Subgroups — C₂×#19136698 #25251043

Order 11008.

786 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583135	C₂	1	2	yes	maximal, complemented
3	#19136698	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582727	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582701	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₄₄	C₂²	1	2	yes	—
8	C₂×#13063299	—	2	2	no	—
9	C₂×D₆₈₈	—	2	4	no	—
10	C₂×D₆₈₈	C₂²	1	4	yes	complemented
11	#14583135	—	2	4	no	—
12	#14583135	C₂²	1	8	yes	complemented
13	C₂×#13063259	C₂²	1	2	yes	—
14	C₂×#13063247	—	2	2	no	—
15	#14582727	C₂²	1	4	yes	complemented
16	#14582701	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₄₃	—	2	4	no	—
19	C₂×M₃₂×C₄₃	C₂²	1	4	yes	complemented
20	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²×D₁₇₂	—	2	2	no	—
23	C₂²×C₂²⋊D₄₃	—	4	2	no	—
24	C₂×D₃₄₄	—	2	4	no	—
25	C₂×D₃₄₄	—	2	4	no	—
26	C₂×D₃₄₄	—	2	2	no	—
27	C₂×D₃₄₄	C₂³	1	4	yes	—
28	#13063299	—	2	8	no	—
29	#13063299	—	4	4	no	—
30	C₂×C₄⋊Dic₄₃	—	2	2	no	—
31	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
32	D₆₈₈	—	2	8	no	—
33	#13063259	—	2	2	no	—
34	#13063259	C₂³	1	4	yes	—
35	#13063247	—	2	8	no	—
36	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₃	—	2	2	no	—
38	C₂²×C₈×C₄₃	—	2	2	no	—
39	C₂²×C₈×C₄₃	D₄	1	2	yes	—
40	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
43	M₃₂×C₄₃	—	2	8	no	—
44	C₂×C₁₆×C₄₃	—	2	2	no	—
45	C₂×C₁₆×C₄₃	D₄	1	4	yes	complemented
46	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
47	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
48	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
49	C₂³×D₄₃	—	4	2	no	—
50	C₂²×C₄×C₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	—	2	2	no	—
52	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	D₈	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₂²⋊D₄₃	—	4	8	no	—
57	C₂×C₂²⋊D₄₃	—	8	8	no	—
58	C₂²×Dic₄₃	—	4	2	no	—
59	C₂×D₁₇₂	—	2	4	no	—
60	C₂×D₁₇₂	—	4	2	no	—
61	C₂×D₁₇₂	—	4	4	no	—
62	C₂×D₁₇₂	—	4	4	no	—
63	C₂×C₈×C₄₃	—	2	2	no	—
64	C₂×C₈×C₄₃	—	2	4	no	—
65	C₂×C₈×C₄₃	—	2	2	no	—
66	C₂×C₈×C₄₃	—	2	2	no	—
67	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	#41	1	2	yes	—
71	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
72	(C₂×D₄₃)⋊C₄	—	4	8	no	—
73	C₄⋊Dic₄₃	—	2	4	no	—
74	C₄⋊Dic₄₃	—	4	2	no	—
75	D₃₄₄	—	2	4	no	—
76	D₃₄₄	—	2	4	no	—
77	D₃₄₄	—	4	4	no	—
78	C₁₆×C₄₃	—	2	2	no	—
79	C₁₆×C₄₃	C₂×D₄	1	4	yes	complemented
80	C₂³×C₄₃	—	2	2	no	—
81	C₂³×C₄₃	—	2	4	no	—
82	C₂³×C₄₃	C₂×D₈	1	2	yes	—
83	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
84	C₂²×D₄₃	—	4	4	no	—
85	C₂²×D₄₃	—	8	4	no	—
86	C₂²×D₄₃	—	8	4	no	—
87	C₂²×D₄₃	—	8	4	no	—
88	C₂×C₄×C₄₃	—	2	2	no	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	—	2	4	no	—
92	C₂×C₄×C₄₃	#121	1	2	yes	—
93	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₄₃	#135	1	2	yes	—
96	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
97	C₂²⋊D₄₃	—	8	16	no	—
98	C₂×Dic₄₃	—	4	4	no	—
99	C₂×Dic₄₃	—	8	4	no	—
100	D₁₇₂	—	4	4	no	—
101	D₁₇₂	—	4	4	no	—
102	D₁₇₂	—	8	4	no	—
103	C₈×C₄₃	—	2	2	no	—
104	C₈×C₄₃	#121	1	2	yes	—
105	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
107	C₂×#2218	—	43	1	no	maximal, Sylow
108	C₂²×C₄₃	—	2	4	no	—
109	C₂²×C₄₃	—	2	2	no	—
110	C₂²×C₄₃	—	4	4	no	—
111	C₂²×C₄₃	#466	1	2	yes	—
112	C₂²×C₄₃	#509	1	4	yes	—
113	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
114	C₂×D₄₃	—	8	4	no	—
115	C₂×D₄₃	—	8	4	no	—
116	C₂×D₄₃	—	8	4	no	—
117	C₂×D₄₃	—	16	8	no	—
118	C₄×C₄₃	—	2	2	no	—
119	C₄×C₄₃	#466	1	2	yes	—
120	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
121	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
122	Dic₄₃	—	8	4	no	—
123	C₂²×D₁₆	—	43	1	no	—
124	C₂²×M₃₂	D₄₃	1	1	yes	characteristic, complemented
125	C₂×#466	—	43	2	no	—
126	#2218	—	43	8	no	—
127	C₂×#367	—	43	1	no	—
128	C₂×#357	—	43	2	no	—
129	C₂×C₄₃	—	4	2	no	—
130	C₂×C₄₃	#2218	1	4	yes	complemented
131	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
132	C₂×C₄₃	C₂×#509	1	2	yes	—
133	D₄₃	—	16	4	no	—
134	C₂²×D₈	—	43	2	no	—
135	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
136	C₂×#121	—	86	2	no	—
137	C₂×D₁₆	—	43	4	no	—
138	C₂×D₁₆	—	86	4	no	—
139	C₂×M₃₂	—	2	4	no	—
140	C₂×M₃₂	C₂×D₄₃	1	4	yes	complemented
141	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
