Subgroups — C₂×#22586527 #31917911

Order 18176.

302 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985833	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17985827	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586527	C₂	1	4	yes	maximal, direct product factor
4	C₂×#508×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556062	—	2	1	no	—
6	#17985833	—	2	1	no	—
7	#17985833	C₂²	1	2	yes	complemented
8	C₂×C₇₁⋊C₃₂	—	2	1	no	—
9	#17985827	—	2	1	no	—
10	#17985827	C₂²	1	2	yes	complemented
11	C₂×#135×C₇₁	—	2	1	no	—
12	#508×C₇₁	—	2	1	no	—
13	#508×C₇₁	—	2	1	no	—
14	#508×C₇₁	C₄	1	2	yes	complemented
15	#508×C₇₁	C₂²	1	2	yes	—
16	C₂×Q₃₂×C₇₁	C₄	1	1	yes	characteristic, complemented
17	C₂×SD₃₂×C₇₁	—	2	1	no	—
18	C₂×D₁₆×C₇₁	C₄	1	1	yes	characteristic, complemented
19	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₇₁	—	4	1	no	—
21	#14556062	—	2	2	no	—
22	#14556062	—	4	1	no	—
23	C₇₁⋊C₃₂	—	2	2	no	—
24	C₂×#41×C₇₁	—	4	1	no	—
25	#135×C₇₁	—	2	2	no	—
26	#135×C₇₁	—	2	2	no	—
27	#135×C₇₁	—	4	1	no	—
28	#135×C₇₁	—	4	1	no	—
29	C₂×Q₁₆×C₇₁	—	2	1	no	—
30	C₂×SD₁₆×C₇₁	—	4	1	no	—
31	C₂×D₈×C₇₁	—	2	1	no	—
32	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
33	Q₃₂×C₇₁	—	2	1	no	—
34	Q₃₂×C₇₁	C₂×C₄	1	2	yes	complemented
35	SD₃₂×C₇₁	—	2	2	no	—
36	SD₃₂×C₇₁	—	2	2	no	—
37	D₁₆×C₇₁	—	2	1	no	—
38	D₁₆×C₇₁	C₂×C₄	1	2	yes	complemented
39	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₇₁	—	8	1	no	—
46	C₄⋊Dic₇₁	—	4	2	no	—
47	C₄⋊Dic₇₁	—	8	1	no	—
48	#41×C₇₁	—	4	2	no	—
49	#41×C₇₁	—	4	2	no	—
50	#41×C₇₁	—	8	1	no	—
51	#41×C₇₁	—	8	1	no	—
52	C₂×Q₈×C₇₁	—	4	1	no	—
53	C₂×D₄×C₇₁	—	4	1	no	—
54	C₂×D₄×C₇₁	—	8	1	no	—
55	C₂²×C₄×C₇₁	—	8	1	no	—
56	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
57	Q₁₆×C₇₁	—	2	2	no	—
58	Q₁₆×C₇₁	—	4	1	no	—
59	SD₁₆×C₇₁	—	4	2	no	—
60	SD₁₆×C₇₁	—	4	2	no	—
61	D₈×C₇₁	—	2	2	no	—
62	D₈×C₇₁	—	4	1	no	—
63	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
66	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
67	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
69	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
72	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₇₁	—	8	1	no	—
74	C₂³×C₇₁	D₁₆	1	1	yes	characteristic
75	Q₈×C₇₁	—	4	2	no	—
76	Q₈×C₇₁	—	8	1	no	—
77	D₄×C₇₁	—	4	2	no	—
78	D₄×C₇₁	—	8	2	no	—
79	D₄×C₇₁	—	8	2	no	—
80	D₄×C₇₁	—	8	1	no	—
81	C₂×C₄×C₇₁	—	8	1	no	—
82	C₂×C₄×C₇₁	—	8	2	no	—
83	C₂×C₄×C₇₁	—	8	2	no	—
84	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
88	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
90	C₂×Dic₇₁	—	8	2	no	—
91	C₂×Dic₇₁	—	16	2	no	—
92	C₈×C₇₁	#102	1	1	yes	characteristic
93	C₈×C₇₁	#102	1	1	yes	characteristic
94	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
96	C₂²×C₇₁	—	2	2	no	—
97	C₂²×C₇₁	—	8	2	no	—
98	C₂²×C₇₁	—	16	2	no	—
99	C₂²×C₇₁	#357	1	1	yes	characteristic
100	C₂²×C₇₁	#357	1	1	yes	characteristic
101	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
102	C₄×C₇₁	—	8	2	no	—
103	C₄×C₇₁	#357	1	1	yes	characteristic
104	C₄×C₇₁	#357	1	1	yes	characteristic
105	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
106	C₄×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
107	Dic₇₁	—	16	2	no	—
108	C₂×#1418	—	71	1	no	maximal, Sylow
109	C₂×C₇₁	—	2	1	no	—
110	C₂×C₇₁	—	2	1	no	—
111	C₂×C₇₁	—	16	2	no	—
112	C₂×C₇₁	#1418	1	2	yes	complemented
113	C₂×C₇₁	C₂×#357	1	1	yes	characteristic
114	C₂×#508	D₇₁	1	1	yes	characteristic
115	C₂×M₆₄	—	71	1	no	—
116	C₂×#367	—	71	1	no	—
117	#1418	—	71	4	no	—
118	C₇₁	C₂×#1418	1	1	yes	characteristic, complemented, Sylow
119	C₂×#135	—	2	1	no	—
120	#508	—	2	1	no	—
121	#508	—	2	1	no	—
122	#508	Dic₇₁	1	2	yes	complemented
123	#508	C₂×D₇₁	1	2	yes	—
124	C₂×Q₃₂	Dic₇₁	1	1	yes	characteristic, complemented
125	C₂×SD₃₂	—	2	1	no	—
126	C₂×D₁₆	Dic₇₁	1	1	yes	characteristic, complemented
127	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
128	C₂×#107	—	142	1	no	—
129	M₆₄	—	71	2	no	—
130	M₆₄	—	142	1	no	—
131	C₂×C₃₂	—	142	1	no	—
132	#367	—	71	2	no	—
133	#367	—	142	1	no	—
134	C₂×#41	—	4	1	no	—
