Subgroups — C₂×#19136856 #25251996

Order 11008.

854 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13063270	C₂	1	2	yes	maximal, complemented
3	C₂²×D₆₈₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583283	C₂	1	2	yes	maximal, complemented
5	#19136856	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
8	C₂²×#10621873	—	2	2	no	—
9	C₂²×D₃₄₄	C₂²	1	2	yes	—
10	C₂×#13063313	—	2	2	no	—
11	C₂×Dic₃₄₄	—	2	4	no	—
12	C₂×Dic₃₄₄	C₂²	1	4	yes	complemented
13	C₂×#13063270	—	2	8	no	—
14	C₂×#13063270	C₂²	1	8	yes	complemented
15	C₂×D₆₈₈	—	2	4	no	—
16	C₂×D₆₈₈	C₂²	1	4	yes	—
17	#14583283	—	2	4	no	—
18	#14583283	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₄₃	C₂²	1	4	yes	complemented
23	C₂²×Dic₈₆	—	2	2	no	—
24	C₂²×D₁₇₂	—	2	2	no	—
25	C₂×Dic₁₇₂	—	2	4	no	—
26	C₂×Dic₁₇₂	—	2	4	no	—
27	C₂×Dic₁₇₂	—	2	2	no	—
28	C₂×Dic₁₇₂	C₂³	1	4	yes	—
29	C₂×#10621873	—	2	8	no	—
30	C₂×#10621873	—	4	8	no	—
31	C₂×D₃₄₄	—	2	4	no	—
32	C₂×D₃₄₄	—	2	4	no	—
33	C₂×D₃₄₄	—	2	2	no	—
34	C₂×D₃₄₄	C₂³	1	4	yes	—
35	#13063313	—	2	8	no	—
36	#13063313	—	4	4	no	—
37	C₂×(C₂×D₄₃)⋊C₄	—	4	4	no	—
38	Dic₃₄₄	—	2	8	no	—
39	#13063270	—	2	16	no	—
40	D₆₈₈	—	2	8	no	—
41	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
45	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
46	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
47	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
48	C₄×C₈×C₄₃	D₄	1	2	yes	—
49	C₄×C₈×C₄₃	C₂³	1	2	yes	—
50	C₂³×D₄₃	—	4	2	no	—
51	C₂²×C₄×C₄₃	D₈	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₄₃	—	4	2	no	—
54	C₂×D₁₇₂	—	2	4	no	—
55	C₂×D₁₇₂	—	4	2	no	—
56	C₂×D₁₇₂	—	4	4	no	—
57	C₂×D₁₇₂	—	4	4	no	—
58	C₂×Dic₈₆	—	2	4	no	—
59	C₂×Dic₈₆	—	4	4	no	—
60	C₂×Dic₈₆	—	4	4	no	—
61	C₂×Dic₈₆	—	4	2	no	—
62	C₂×C₈×C₄₃	D₈	1	4	yes	—
63	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₄²×C₄₃	D₈	1	2	yes	—
69	C₄²×C₄₃	C₂×D₄	1	2	yes	—
70	(C₂×D₄₃)⋊C₄	—	4	16	no	—
71	Dic₁₇₂	—	2	4	no	—
72	Dic₁₇₂	—	2	4	no	—
73	Dic₁₇₂	—	4	4	no	—
74	D₃₄₄	—	2	4	no	—
75	D₃₄₄	—	2	4	no	—
76	D₃₄₄	—	4	4	no	—
77	#10621873	—	4	16	no	—
78	C₁₆×C₄₃	C₂×D₄	1	4	yes	—
79	C₁₆×C₄₃	C₂×D₄	1	4	yes	—
80	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₄₃	—	4	4	no	—
82	C₂²×D₄₃	—	8	4	no	—
83	C₂²×D₄₃	—	8	8	no	—
84	C₂×C₄×C₄₃	—	2	4	no	—
85	C₂×C₄×C₄₃	#127	1	2	yes	—
86	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
87	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×Dic₄₃	—	4	4	no	—
91	C₂×Dic₄₃	—	8	4	no	—
92	D₁₇₂	—	4	4	no	—
93	D₁₇₂	—	4	4	no	—
94	D₁₇₂	—	8	4	no	—
95	Dic₈₆	—	4	4	no	—
96	Dic₈₆	—	4	4	no	—
97	Dic₈₆	—	8	4	no	—
