	C₂×#384	1	1	yes	characteristic
98	C₂×C₇₁	C₂²×#107	1	2	yes	—
99	C₂×C₇₁	C₂×#493	1	1	yes	characteristic
100	C₂×C₇₁	C₂×#494	1	1	yes	characteristic
101	C₂×C₇₁	C₂×#500	1	2	yes	—
102	C₂²×#107	—	71	4	no	—
103	C₂²×C₄×C₈	D₇₁	1	1	yes	characteristic
104	C₂×#384	—	71	2	no	—
105	#1845	—	71	8	no	—
106	C₇₁	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₈	C₂×D₇₁	1	2	yes	—
108	C₂²×C₄⋊C₄	—	71	4	no	—
109	C₂²×C₄⋊C₄	—	142	4	no	—
110	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
111	C₂×#107	—	71	16	no	—
112	C₂×#107	—	142	16	no	—
113	C₂×C₄×C₈	Dic₇₁	1	8	yes	complemented
114	C₂×C₄×C₈	C₂×D₇₁	1	4	yes	—
115	#384	—	71	8	no	—
116	#384	—	142	4	no	—
117	C₂³×C₄	—	142	4	no	—
118	C₂³×C₄	D₂₈₄	1	2	yes	—
119	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
120	C₂²×C₈	Dic₁₄₂	1	8	yes	—
121	C₂²×C₈	D₂₈₄	1	8	yes	—
122	C₂²×C₈	C₂×Dic₇₁	1	8	yes	—
123	C₂²×C₈	C₂²×D₇₁	1	4	yes	—
124	C₂×C₄⋊C₄	—	284	16	no	—
125	C₂×C₄⋊C₄	—	71	8	no	—
126	C₂×C₄⋊C₄	—	142	4	no	—
127	C₂×C₄⋊C₄	—	142	16	no	—
128	C₂×C₄⋊C₄	—	142	16	no	—
129	C₂×C₄²	Dic₁₄₂	1	4	yes	—
130	C₂×C₄²	D₂₈₄	1	2	yes	—
131	C₂×C₄²	C₂×Dic₇₁	1	4	yes	—
132	C₂×C₄²	C₂²×D₇₁	1	2	yes	—
133	#107	—	142	32	no	—
134	C₄×C₈	C₂×Dic₇₁	1	16	yes	complemented
135	C₂⁴	C₂×D₂₈₄	1	1	yes	characteristic
136	C₂²×C₄	—	284	8	no	—
137	C₂²×C₄	—	284	16	no	—
138	C₂²×C₄	—	142	8	no	—
139	C₂²×C₄	C₄⋊Dic₇₁	1	8	yes	—
140	C₂²×C₄	D₅₆₈	1	8	yes	—
141	C₂²×C₄	Dic₂₈₄	1	8	yes	—
142	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
143	C₂²×C₄	C₂×D₂₈₄	1	4	yes	—
144	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₂₈₄	1	4	yes	—
146	C₂²×C₄	C₂×Dic₁₄₂	1	4	yes	—
147	C₂²×C₄	C₂×Dic₁₄₂	1	2	yes	—
148	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
149	C₂×C₈	C₄⋊Dic₇₁	1	32	yes	complemented
150	C₂×C₈	C₂²×Dic₇₁	1	8	yes	—
151	C₂×C₈	C₂×D₂₈₄	1	8	yes	—
152	C₂×C₈	C₂×Dic₁₄₂	1	8	yes	—
153	C₄⋊C₄	—	284	8	no	—
154	C₄⋊C₄	—	284	32	no	—
155	C₄⋊C₄	—	142	16	no	—
156	C₄²	C₄⋊Dic₇₁	1	8	yes	—
157	C₄²	C₂²×Dic₇₁	1	4	yes	—
158	C₄²	C₂×Dic₁₄₂	1	4	yes	—
159	C₂³	C₂×C₄⋊Dic₇₁	1	2	yes	—
160	C₂³	#14556125	1	2	yes	—
161	C₂³	#14556128	1	2	yes	—
162	C₂³	C₂×D₅₆₈	1	4	yes	—
163	C₂³	C₂×Dic₂₈₄	1	4	yes	—
164	C₂³	C₂²×D₂₈₄	1	1	yes	characteristic
165	C₂×C₄	—	284	16	no	—
166	C₂×C₄	—	284	8	no	—
167	C₂×C₄	—	568	16	no	—
168	C₂×C₄	#14556062	1	32	yes	complemented
169	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
170	C₂×C₄	C₂×C₄⋊Dic₇₁	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₇₁	1	8	yes	—
172	C₂×C₄	#14556125	1	2	yes	—
173	C₂×C₄	#14556128	1	2	yes	—
174	C₂×C₄	#14556128	1	8	yes	—
175	C₂×C₄	C₂×D₅₆₈	1	8	yes	—
176	C₂×C₄	C₂×Dic₂₈₄	1	8	yes	—
177	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₂₈₄	1	2	yes	—
179	C₂×C₄	C₂²×Dic₁₄₂	1	2	yes	—
180	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
181	C₈	C₂×C₄⋊Dic₇₁	1	16	yes	complemented
182	C₂²	#17985869	1	4	yes	—
183	C₂²	C₂×#14556062	1	8	yes	—
184	C₂²	#17986376	1	4	yes	—
185	C₂²	#17986379	1	4	yes	—
186	C₂²	#17986404	1	8	yes	—
187	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
188	C₂²	C₂×#14556125	1	1	yes	characteristic
189	C₂²	C₂×#14556128	1	1	yes	characteristic
190	C₂²	C₂²×D₅₆₈	1	2	yes	—
191	C₂²	C₂²×Dic₂₈₄	1	2	yes	—
192	C₄	—	568	8	no	—
193	C₄	#17985869	1	4	yes	—
194	C₄	C₂×#14556062	1	16	yes	complemented
195	C₄	C₂²×C₄⋊Dic₇₁	1	2	yes	—
196	C₄	C₂×#14556128	1	2	yes	—
197	C₂	#22588500	1	8	yes	direct product factor
198	C₂	C₂×#17985869	1	1	yes	characteristic
199	C₂	C₂²×#14556062	1	2	yes	—
200	C₂	C₂×#17986376	1	1	yes	characteristic
201	C₂	C₂×#17986379	1	1	yes	characteristic
202	C₂	C₂×#17986404	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂²×C₄×C₈×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂×D₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇₁
hypocenter	C₇₁	C₂×#1845
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #22588500	C₂	direct (N × H)
13. C₂×C₄×C₈×C₇₁	C₄	semidirect
32. C₄×C₈×C₇₁	C₂×C₄	semidirect
50. C₂×C₈×C₇₁	C₄⋊C₄	semidirect
63. C₂×C₄×C₇₁	#107	semidirect
79. C₈×C₇₁	C₂×C₄⋊C₄	semidirect
91. C₄×C₇₁	C₂×#107	semidirect
96. C₂×C₇₁	#1845	semidirect
106. C₇₁	C₂×#1845	semidirect
113. C₂×C₄×C₈	Dic₇₁	semidirect
134. C₄×C₈	C₂×Dic₇₁	semidirect
149. C₂×C₈	C₄⋊Dic₇₁	semidirect
168. C₂×C₄	#14556062	semidirect
181. C₈	C₂×C₄⋊Dic₇₁	semidirect
194. C₄	C₂×#14556062	semidirect
197. C₂	#22588500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556062	2
C₂×#17985869	2
#22588500	2
C₂²×C₄×C₈×C₇₁	2
C₂×#1845	71

Subgroups — C2×#22588500 #31997854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22588500 #31997854