Subgroups — C₂×#19147269 #25976197

Order 11008.

1830 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	#19147269	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583249	C₂	1	3	yes	maximal
4	C₂×#14583194	C₂	1	3	yes	maximal
5	C₂×#14583106	C₂	1	3	yes	maximal
6	C₂×Q₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×#10621885	C₂	1	3	yes	maximal
8	C₂×C₄₃×#557	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10621885	C₂²	1	3	yes	—
10	Q₈×Dic₈₆	—	2	4	no	—
11	Q₈×Dic₈₆	C₂²	1	8	yes	—
12	C₂×#13063325	C₂²	1	3	yes	—
13	C₂×#13063324	—	2	3	no	—
14	C₂×Q₈×Dic₄₃	—	2	1	no	—
15	C₂×C₄×Dic₈₆	C₂²	1	3	yes	—
16	#14583249	C₂²	1	24	yes	—
17	#14583194	C₂²	1	24	yes	—
18	#14583106	C₂²	1	24	yes	—
19	Q₈×C₄₃⋊C₈	C₂²	1	8	yes	—
20	C₄×#10621885	C₂²	1	24	yes	—
21	C₂×#13063258	C₂²	1	3	yes	—
22	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	3	yes	—
23	C₂×#13063249	C₂²	1	6	yes	—
24	C₂×#13063249	C₂²	1	3	yes	—
25	C₂×#13063248	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
27	C₄₃×#557	—	2	4	no	—
28	C₄₃×#557	C₂²	1	8	yes	—
29	C₂×C₄⋊Q₈×C₄₃	C₂²	1	3	yes	—
30	C₂×#125×C₄₃	—	2	3	no	—
31	C₂×C₄×Q₈×C₄₃	—	2	1	no	—
32	C₂×C₄×Q₈×C₄₃	C₂²	1	3	yes	—
33	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₂²×Dic₈₆	—	2	3	no	—
35	C₂²×Dic₈₆	C₂³	1	1	yes	characteristic
36	C₂×#10621885	—	2	12	no	—
37	C₂×#10621885	C₂³	1	12	yes	—
38	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
40	#13063325	—	2	6	no	—
41	#13063325	C₂³	1	12	yes	—
42	#13063324	—	2	24	no	—
43	Q₈×Dic₄₃	—	2	8	no	—
44	C₄×Dic₈₆	—	2	6	no	—
45	C₄×Dic₈₆	C₂³	1	12	yes	—
46	C₂×C₄⋊Dic₄₃	—	2	3	no	—
47	C₂×C₄⋊Dic₄₃	C₂³	1	3	yes	—
48	C₂×Dic₄₃⋊C₄	—	2	3	no	—
49	C₂×C₄×Dic₄₃	—	2	3	no	—
50	#13063258	C₂³	1	12	yes	—
51	C₄⋊(C₄₃⋊C₈)	C₂³	1	12	yes	—
52	#13063249	C₂³	1	24	yes	—
53	#13063249	C₂³	1	12	yes	—
54	#13063248	C₂³	1	4	yes	—
55	C₄×C₄₃⋊C₈	C₂³	1	12	yes	—
56	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₄₃	C₂³	1	3	yes	—
58	C₄⋊Q₈×C₄₃	—	2	6	no	—
59	C₄⋊Q₈×C₄₃	C₂³	1	12	yes	—
60	#125×C₄₃	—	2	24	no	—
61	C₄×Q₈×C₄₃	—	2	8	no	—
62	C₄×Q₈×C₄₃	—	2	6	no	—
63	C₄×Q₈×C₄₃	D₄	1	4	yes	—
64	C₄×Q₈×C₄₃	C₂³	1	12	yes	—
65	C₄×Q₈×C₄₃	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
67	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
68	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
69	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
