Subgroups — C₂×#19136798 #25251389

Order 11008.

630 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583140	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13063259	C₂	1	1	yes	maximal, characteristic
3	#19136798	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582725	C₂	1	2	yes	maximal, complemented
5	C₂×#14582704	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
8	C₂×#13063304	—	2	1	no	—
9	C₂²×C₄⋊Dic₄₃	—	2	1	no	—
10	#14583140	—	2	2	no	—
11	#14583140	C₂²	1	4	yes	—
12	C₂×#13063259	—	2	2	no	—
13	C₂×#13063259	—	2	1	no	—
14	C₂×#13063259	C₂²	1	4	yes	—
15	C₂×#13063259	C₂²	1	2	yes	—
16	C₂×#13063259	C₂²	1	1	yes	characteristic
17	C₂×#13063250	—	2	1	no	—
18	#14582725	C₂²	1	8	yes	complemented
19	#14582704	C₂²	1	8	yes	complemented
20	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
22	C₄₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
23	C₂²×Dic₈₆	—	2	1	no	—
24	C₂³×Dic₄₃	—	4	1	no	—
25	C₂×Dic₁₇₂	—	2	4	no	—
26	C₂×Dic₁₇₂	—	2	1	no	—
27	C₂×Dic₁₇₂	C₂³	1	2	yes	—
28	#13063304	—	2	4	no	—
29	#13063304	—	4	2	no	—
30	C₂×C₄⋊Dic₄₃	—	2	4	no	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	1	no	—
33	C₂×C₄⋊Dic₄₃	—	4	1	no	—
34	C₂×C₄⋊Dic₄₃	—	4	2	no	—
35	C₂×Dic₄₃⋊C₄	—	4	1	no	—
36	C₂×C₂²⋊Dic₄₃	—	4	1	no	—
37	#13063259	—	2	1	no	—
38	#13063259	—	2	4	no	—
39	#13063259	—	2	2	no	—
40	#13063259	C₂³	1	2	yes	—
41	#13063259	C₂³	1	4	yes	—
42	#13063250	—	2	4	no	—
43	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	—	2	4	no	—
45	C₂²×C₈×C₄₃	D₄	1	2	yes	—
46	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₄₃	—	2	4	no	—
50	C₂×C₁₆×C₄₃	C₂³	1	4	yes	—
51	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	—	2	2	no	—
53	C₂²×C₄×C₄₃	—	2	2	no	—
54	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	D₈	1	2	yes	—
56	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₄₃	—	4	1	no	—
59	C₂²×Dic₄₃	—	4	1	no	—
60	C₂²×Dic₄₃	—	4	4	no	—
61	C₂²×Dic₄₃	—	4	2	no	—
62	C₂²×Dic₄₃	—	8	4	no	—
63	C₂×Dic₈₆	—	2	2	no	—
64	C₂×Dic₈₆	—	4	2	no	—
65	C₂×Dic₈₆	—	4	2	no	—
66	C₂×Dic₈₆	—	4	1	no	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	2	4	no	—
69	C₂×C₈×C₄₃	—	2	4	no	—
70	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₄₃	#41	1	4	yes	—
73	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂²⋊Dic₄₃	—	4	4	no	—
75	C₄⋊Dic₄₃	—	2	2	no	—
76	C₄⋊Dic₄₃	—	2	4	no	—
77	C₄⋊Dic₄₃	—	4	4	no	—
78	C₄⋊Dic₄₃	—	4	2	no	—
79	C₄⋊Dic₄₃	—	4	1	no	—
80	Dic₄₃⋊C₄	—	4	4	no	—
81	Dic₁₇₂	—	2	4	no	—
82	Dic₁₇₂	—	4	2	no	—
83	C₁₆×C₄₃	—	2	4	no	—
84	C₂³×C₄₃	—	2	2	no	—
85	C₂³×C₄₃	—	2	2	no	—
86	C₂³×C₄₃	Q₃₂	1	4	yes	—
87	C₂³×C₄₃	C₂×D₈	1	2	yes	—
88	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	—	2	4	no	—
93	C₂×C₄×C₄₃	#123	1	2	yes	—
