Subgroups — C₂×#19135398 #25100218

Order 11008.

1726 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063249	C₂	1	4	yes	maximal
2	C₂²×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582769	C₂	1	1	yes	maximal, characteristic
4	#19135398	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₃	—	2	2	no	—
8	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
9	C₂×#13063249	—	2	16	no	—
10	C₂×#13063249	C₂²	1	16	yes	—
11	C₂×C₄×C₄₃⋊C₈	—	2	4	no	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
13	#14582769	—	2	2	no	—
14	#14582769	C₂²	1	4	yes	—
15	C₂³×Q₈×C₄₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₄₃	—	2	2	no	—
17	C₂×C₄⋊Q₈×C₄₃	—	2	4	no	—
18	C₂×C₄⋊Q₈×C₄₃	C₄	1	8	yes	complemented
19	C₂×C₄⋊Q₈×C₄₃	C₂²	1	4	yes	—
20	C₂²×C₄⋊C₄×C₄₃	—	2	2	no	—
21	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	C₂³×Dic₄₃	—	2	2	no	—
23	C₂²×C₄₃⋊C₈	—	2	4	no	—
24	C₂²×C₄₃⋊C₈	—	2	4	no	—
25	C₂²×C₄₃⋊C₈	D₄	1	8	yes	—
26	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
27	C₂×C₄⋊Dic₄₃	—	2	8	no	—
28	C₂×C₄⋊Dic₄₃	—	2	8	no	—
29	C₂×C₄⋊Dic₄₃	—	2	2	no	—
30	C₂×C₄⋊Dic₄₃	—	4	8	no	—
31	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
32	#13063249	—	2	32	no	—
33	C₄×C₄₃⋊C₈	—	2	8	no	—
34	C₂²×Q₈×C₄₃	—	2	4	no	—
35	C₂²×Q₈×C₄₃	—	2	2	no	—
36	C₂²×Q₈×C₄₃	—	2	8	no	—
37	C₂²×Q₈×C₄₃	—	2	8	no	—
38	C₂²×Q₈×C₄₃	C₂×C₄	1	8	yes	—
39	C₂²×Q₈×C₄₃	C₂³	1	4	yes	—
40	C₂³×C₄×C₄₃	—	2	2	no	—
41	C₂³×C₄×C₄₃	D₄	1	2	yes	—
42	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₄₃	—	2	16	no	—
44	C₄⋊Q₈×C₄₃	—	2	8	no	—
45	C₄⋊Q₈×C₄₃	C₂×C₄	1	16	yes	complemented
46	C₂×C₄⋊C₄×C₄₃	—	2	16	no	—
47	C₂×C₄⋊C₄×C₄₃	—	2	8	no	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₃	D₄	1	2	yes	—
50	C₂×C₄²×C₄₃	D₄	1	4	yes	—
51	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
52	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×Q₈×C₄₃	—	2	8	no	—
54	C₂×Q₈×C₄₃	—	2	4	no	—
55	C₂×Q₈×C₄₃	—	2	32	no	—
56	C₂×Q₈×C₄₃	—	2	8	no	—
57	C₂×Q₈×C₄₃	—	2	8	no	—
58	C₂×Q₈×C₄₃	—	4	16	no	—
59	C₂×Q₈×C₄₃	—	4	4	no	—
60	C₂×Q₈×C₄₃	—	4	4	no	—
61	C₂×Q₈×C₄₃	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₄₃	—	2	8	no	—
63	C₂²×C₄×C₄₃	—	2	4	no	—
64	C₂²×C₄×C₄₃	—	4	4	no	—
65	C₂²×C₄×C₄₃	—	4	4	no	—
66	C₂²×C₄×C₄₃	C₂²⋊C₄	1	8	yes	—
67	C₂²×C₄×C₄₃	SD₁₆	1	8	yes	—
68	C₂²×C₄×C₄₃	Q₁₆	1	8	yes	—
69	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₄₃	#41	1	4	yes	—
75	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₃	—	2	4	no	—
77	C₂²×Dic₄₃	—	4	4	no	—
78	C₂²×Dic₄₃	—	4	8	no	—
79	C₄⋊C₄×C₄₃	—	2	32	no	—
80	C₄²×C₄₃	C₂²⋊C₄	1	8	yes	—
81	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
82	C₄²×C₄₃	C₂×D₄	1	4	yes	—
83	C₄⋊Dic₄₃	—	2	8	no	—
84	C₄⋊Dic₄₃	—	4	4	no	—
85	C₄⋊Dic₄₃	—	4	16	no	—
86	C₂×C₄₃⋊C₈	—	2	16	no	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	C₂×D₄	1	8	yes	—
90	C₂³×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂³×C₄₃	#124	1	2	yes	—
92	C₂³×C₄₃	#127	1	2	yes	—
93	C₂³×C₄₃	C₂×SD₁₆	1	4	yes	—
94	C₂³×C₄₃	C₂×Q₁₆	1	4	yes	—
95	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₄₃	—	2	16	no	—
97	Q₈×C₄₃	—	4	8	no	—
98	Q₈×C₄₃	—	4	16	no	—
99	C₂×C₄×C₄₃	—	2	8	no	—
100	C₂×C₄×C₄₃	—	4	4	no	—
101	C₂×C₄×C₄₃	—	4	4	no	—
102	C₂×C₄×C₄₃	—	4	16	no	—