70	C₂×C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₃	—	2	3	no	—
72	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
73	C₂×Q₈×C₄₃	—	2	3	no	—
74	C₂×Q₈×C₄₃	—	2	6	no	—
75	C₂×Q₈×C₄₃	—	2	6	no	—
76	C₂×Q₈×C₄₃	Q₁₆	1	8	yes	complemented
77	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
78	C₂×Q₈×C₄₃	C₂⁴	1	6	yes	—
79	C₂²×C₄×C₄₃	—	2	1	no	—
80	C₂²×C₄×C₄₃	—	2	3	no	—
81	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
83	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₄₃	—	2	1	no	—
85	C₂²×Dic₄₃	—	2	3	no	—
86	C₂×Dic₈₆	—	2	1	no	—
87	C₂×Dic₈₆	—	2	2	no	—
88	C₂×Dic₈₆	—	2	12	no	—
89	C₂×Dic₈₆	—	8	6	no	—
90	C₂×Dic₈₆	#41	1	4	yes	—
91	C₂×Dic₈₆	C₂⁴	1	2	yes	—
92	C₄⋊C₄×C₄₃	—	2	12	no	—
93	C₄⋊C₄×C₄₃	—	2	12	no	—
94	C₄⋊C₄×C₄₃	—	2	12	no	—
95	C₄⋊C₄×C₄₃	—	2	1	no	—
96	C₄⋊C₄×C₄₃	C₂×D₄	1	12	yes	—
97	C₄⋊C₄×C₄₃	C₂⁴	1	2	yes	—
98	C₄²×C₄₃	—	2	12	no	—
99	C₄²×C₄₃	C₂×D₄	1	6	yes	—
100	C₄²×C₄₃	C₂⁴	1	6	yes	—
101	#10621885	—	2	24	no	—
102	C₄⋊Dic₄₃	—	2	12	no	—
103	C₄⋊Dic₄₃	—	2	3	no	—
104	C₄⋊Dic₄₃	C₂⁴	1	6	yes	—
105	Dic₄₃⋊C₄	—	2	12	no	—
106	C₄×Dic₄₃	—	2	12	no	—
107	C₂×C₄₃⋊C₈	—	2	6	no	—
108	C₂×C₄₃⋊C₈	—	2	2	no	—
109	C₂×C₄₃⋊C₈	C₂⁴	1	2	yes	—
110	C₂×C₄₃⋊C₈	C₂⁴	1	6	yes	—
111	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₄₃	—	2	6	no	—
113	Q₈×C₄₃	—	2	6	no	—
114	Q₈×C₄₃	—	4	6	no	—
115	Q₈×C₄₃	C₂×Q₁₆	1	16	yes	complemented
116	C₂×C₄×C₄₃	—	2	6	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	6	no	—
119	C₂×C₄×C₄₃	—	4	6	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	C₂×Q₁₆	1	12	yes	—
122	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
123	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
125	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
127	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
128	C₂×Dic₄₃	—	2	4	no	—
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	—	2	6	no	—
131	C₂×Dic₄₃	—	4	6	no	—
132	Dic₈₆	—	2	4	no	—
133	Dic₈₆	—	2	2	no	—
134	Dic₈₆	—	8	12	no	—
135	Dic₈₆	C₂×#41	1	4	yes	—
136	C₄₃⋊C₈	—	2	6	no	—
137	C₄₃⋊C₈	—	2	2	no	—
138	C₂×#3363	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	#548	1	2	yes	—
140	C₂²×C₄₃	C₂²×Q₁₆	1	2	yes	—
141	C₂²×C₄₃	#577	1	2	yes	—
142	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
143	C₄×C₄₃	—	2	6	no	—
144	C₄×C₄₃	—	4	6	no	—
145	C₄×C₄₃	—	4	2	no	—
146	C₄×C₄₃	#548	1	2	yes	—
147	C₄×C₄₃	C₂²×Q₁₆	1	12	yes	—