94	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₄₃	#137	1	2	yes	—
97	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₄₃	—	4	2	no	—
99	C₂×Dic₄₃	—	4	2	no	—
100	C₂×Dic₄₃	—	4	4	no	—
101	C₂×Dic₄₃	—	8	8	no	—
102	C₂×Dic₄₃	—	8	2	no	—
103	C₂×Dic₄₃	—	8	2	no	—
104	C₂×Dic₄₃	—	8	4	no	—
105	Dic₈₆	—	4	2	no	—
106	Dic₈₆	—	4	2	no	—
107	Dic₈₆	—	8	2	no	—
108	C₈×C₄₃	—	2	2	no	—
109	C₈×C₄₃	#123	1	2	yes	—
110	C₈×C₄₃	C₂×#41	1	2	yes	—
111	C₂×#2236	—	43	1	no	maximal, Sylow
112	C₂²×C₄₃	—	2	2	no	—
113	C₂²×C₄₃	—	2	2	no	—
114	C₂²×C₄₃	—	2	8	no	—
115	C₂²×C₄₃	#480	1	2	yes	—
116	C₂²×C₄₃	C₂×Q₃₂	1	4	yes	—
117	C₂²×C₄₃	C₂×Q₃₂	1	2	yes	—
118	C₂²×C₄₃	#509	1	2	yes	—
119	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₄₃	—	2	2	no	—
121	C₄×C₄₃	#480	1	2	yes	—
122	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
123	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
124	Dic₄₃	—	8	2	no	—
125	Dic₄₃	—	8	2	no	—
126	Dic₄₃	—	8	4	no	—
127	C₂×#467	—	43	1	no	—
128	C₂²×#107	—	43	1	no	—
129	#2236	—	43	8	no	—
130	C₂×#366	—	43	2	no	—
131	C₂×#358	—	43	2	no	—
132	C₂×C₂²⋊C₁₆	D₄₃	1	1	yes	characteristic
133	C₂×C₄₃	—	2	4	no	—
134	C₂×C₄₃	#2236	1	4	yes	complemented
135	C₂×C₄₃	C₂×#480	1	1	yes	characteristic
136	C₂×C₄₃	C₂²×Q₃₂	1	1	yes	characteristic
137	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
138	C₂²×Q₁₆	—	43	1	no	—
139	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
140	C₂×C₂²⋊Q₈	—	86	1	no	—
141	C₂²×C₄⋊C₄	—	86	1	no	—
142	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
143	#467	—	43	4	no	—
144	#467	—	86	2	no	—
145	C₂×#107	—	43	2	no	—
146	C₂×#107	—	43	4	no	—
147	C₂×#107	—	43	1	no	—
148	C₂×#107	—	86	1	no	—
149	C₂×#107	—	86	2	no	—
150	C₂×#103	—	86	1	no	—
151	#366	—	43	8	no	—
152	#358	—	43	8	no	—
153	C₂²⋊C₁₆	C₂×D₄₃	1	4	yes	—
154	C₄₃	C₂×#2236	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	86	1	no	—
156	C₂³×C₄	—	172	1	no	—
157	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
158	C₂×Q₁₆	—	43	2	no	—
159	C₂×Q₁₆	—	86	1	no	—
160	C₂×Q₁₆	—	86	4	no	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
163	C₂²×C₈	D₁₇₂	1	2	yes	—
164	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
165	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
166	C₂²⋊Q₈	—	86	4	no	—
167	C₂²⋊Q₈	—	172	2	no	—
168	C₂×C₄⋊C₄	—	86	4	no	—
169	C₂×C₄⋊C₄	—	86	2	no	—
170	C₂×C₄⋊C₄	—	86	1	no	—
171	C₂×C₄⋊C₄	—	172	2	no	—
172	C₂×C₄⋊C₄	—	172	1	no	—
173	C₂×C₄⋊C₄	—	172	1	no	—
174	C₂×C₂²⋊C₄	—	172	1	no	—
175	C₂×C₁₆	—	2	4	no	—
176	C₂×C₁₆	C₂²×D₄₃	1	4	yes	—
177	#107	—	43	4	no	—
178	#107	—	43	2	no	—
179	#107	—	86	4	no	—
180	#107	—	86	2	no	—
181	#107	—	86	1	no	—
182	#103	—	86	4	no	—
183	C₂⁴	D₃₄₄	1	1	yes	characteristic
184	C₂×Q₈	—	86	2	no	—
185	C₂×Q₈	—	172	2	no	—