98	C₈×C₄₃	#127	1	2	yes	—
99	C₈×C₄₃	C₂×D₈	1	4	yes	—
100	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×#2249	—	43	1	no	maximal, Sylow
103	C₂²×C₄₃	#493	1	2	yes	—
104	C₂²×C₄₃	#508	1	4	yes	—
105	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
106	C₂×D₄₃	—	8	4	no	—
107	C₂×D₄₃	—	8	8	no	—
108	C₂×D₄₃	—	16	8	no	—
109	C₄×C₄₃	—	2	4	no	—
110	C₄×C₄₃	#493	1	2	yes	—
111	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
112	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
113	Dic₄₃	—	8	4	no	—
114	C₂²×Q₃₂	—	43	1	no	—
115	C₂²×SD₃₂	—	43	2	no	—
116	C₂²×D₁₆	—	43	1	no	—
117	C₂×#495	—	43	2	no	—
118	#2249	—	43	8	no	—
119	C₂×C₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
120	C₂×C₄₃	#2249	1	4	yes	complemented
121	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
122	C₂×C₄₃	C₂×#508	1	2	yes	—
123	D₄₃	—	16	4	no	—
124	C₂²×Q₁₆	—	43	2	no	—
125	C₂²×SD₁₆	—	86	2	no	—
126	C₂²×D₈	—	43	2	no	—
127	C₂×#124	—	86	2	no	—
128	C₂×Q₃₂	—	43	4	no	—
129	C₂×Q₃₂	—	86	4	no	—
130	C₂×SD₃₂	—	43	8	no	—
131	C₂×SD₃₂	—	86	8	no	—
132	C₂×D₁₆	—	43	4	no	—
133	C₂×D₁₆	—	86	4	no	—
134	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
135	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
136	#495	—	43	8	no	—
137	#495	—	86	4	no	—
138	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
139	C₄×C₁₆	C₂×D₄₃	1	4	yes	complemented
140	C₄₃	C₂×#2249	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	86	2	no	—
142	C₂²×D₄	—	86	2	no	—
143	C₂×Q₁₆	—	43	4	no	—
144	C₂×Q₁₆	—	86	4	no	—
145	C₂×Q₁₆	—	86	4	no	—
146	C₂×Q₁₆	—	86	2	no	—
147	C₂×SD₁₆	—	86	8	no	—
148	C₂×SD₁₆	—	172	8	no	—
149	C₂×D₈	—	43	4	no	—
150	C₂×D₈	—	86	4	no	—
151	C₂×D₈	—	86	4	no	—
152	C₂×D₈	—	86	2	no	—
153	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
155	#124	—	86	8	no	—
156	#124	—	172	4	no	—
157	C₂×C₂²⋊C₄	—	172	4	no	—
158	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
159	Q₃₂	—	86	8	no	—
160	SD₃₂	—	86	16	no	—
161	D₁₆	—	86	8	no	—
162	C₂×C₁₆	D₁₇₂	1	4	yes	—
163	C₂×C₁₆	D₁₇₂	1	4	yes	—
164	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
165	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
166	C₄×C₈	D₁₇₂	1	2	yes	—
167	C₄×C₈	C₂²×D₄₃	1	2	yes	—
168	C₂⁴	—	172	2	no	—
169	C₂×Q₈	—	86	4	no	—
170	C₂×Q₈	—	172	4	no	—
171	C₂×Q₈	—	172	4	no	—
172	C₂×Q₈	—	172	2	no	—
173	C₂×D₄	—	86	4	no	—
174	C₂×D₄	—	172	4	no	—
175	C₂×D₄	—	172	2	no	—
176	C₂×D₄	—	172	4	no	—
177	C₂²×C₄	—	172	2	no	—
178	C₂²×C₄	D₃₄₄	1	2	yes	—
179	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
180	Q₁₆	—	86	4	no	—
181	Q₁₆	—	86	4	no	—