148	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
149	C₄×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₄₃	—	2	4	no	—
151	Dic₄₃	—	4	6	no	—
152	C₂×Q₈²	—	43	1	no	—
153	C₂×#557	D₄₃	1	1	yes	characteristic
154	#3363	—	43	16	no	—
155	C₂×#487	—	43	3	no	—
156	C₂×#477	—	43	3	no	—
157	C₂×#462	—	43	3	no	—
158	C₂×C₈×Q₈	—	43	1	no	—
159	C₂×C₄×Q₁₆	—	43	3	no	—
160	C₂×C₄₃	#3363	1	4	yes	complemented
161	C₂×C₄₃	C₂×#548	1	1	yes	characteristic
162	C₂×C₄₃	C₂³×Q₁₆	1	1	yes	characteristic
163	C₂×C₄₃	C₂×#577	1	1	yes	characteristic
164	C₂²×Q₁₆	—	43	3	no	—
165	Q₈²	—	43	8	no	—
166	Q₈²	—	86	4	no	—
167	#557	—	2	4	no	—
168	#557	C₂×D₄₃	1	8	yes	—
169	C₂×C₄⋊Q₈	—	43	3	no	—
170	C₂×C₄⋊Q₈	—	86	3	no	—
171	C₂×C₄⋊Q₈	C₂×D₄₃	1	3	yes	—
172	C₂×#125	—	2	3	no	—
173	C₂×C₄×Q₈	—	2	1	no	—
174	C₂×C₄×Q₈	—	43	3	no	—
175	C₂×C₄×Q₈	—	86	1	no	—
176	C₂×C₄×Q₈	C₂×D₄₃	1	3	yes	—
177	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
178	#487	—	43	24	no	—
179	#477	—	43	24	no	—
180	#462	—	43	24	no	—
181	C₈×Q₈	—	43	8	no	—
182	C₄×Q₁₆	—	43	24	no	—
183	C₂×#107	—	43	3	no	—
184	C₂×C₄⋊C₈	—	43	3	no	—
185	C₂×#103	—	43	6	no	—
186	C₂×#103	—	43	3	no	—
187	C₂×#103	—	43	1	no	—
188	C₂×C₄×C₈	—	43	3	no	—
189	C₄₃	C₂×#3363	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	43	1	no	—
191	C₂²×Q₈	—	86	3	no	—
192	C₂²×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
193	C₂²×Q₈	C₂²×D₄₃	1	3	yes	—
194	C₂×Q₁₆	—	43	12	no	—
195	C₂×Q₁₆	—	86	12	no	—
196	C₂²×C₈	—	43	3	no	—
197	C₂²×C₈	—	43	1	no	—
198	C₄⋊Q₈	—	2	6	no	—
199	C₄⋊Q₈	—	43	12	no	—
200	C₄⋊Q₈	—	86	24	no	—
201	C₄⋊Q₈	—	86	6	no	—
202	C₄⋊Q₈	C₂²×D₄₃	1	12	yes	—
203	#125	—	2	24	no	—
204	C₄×Q₈	—	2	8	no	—
205	C₄×Q₈	—	2	6	no	—
206	C₄×Q₈	—	43	12	no	—
207	C₄×Q₈	—	86	8	no	—
208	C₄×Q₈	—	86	6	no	—
209	C₄×Q₈	C₂²⋊D₄₃	1	4	yes	—
210	C₄×Q₈	C₂²×D₄₃	1	12	yes	—
211	C₄×Q₈	C₂²×D₄₃	1	4	yes	—
212	C₂×C₄⋊C₄	—	2	3	no	—
213	C₂×C₄⋊C₄	—	2	3	no	—
214	C₂×C₄⋊C₄	—	2	3	no	—
215	C₂×C₄⋊C₄	—	43	3	no	—
216	C₂×C₄⋊C₄	—	86	3	no	—
217	C₂×C₄⋊C₄	—	86	3	no	—
218	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	3	yes	—
219	C₂×C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
220	C₂×C₄²	—	2	3	no	—
221	C₂×C₄²	—	86	3	no	—
222	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
223	#107	—	43	12	no	—
224	C₄⋊C₈	—	43	12	no	—
225	#103	—	43	12	no	—
226	#103	—	43	24	no	—
227	#103	—	43	4	no	—
