Subgroups — C₂×#22588497 #31997850

Order 18176.

1358 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556060	C₂	1	4	yes	maximal
2	C₂×#17985869	C₂	1	2	yes	maximal
3	#22588497	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×#14556060	—	2	16	no	—
8	C₂×#14556060	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	#14556060	—	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	SD₁₆	1	16	yes	—
42	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
45	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₇₁	C₂×Q₈	1	4	yes	—
47	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
48	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₄⋊C₄	1	32	yes	complemented
50	C₂×C₈×C₇₁	C₂²×C₄	1	8	yes	—
51	C₂×C₈×C₇₁	C₂×D₄	1	8	yes	—
52	C₂×C₈×C₇₁	C₂×Q₈	1	8	yes	—
53	C₄²×C₇₁	C₄⋊C₄	1	8	yes	—
54	C₄²×C₇₁	C₂²×C₄	1	4	yes	—
55	C₄²×C₇₁	C₂×Q₈	1	4	yes	—
56	C₂³×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
57	C₂³×C₇₁	#127	1	2	yes	—
58	C₂³×C₇₁	C₄⋊Q₈	1	2	yes	—
59	C₂³×C₇₁	C₂×SD₁₆	1	8	yes	—
60	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#106	1	32	yes	complemented
62	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	8	yes	—
63	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	8	yes	—
65	C₂×C₄×C₇₁	#127	1	2	yes	—
66	C₂×C₄×C₇₁	C₄⋊Q₈	1	8	yes	—
67	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
68	C₂×C₄×C₇₁	C₂×SD₁₆	1	16	yes	—
69	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
71	C₂×C₄×C₇₁	C₂²×Q₈	1	2	yes	—
72	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
73	C₂×Dic₇₁	—	4	8	no	—
74	C₂×Dic₇₁	—	4	16	no	—
75	C₂×Dic₇₁	—	8	16	no	—
76	C₈×C₇₁	C₂×C₄⋊C₄	1	16	yes	complemented
77	C₂²×C₇₁	#384	1	4	yes	—
78	C₂²×C₇₁	C₂×#106	1	8	yes	—
79	C₂²×C₇₁	#492	1	8	yes	—
80	C₂²×C₇₁	#498	1	8	yes	—
81	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₇₁	C₂×#127	1	1	yes	characteristic
83	C₂²×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
84	C₂²×C₇₁	C₂²×SD₁₆	1	4	yes	—
85	C₄×C₇₁	#384	1	4	yes	—
86	C₄×C₇₁	C₂×#106	1	16	yes	complemented
87	C₄×C₇₁	C₂²×C₄⋊C₄	1	2	yes	—
88	C₄×C₇₁	C₂×C₄⋊Q₈	1	2	yes	—
89	Dic₇₁	—	8	8	no	—
90	C₂×#1844	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	#1844	1	8	yes	complemented
92	C₂×C₇₁	C₂×#384	1	1	yes	characteristic
93	C₂×C₇₁	C₂²×#106	1	2	yes	—
94	C₂×C₇₁	C₂×#492	1	2	yes	—
95	C₂×C₇₁	C₂×#498	1	2	yes	—
96	C₂²×#106	—	71	4	no	—
97	C₂²×C₄×C₈	D₇₁	1	1	yes	characteristic
98	C₂×#384	—	71	2	no	—
99	#1844	—	71	8	no	—
100	C₇₁	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
101	C₂³×C₈	C₂×D₇₁	1	2	yes	—
102	C₂²×C₄⋊C₄	—	71	4	no	—
103	C₂²×C₄⋊C₄	—	142	4	no	—
104	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
105	C₂×#106	—	71	16	no	—
106	C₂×#106	—	142	16	no	—
107	C₂×C₄×C₈	Dic₇₁	1	8	yes	complemented
108	C₂×C₄×C₈	C₂×D₇₁	1	4	yes	—
109	#384	—	71	8	no	—
110	#384	—	142	4	no	—
111	C₂³×C₄	—	142	4	no	—
112	C₂³×C₄	D₂₈₄	1	2	yes	—
113	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
114	C₂²×C₈	Dic₁₄₂	1	8	yes	—
115	C₂²×C₈	D₂₈₄	1	8	yes	—
116	C₂²×C₈	C₂×Dic₇₁	1	8	yes	—
117	C₂²×C₈	C₂²×D₇₁	1	4	yes	—
118	C₂×C₄⋊C₄	—	284	16	no	—
119	C₂×C₄⋊C₄	—	71	8	no	—
120	C₂×C₄⋊C₄	—	142	4	no	—
121	C₂×C₄⋊C₄	—	142	16	no	—
122	C₂×C₄⋊C₄	—	142	16	no	—