103	C₂×C₄×C₄₃	#103	1	32	yes	complemented
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	8	yes	—
106	C₂×C₄×C₄₃	C₄×D₄	1	8	yes	—
107	C₂×C₄×C₄₃	#121	1	8	yes	—
108	C₂×C₄×C₄₃	#124	1	2	yes	—
109	C₂×C₄×C₄₃	#127	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×SD₁₆	1	8	yes	—
111	C₂×C₄×C₄₃	C₂×Q₁₆	1	8	yes	—
112	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
114	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
116	C₂×Dic₄₃	—	4	4	no	—
117	C₂×Dic₄₃	—	4	8	no	—
118	C₂×Dic₄₃	—	8	8	no	—
119	C₄₃⋊C₈	—	2	8	no	—
120	C₂×#1956	—	43	1	no	maximal, Sylow
121	C₂²×C₄₃	#390	1	4	yes	—
122	C₂²×C₄₃	C₂×#103	1	8	yes	—
123	C₂²×C₄₃	#487	1	8	yes	—
124	C₂²×C₄₃	#492	1	4	yes	—
125	C₂²×C₄₃	#494	1	4	yes	—
126	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
128	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
129	C₂²×C₄₃	C₂²×SD₁₆	1	2	yes	—
130	C₂²×C₄₃	C₂²×Q₁₆	1	2	yes	—
131	C₄×C₄₃	—	4	8	no	—
132	C₄×C₄₃	#390	1	4	yes	—
133	C₄×C₄₃	C₂×#103	1	16	yes	complemented
134	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
135	C₄×C₄₃	C₂×#121	1	2	yes	—
136	Dic₄₃	—	8	4	no	—
137	C₂²×C₄⋊Q₈	D₄₃	1	1	yes	characteristic
138	C₂²×#103	—	43	4	no	—
139	C₂²×C₄×C₈	—	43	1	no	—
140	C₂×#384	—	43	1	no	—
141	#1956	—	43	8	no	—
142	C₂×C₄₃	#1956	1	8	yes	complemented
143	C₂×C₄₃	C₂×#390	1	1	yes	characteristic
144	C₂×C₄₃	C₂²×#103	1	2	yes	—
145	C₂×C₄₃	C₂×#487	1	2	yes	—
146	C₂×C₄₃	C₂×#492	1	1	yes	characteristic
147	C₂×C₄₃	C₂×#494	1	1	yes	characteristic
148	C₂³×Q₈	C₂×D₄₃	1	2	yes	—
149	C₂³×C₈	—	43	2	no	—
150	C₂×C₄⋊Q₈	—	2	2	no	—
151	C₂×C₄⋊Q₈	—	2	4	no	—
152	C₂×C₄⋊Q₈	Dic₄₃	1	8	yes	complemented
153	C₂×C₄⋊Q₈	C₂×D₄₃	1	4	yes	—
154	C₂²×C₄⋊C₄	—	2	2	no	—
155	C₂²×C₄⋊C₄	—	43	2	no	—
156	C₂²×C₄⋊C₄	—	86	2	no	—
157	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
158	C₂×#103	—	43	16	no	—
159	C₂×#103	—	86	16	no	—
160	C₂×C₄×C₈	—	43	4	no	—
161	C₂×C₄×C₈	—	86	4	no	—
162	#384	—	43	4	no	—
163	#384	—	86	2	no	—
164	C₄₃	C₂×#1956	1	1	yes	characteristic, complemented, Sylow
165	C₂²×Q₈	—	2	4	no	—
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×Q₈	—	2	8	no	—
168	C₂²×Q₈	—	2	8	no	—
169	C₂²×Q₈	C₂×Dic₄₃	1	8	yes	—
170	C₂²×Q₈	C₂²×D₄₃	1	4	yes	—
171	C₂³×C₄	—	2	2	no	—
172	C₂³×C₄	—	86	2	no	—
173	C₂³×C₄	C₂²⋊D₄₃	1	2	yes	—
174	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
175	C₂²×C₈	—	43	8	no	—
176	C₂²×C₈	—	43	4	no	—
177	C₂²×C₈	—	86	4	no	—
178	C₂²×C₈	—	86	4	no	—
179	C₄⋊Q₈	—	2	16	no	—
180	C₄⋊Q₈	—	2	8	no	—
181	C₄⋊Q₈	C₂×Dic₄₃	1	16	yes	complemented
182	C₂×C₄⋊C₄	—	2	16	no	—
183	C₂×C₄⋊C₄	—	2	8	no	—
184	C₂×C₄⋊C₄	—	43	4	no	—
185	C₂×C₄⋊C₄	—	86	8	no	—
186	C₂×C₄⋊C₄	—	86	2	no	—
187	C₂×C₄⋊C₄	—	86	8	no	—
188	C₂×C₄⋊C₄	—	172	8	no	—
189	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
190	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₄₃	1	4	yes	—
192	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
193	#103	—	86	32	no	—
194	C₄×C₈	—	86	8	no	—
195	C₂⁴	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
196	C₂×Q₈	—	2	8	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	2	32	no	—
199	C₂×Q₈	—	2	8	no	—
200	C₂×Q₈	—	2	8	no	—
201	C₂×Q₈	—	4	16	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	C₂²×Dic₄₃	1	8	yes	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	86	4	no	—