182	Q₁₆	—	172	4	no	—
183	SD₁₆	—	172	16	no	—
184	D₈	—	86	4	no	—
185	D₈	—	86	4	no	—
186	D₈	—	172	4	no	—
187	C₂×C₈	D₃₄₄	1	4	yes	—
188	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
189	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
190	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
191	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
192	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
193	C₂²⋊C₄	—	172	16	no	—
194	C₄²	D₃₄₄	1	2	yes	—
195	C₄²	C₂×D₁₇₂	1	2	yes	—
196	C₁₆	C₂×D₁₇₂	1	4	yes	—
197	C₁₆	C₂×D₁₇₂	1	4	yes	—
198	C₂³	—	344	8	no	—
199	C₂³	—	344	4	no	—
200	C₂³	—	172	4	no	—
201	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
202	Q₈	—	344	4	no	—
203	Q₈	—	172	4	no	—
204	Q₈	—	172	4	no	—
205	D₄	—	344	4	no	—
206	D₄	—	172	4	no	—
207	D₄	—	172	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	344	4	no	—
210	C₂×C₄	—	172	4	no	—
211	C₂×C₄	#13063322	1	2	yes	—
212	C₂×C₄	C₂×D₃₄₄	1	4	yes	—
213	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
214	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
216	C₈	#13063322	1	2	yes	—
217	C₈	C₂×D₃₄₄	1	4	yes	—
218	C₈	C₂²×D₁₇₂	1	1	yes	characteristic
219	C₈	C₂²×D₁₇₂	1	1	yes	characteristic
220	C₂²	—	344	4	no	—
221	C₂²	—	344	8	no	—
222	C₂²	—	688	8	no	—
223	C₂²	#14583276	1	2	yes	—
224	C₂²	#14583331	1	4	yes	—
225	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	344	4	no	—
228	C₄	#14583276	1	2	yes	—
229	C₄	C₂×#13063322	1	1	yes	characteristic
230	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
231	C₂	—	688	4	no	—
232	C₂	#19136856	1	4	yes	direct product factor
233	C₂	C₂×#14583276	1	1	yes	characteristic
234	C₂	C₂×#14583331	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄×C₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₆₈₈	C₂
hypercenter	C₂×C₄×C₁₆	D₄₃
hypocenter	C₄₃	C₂×#2249
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₃₄₄	C₂	semidirect
2. C₂²×#13063270	C₂	semidirect
4. C₂×#14583283	C₂	semidirect
5. #19136856	C₂	direct (N × H)
6. C₂×C₄×C₁₆×C₄₃	C₂	semidirect
12. C₂×Dic₃₄₄	C₂²	semidirect
14. C₂×#13063270	C₂²	semidirect
18. #14583283	C₂²	semidirect
22. C₄×C₁₆×C₄₃	C₂²	semidirect
119. C₂×C₄×C₁₆	D₄₃	semidirect
120. C₂×C₄₃	#2249	semidirect
139. C₄×C₁₆	C₂×D₄₃	semidirect
140. C₄₃	C₂×#2249	semidirect
232. C₂	#19136856	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×Dic₃₄₄	2
C₂²×#13063270	2
C₂²×D₆₈₈	2
C₂×#14583283	2
#19136856	2
C₂×C₄×C₁₆×C₄₃	2
C₂×#2249	43

Subgroups — C2×#19136856 #25251996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136856 #25251996