142	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
143	#466	—	43	8	no	—
144	#466	—	86	4	no	—
145	C₂×#107	—	43	2	no	—
146	C₂×#102	—	86	2	no	—
147	#367	—	43	4	no	—
148	#357	—	43	8	no	—
149	C₄₃	C₂×#2218	1	1	yes	characteristic, complemented, Sylow
150	C₂²×D₄	—	86	2	no	—
151	C₂²×D₄	—	172	2	no	—
152	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
153	C₂×D₈	—	43	4	no	—
154	C₂×D₈	—	86	2	no	—
155	C₂×D₈	—	86	4	no	—
156	C₂×D₈	—	86	4	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
160	C₂²×C₈	D₁₇₂	1	2	yes	—
161	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
162	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
163	#121	—	86	8	no	—
164	#121	—	172	4	no	—
165	C₂×C₄⋊C₄	—	86	2	no	—
166	C₂×C₂²⋊C₄	—	172	2	no	—
167	D₁₆	—	86	8	no	—
168	M₃₂	—	2	8	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	C₂²⋊D₄₃	1	4	yes	complemented
171	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
172	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
173	#107	—	43	4	no	—
174	#107	—	86	2	no	—
175	#102	—	86	8	no	—
176	C₂⁴	—	172	2	no	—
177	C₂⁴	D₃₄₄	1	1	yes	characteristic
178	C₂×D₄	—	86	4	no	—
179	C₂×D₄	—	344	8	no	—
180	C₂×D₄	—	172	8	no	—
181	C₂×D₄	—	172	4	no	—
182	C₂×D₄	—	172	2	no	—
183	C₂×D₄	—	172	4	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	172	2	no	—
187	C₂²×C₄	D₃₄₄	1	2	yes	—
188	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
190	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
191	D₈	—	86	4	no	—
192	D₈	—	86	4	no	—
193	D₈	—	172	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
199	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
200	C₂×C₈	#10621898	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
202	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
203	C₄⋊C₄	—	86	4	no	—
204	C₄⋊C₄	—	172	2	no	—
205	C₂²⋊C₄	—	172	8	no	—
206	C₁₆	—	2	2	no	—
207	C₁₆	C₂×C₂²⋊D₄₃	1	4	yes	complemented
208	C₂³	—	2	4	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	344	4	no	—
211	C₂³	—	344	4	no	—
212	C₂³	—	344	4	no	—
213	C₂³	—	172	4	no	—
214	C₂³	C₂×D₃₄₄	1	2	yes	—
215	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
216	D₄	—	344	16	no	—
217	D₄	—	344	4	no	—
218	D₄	—	172	4	no	—
219	D₄	—	172	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	344	4	no	—
225	C₂×C₄	—	172	4	no	—
226	C₂×C₄	#13063299	1	2	yes	—
227	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
228	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
229	C₂×C₄	#13063347	1	2	yes	—
230	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	#13063299	1	2	yes	—
233	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
234	C₈	C₂×#10621898	1	1	yes	characteristic
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	344	4	no	—
239	C₂²	—	344	4	no	—
240	C₂²	—	344	4	no	—
241	C₂²	—	688	8	no	—
242	C₂²	#14583135	1	2	yes	—
243	C₂²	#14583337	1	4	yes	—
244	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	344	4	no	—
247	C₄	#14583135	1	2	yes	—
248	C₄	C₂×#13063299	1	1	yes	characteristic
249	C₄	C₂×#13063347	1	1	yes	characteristic
250	C₂	—	4	2	no	—
251	C₂	—	688	4	no	—
252	C₂	#19136698	1	4	yes	direct product factor
253	C₂	C₂×#14583135	1	1	yes	characteristic
254	C₂	C₂×#14583337	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂²×M₃₂×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
hypercenter	C₂²×M₃₂	D₄₃
hypocenter	C₄₃	C₂×#2218
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂²×D₆₈₈	C₂	semidirect
2. C₂×#14583135	C₂	semidirect
3. #19136698	C₂	direct (N × H)
4. C₂×#14582727	C₂	semidirect
5. C₂×#14582701	C₂	semidirect
6. C₂²×M₃₂×C₄₃	C₂	semidirect
10. C₂×D₆₈₈	C₂²	semidirect
12. #14583135	C₂²	semidirect
15. #14582727	C₂²	semidirect
16. #14582701	C₂²	semidirect
19. C₂×M₃₂×C₄₃	C₂²	semidirect
45. C₂×C₁₆×C₄₃	D₄	semidirect
79. C₁₆×C₄₃	C₂×D₄	semidirect
124. C₂²×M₃₂	D₄₃	semidirect
130. C₂×C₄₃	#2218	semidirect
140. C₂×M₃₂	C₂×D₄₃	semidirect
149. C₄₃	C₂×#2218	semidirect
170. C₂×C₁₆	C₂²⋊D₄₃	semidirect
207. C₁₆	C₂×C₂²⋊D₄₃	semidirect
252. C₂	#19136698	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×D₆₈₈	2
C₂×#14583135	2
#19136698	2
C₂×#14582727	2
C₂×#14582701	2
C₂²×M₃₂×C₄₃	2
C₂×#2218	43

Subgroups — C2×#19136698 #25251043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136698 #25251043