210	C₂²×C₄	—	172	4	no	—
211	C₂²×C₄	—	172	8	no	—
212	C₂²×C₄	#10621884	1	8	yes	—
213	C₂²×C₄	#10621885	1	8	yes	—
214	C₂²×C₄	C₂²⋊Dic₄₃	1	8	yes	—
215	C₂²×C₄	D₄×D₄₃	1	4	yes	—
216	C₂²×C₄	#10621900	1	4	yes	—
217	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
221	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
222	C₂×C₈	—	43	8	no	—
223	C₂×C₈	—	86	16	no	—
224	C₂×C₈	—	86	4	no	—
225	C₂×C₈	—	86	4	no	—
226	C₄⋊C₄	—	2	32	no	—
227	C₄⋊C₄	—	86	8	no	—
228	C₄⋊C₄	—	172	4	no	—
229	C₄⋊C₄	—	172	16	no	—
230	C₄²	C₂²⋊Dic₄₃	1	8	yes	—
231	C₄²	C₂²×Dic₄₃	1	4	yes	—
232	C₄²	C₂×C₂²⋊D₄₃	1	4	yes	—
233	C₂³	C₂×C₂²⋊Dic₄₃	1	2	yes	—
234	C₂³	#13063312	1	2	yes	—
235	C₂³	#13063321	1	2	yes	—
236	C₂³	C₂×#10621884	1	4	yes	—
237	C₂³	C₂×#10621885	1	4	yes	—
238	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
239	Q₈	—	2	16	no	—
240	Q₈	—	4	16	no	—
241	Q₈	—	4	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	4	16	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	344	8	no	—
247	C₂×C₄	—	172	8	no	—
248	C₂×C₄	—	172	4	no	—
249	C₂×C₄	#13063249	1	32	yes	complemented
250	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	8	yes	—
252	C₂×C₄	D₄×Dic₄₃	1	8	yes	—
253	C₂×C₄	#13063296	1	8	yes	—
254	C₂×C₄	#13063312	1	2	yes	—
255	C₂×C₄	#13063321	1	2	yes	—
256	C₂×C₄	C₂×#10621884	1	8	yes	—
257	C₂×C₄	C₂×#10621885	1	8	yes	—
258	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
259	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
260	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#10621900	1	2	yes	—
262	C₈	—	86	8	no	—
263	C₂²	#14582802	1	4	yes	—
264	C₂²	C₂×#13063249	1	8	yes	—
265	C₂²	#14583249	1	8	yes	—
266	C₂²	#14583273	1	4	yes	—
267	C₂²	#14583278	1	4	yes	—
268	C₂²	C₂²×C₂²⋊Dic₄₃	1	1	yes	characteristic
269	C₂²	C₂×#13063312	1	1	yes	characteristic
270	C₂²	C₂×#13063321	1	1	yes	characteristic
271	C₂²	C₂²×#10621884	1	2	yes	—
272	C₂²	C₂²×#10621885	1	2	yes	—
273	C₄	—	4	8	no	—
274	C₄	—	344	4	no	—
275	C₄	#14582802	1	4	yes	—
276	C₄	C₂×#13063249	1	16	yes	complemented
277	C₄	C₂×D₄×Dic₄₃	1	2	yes	—
278	C₄	C₂×#13063296	1	2	yes	—
279	C₂	#19135398	1	8	yes	direct product factor
280	C₂	C₂×#14582802	1	1	yes	characteristic
281	C₂	C₂²×#13063249	1	2	yes	—
282	C₂	C₂×#14583249	1	2	yes	—
283	C₂	C₂×#14583273	1	1	yes	characteristic
284	C₂	C₂×#14583278	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂²×C₄⋊Q₈×C₄₃	C₂
socle, prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	C₂×#1956
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. #19135398	C₂	direct (N × H)
18. C₂×C₄⋊Q₈×C₄₃	C₄	semidirect
45. C₄⋊Q₈×C₄₃	C₂×C₄	semidirect
103. C₂×C₄×C₄₃	#103	semidirect
133. C₄×C₄₃	C₂×#103	semidirect
142. C₂×C₄₃	#1956	semidirect
152. C₂×C₄⋊Q₈	Dic₄₃	semidirect
164. C₄₃	C₂×#1956	semidirect
181. C₄⋊Q₈	C₂×Dic₄₃	semidirect
249. C₂×C₄	#13063249	semidirect
276. C₄	C₂×#13063249	semidirect
279. C₂	#19135398	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×#13063249	2
C₂²×C₄×C₄₃⋊C₈	2
C₂×#14582769	2
#19135398	2
C₂²×C₄⋊Q₈×C₄₃	2
C₂×#1956	43

Subgroups — C2×#19135398 #25100218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19135398 #25100218