123	C₂×C₄²	Dic₁₄₂	1	4	yes	—
124	C₂×C₄²	D₂₈₄	1	2	yes	—
125	C₂×C₄²	C₂×Dic₇₁	1	4	yes	—
126	C₂×C₄²	C₂²×D₇₁	1	2	yes	—
127	#106	—	142	32	no	—
128	C₄×C₈	C₂×Dic₇₁	1	16	yes	complemented
129	C₂⁴	C₂×D₂₈₄	1	1	yes	characteristic
130	C₂²×C₄	—	284	8	no	—
131	C₂²×C₄	—	284	16	no	—
132	C₂²×C₄	—	142	8	no	—
133	C₂²×C₄	C₄⋊Dic₇₁	1	8	yes	—
134	C₂²×C₄	#11765854	1	16	yes	—
135	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
136	C₂²×C₄	C₂×D₂₈₄	1	4	yes	—
137	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₂₈₄	1	4	yes	—
139	C₂²×C₄	C₂×Dic₁₄₂	1	4	yes	—
140	C₂²×C₄	C₂×Dic₁₄₂	1	2	yes	—
141	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₇₁	1	32	yes	complemented
143	C₂×C₈	C₂²×Dic₇₁	1	8	yes	—
144	C₂×C₈	C₂×D₂₈₄	1	8	yes	—
145	C₂×C₈	C₂×Dic₁₄₂	1	8	yes	—
146	C₄⋊C₄	—	284	8	no	—
147	C₄⋊C₄	—	284	32	no	—
148	C₄⋊C₄	—	142	16	no	—
149	C₄²	C₄⋊Dic₇₁	1	8	yes	—
150	C₄²	C₂²×Dic₇₁	1	4	yes	—
151	C₄²	C₂×Dic₁₄₂	1	4	yes	—
152	C₂³	C₂×C₄⋊Dic₇₁	1	2	yes	—
153	C₂³	#14556125	1	2	yes	—
154	C₂³	#14556128	1	2	yes	—
155	C₂³	C₂×#11765854	1	8	yes	—
156	C₂³	C₂²×D₂₈₄	1	1	yes	characteristic
157	C₂×C₄	—	284	16	no	—
158	C₂×C₄	—	284	8	no	—
159	C₂×C₄	—	568	16	no	—
160	C₂×C₄	#14556060	1	32	yes	complemented
161	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₇₁	1	8	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₇₁	1	8	yes	—
164	C₂×C₄	#14556125	1	2	yes	—
165	C₂×C₄	#14556128	1	2	yes	—
166	C₂×C₄	#14556128	1	8	yes	—
167	C₂×C₄	C₂×#11765854	1	16	yes	—
168	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
169	C₂×C₄	C₂²×D₂₈₄	1	2	yes	—
170	C₂×C₄	C₂²×Dic₁₄₂	1	2	yes	—
171	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
172	C₈	C₂×C₄⋊Dic₇₁	1	16	yes	complemented
173	C₂²	#17985869	1	4	yes	—
174	C₂²	C₂×#14556060	1	8	yes	—
175	C₂²	#17986372	1	8	yes	—
176	C₂²	#17986397	1	8	yes	—
177	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
178	C₂²	C₂×#14556125	1	1	yes	characteristic
179	C₂²	C₂×#14556128	1	1	yes	characteristic
180	C₂²	C₂²×#11765854	1	4	yes	—
181	C₄	—	568	8	no	—
182	C₄	#17985869	1	4	yes	—
183	C₄	C₂×#14556060	1	16	yes	complemented
184	C₄	C₂²×C₄⋊Dic₇₁	1	2	yes	—
185	C₄	C₂×#14556128	1	2	yes	—
186	C₂	#22588497	1	8	yes	direct product factor
187	C₂	C₂×#17985869	1	1	yes	characteristic
188	C₂	C₂²×#14556060	1	2	yes	—
189	C₂	C₂×#17986372	1	2	yes	—
190	C₂	C₂×#17986397	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×#1844
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. #22588497	C₂	direct (N × H)
13. C₂×C₄×C₈×C₇₁	C₄	semidirect
32. C₄×C₈×C₇₁	C₂×C₄	semidirect
49. C₂×C₈×C₇₁	C₄⋊C₄	semidirect
61. C₂×C₄×C₇₁	#106	semidirect
76. C₈×C₇₁	C₂×C₄⋊C₄	semidirect
86. C₄×C₇₁	C₂×#106	semidirect
91. C₂×C₇₁	#1844	semidirect
100. C₇₁	C₂×#1844	semidirect
107. C₂×C₄×C₈	Dic₇₁	semidirect
128. C₄×C₈	C₂×Dic₇₁	semidirect
142. C₂×C₈	C₄⋊Dic₇₁	semidirect
160. C₂×C₄	#14556060	semidirect
172. C₈	C₂×C₄⋊Dic₇₁	semidirect
183. C₄	C₂×#14556060	semidirect
186. C₂	#22588497	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×#14556060	2
C₂×#17985869	2
#22588497	2
C₂²×C₄×C₈×C₇₁	2
C₂×#1844	71

Subgroups — C2×#22588497 #31997850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22588497 #31997850