135	#135	—	2	2	no	—
136	#135	—	2	2	no	—
137	#135	—	4	1	no	—
138	#135	—	4	1	no	—
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×SD₁₆	—	4	1	no	—
141	C₂×D₈	—	2	1	no	—
142	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	284	1	no	—
144	Q₃₂	—	2	1	no	—
145	Q₃₂	C₂×Dic₇₁	1	2	yes	complemented
146	SD₃₂	—	2	2	no	—
147	SD₃₂	—	2	2	no	—
148	D₁₆	—	2	1	no	—
149	D₁₆	C₂×Dic₇₁	1	2	yes	complemented
150	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
151	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
152	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
153	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
154	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
155	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
156	#107	—	284	1	no	—
157	#107	—	142	2	no	—
158	C₃₂	—	142	2	no	—
159	#41	—	4	2	no	—
160	#41	—	4	2	no	—
161	#41	—	8	1	no	—
162	#41	—	8	1	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂×D₄	—	4	1	no	—
165	C₂×D₄	—	8	1	no	—
166	C₂²×C₄	—	8	1	no	—
167	C₂²×C₄	—	568	1	no	—
168	C₂²×C₄	#11765850	1	1	yes	characteristic
169	Q₁₆	—	2	2	no	—
170	Q₁₆	—	4	1	no	—
171	SD₁₆	—	4	2	no	—
172	SD₁₆	—	4	2	no	—
173	D₈	—	2	2	no	—
174	D₈	—	4	1	no	—
175	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
177	C₂×C₈	#11765850	1	1	yes	characteristic
178	C₂×C₈	#11765853	1	1	yes	characteristic
179	C₂×C₈	#11765853	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
181	C₄⋊C₄	—	284	2	no	—
182	C₄⋊C₄	—	568	1	no	—
183	C₁₆	C₂²⋊Dic₇₁	1	1	yes	characteristic
184	C₁₆	C₂²⋊Dic₇₁	1	1	yes	characteristic
185	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
186	C₁₆	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
187	C₂³	—	8	1	no	—
188	C₂³	#14556069	1	1	yes	characteristic
189	Q₈	—	4	2	no	—
190	Q₈	—	8	1	no	—
191	D₄	—	4	2	no	—
192	D₄	—	8	2	no	—
193	D₄	—	8	2	no	—
194	D₄	—	8	1	no	—
195	C₂×C₄	—	8	1	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	568	2	no	—
199	C₂×C₄	—	1136	2	no	—
200	C₂×C₄	#14556049	1	1	yes	characteristic
201	C₂×C₄	#14556049	1	1	yes	characteristic
202	C₂×C₄	#14556069	1	1	yes	characteristic
203	C₂×C₄	#14556072	1	1	yes	characteristic
204	C₂×C₄	#14556072	1	1	yes	characteristic
205	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
206	C₈	#14556049	1	1	yes	characteristic
207	C₈	#14556049	1	1	yes	characteristic
208	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
209	C₈	C₂×#11765853	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	—	8	2	no	—
212	C₂²	—	16	2	no	—
213	C₂²	#17985800	1	1	yes	characteristic
214	C₂²	#17985800	1	1	yes	characteristic
215	C₂²	C₂×#14556069	1	1	yes	characteristic
216	C₄	—	8	2	no	—
217	C₄	—	1136	2	no	—
218	C₄	#17985800	1	1	yes	characteristic
219	C₄	#17985800	1	1	yes	characteristic
220	C₄	C₂×#14556049	1	1	yes	characteristic
221	C₄	C₂×#14556072	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	—	16	2	no	—
225	C₂	#22586527	1	2	yes	direct product factor
226	C₂	C₂×#17985800	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985800
commutator	C₁₆×C₇₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting, prime radical	C₂×#508×C₇₁	C₂
socle	C₂²×C₇₁	#357
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂×#508	D₇₁
hypocenter	C₇₁	C₂×#1418
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#17985833	C₂	semidirect
2. C₂×#17985827	C₂	semidirect
3. #22586527	C₂	direct (N × H)
7. #17985833	C₂²	semidirect
10. #17985827	C₂²	semidirect
14. #508×C₇₁	C₄	semidirect
16. C₂×Q₃₂×C₇₁	C₄	semidirect
18. C₂×D₁₆×C₇₁	C₄	semidirect
34. Q₃₂×C₇₁	C₂×C₄	semidirect
38. D₁₆×C₇₁	C₂×C₄	semidirect
112. C₂×C₇₁	#1418	semidirect
118. C₇₁	C₂×#1418	semidirect
122. #508	Dic₇₁	semidirect
124. C₂×Q₃₂	Dic₇₁	semidirect
126. C₂×D₁₆	Dic₇₁	semidirect
145. Q₃₂	C₂×Dic₇₁	semidirect
149. D₁₆	C₂×Dic₇₁	semidirect
225. C₂	#22586527	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17985833	2
C₂×#17985827	2
#22586527	2
C₂×#508×C₇₁	2
C₂×#1418	71

Subgroups — C2×#22586527 #31917911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586527 #31917911