186	C₂×Q₈	—	172	2	no	—
187	C₂×Q₈	—	172	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	344	4	no	—
191	C₂²×C₄	—	172	4	no	—
192	C₂²×C₄	—	172	1	no	—
193	C₂²×C₄	—	172	1	no	—
194	C₂²×C₄	—	172	2	no	—
195	C₂²×C₄	D₃₄₄	1	2	yes	—
196	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
198	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
199	Q₁₆	—	86	4	no	—
200	Q₁₆	—	172	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
205	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
206	C₂×C₈	#10621900	1	4	yes	—
207	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
208	C₄⋊C₄	—	86	2	no	—
209	C₄⋊C₄	—	86	4	no	—
210	C₄⋊C₄	—	172	4	no	—
211	C₄⋊C₄	—	172	2	no	—
212	C₄⋊C₄	—	172	1	no	—
213	C₄⋊C₄	—	172	4	no	—
214	C₂²⋊C₄	—	172	4	no	—
215	C₁₆	—	2	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	Dic₃₄₄	1	4	yes	—
219	C₂³	C₂×D₃₄₄	1	2	yes	—
220	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
221	Q₈	—	344	2	no	—
222	Q₈	—	172	2	no	—
223	Q₈	—	172	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	344	8	no	—
229	C₂×C₄	—	344	4	no	—
230	C₂×C₄	—	344	2	no	—
231	C₂×C₄	—	344	2	no	—
232	C₂×C₄	—	172	2	no	—
233	C₂×C₄	—	172	2	no	—
234	C₂×C₄	—	172	4	no	—
235	C₂×C₄	#13063309	1	2	yes	—
236	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
237	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
238	C₂×C₄	#13063358	1	2	yes	—
239	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	#13063309	1	2	yes	—
242	C₈	C₂×#10621900	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	2	no	—
246	C₂²	#14583213	1	2	yes	—
247	C₂²	C₂×Dic₃₄₄	1	4	yes	—
248	C₂²	C₂×Dic₃₄₄	1	2	yes	—
249	C₂²	#14583337	1	2	yes	—
250	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	344	2	no	—
253	C₄	—	344	4	no	—
254	C₄	—	344	2	no	—
255	C₄	#14583213	1	2	yes	—
256	C₄	C₂×#13063309	1	1	yes	characteristic
257	C₄	C₂×#13063358	1	1	yes	characteristic
258	C₂	—	2	4	no	—
259	C₂	#19136798	1	4	yes	direct product factor
260	C₂	C₂×#14583213	1	1	yes	characteristic
261	C₂	C₂²×Dic₃₄₄	1	1	yes	characteristic
262	C₂	C₂×#14583337	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂⁴×C₄₃	D₈
hypercenter	C₂×C₂²⋊C₁₆	D₄₃
hypocenter	C₄₃	C₂×#2236
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #19136798	C₂	direct (N × H)
4. C₂×#14582725	C₂	semidirect
5. C₂×#14582704	C₂	semidirect
18. #14582725	C₂²	semidirect
19. #14582704	C₂²	semidirect
134. C₂×C₄₃	#2236	semidirect
154. C₄₃	C₂×#2236	semidirect
259. C₂	#19136798	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×#14583140	2
C₂²×#13063259	2
#19136798	2
C₂×#14582725	2
C₂×#14582704	2
C₂×C₄₃×C₂²⋊C₁₆	2
C₂×#2236	43

Subgroups — C2×#19136798 #25251389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136798 #25251389