228	C₄×C₈	—	43	12	no	—
229	C₂×Q₈	—	2	3	no	—
230	C₂×Q₈	—	2	6	no	—
231	C₂×Q₈	—	2	6	no	—
232	C₂×Q₈	—	43	4	no	—
233	C₂×Q₈	—	43	2	no	—
234	C₂×Q₈	—	86	1	no	—
235	C₂×Q₈	—	86	12	no	—
236	C₂×Q₈	—	86	2	no	—
237	C₂×Q₈	—	344	6	no	—
238	C₂×Q₈	#10621885	1	8	yes	complemented
239	C₂×Q₈	C₂×C₂²⋊D₄₃	1	4	yes	—
240	C₂×Q₈	C₂³×D₄₃	1	6	yes	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	3	no	—
243	C₂²×C₄	—	86	1	no	—
244	C₂²×C₄	—	86	3	no	—
245	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
247	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
248	Q₁₆	—	86	24	no	—
249	C₂×C₈	—	43	2	no	—
250	C₂×C₈	—	43	6	no	—
251	C₂×C₈	—	86	6	no	—
252	C₂×C₈	—	86	2	no	—
253	C₄⋊C₄	—	2	12	no	—
254	C₄⋊C₄	—	2	12	no	—
255	C₄⋊C₄	—	2	12	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	43	6	no	—
258	C₄⋊C₄	—	86	3	no	—
259	C₄⋊C₄	—	86	12	no	—
260	C₄⋊C₄	—	86	12	no	—
261	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	12	yes	—
262	C₄⋊C₄	C₂³×D₄₃	1	2	yes	—
263	C₄²	—	2	12	no	—
264	C₄²	—	86	12	no	—
265	C₄²	C₂×C₂²⋊D₄₃	1	6	yes	—
266	C₄²	C₂³×D₄₃	1	6	yes	—
267	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
268	Q₈	—	2	6	no	—
269	Q₈	—	2	6	no	—
270	Q₈	—	4	6	no	—
271	Q₈	—	43	4	no	—
272	Q₈	—	86	4	no	—
273	Q₈	—	86	2	no	—
274	Q₈	—	344	12	no	—
275	Q₈	C₂×#10621885	1	16	yes	complemented
276	C₂×C₄	—	2	6	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	6	no	—
279	C₂×C₄	—	4	6	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	86	4	no	—
282	C₂×C₄	—	86	2	no	—
283	C₂×C₄	—	86	6	no	—
284	C₂×C₄	—	172	6	no	—
285	C₂×C₄	C₂×#10621885	1	12	yes	—
286	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
287	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
289	C₂×C₄	#41×D₄₃	1	2	yes	—
290	C₂×C₄	#13063372	1	2	yes	—
291	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
292	C₈	—	86	6	no	—
293	C₈	—	86	2	no	—
294	C₂²	#14583556	1	2	yes	—
295	C₂²	C₂²×#10621885	1	2	yes	—
296	C₂²	#14583707	1	2	yes	—
297	C₂²	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
298	C₄	—	2	6	no	—
299	C₄	—	4	6	no	—
300	C₄	—	4	2	no	—
301	C₄	—	86	4	no	—
302	C₄	—	172	6	no	—
303	C₄	#14583556	1	2	yes	—
304	C₄	C₂²×#10621885	1	12	yes	—
305	C₄	C₂×#41×D₄₃	1	1	yes	characteristic
306	C₄	C₂×#13063372	1	1	yes	characteristic
307	C₂	#19147269	1	4	yes	direct product factor
308	C₂	C₂×#14583556	1	1	yes	characteristic
309	C₂	C₂³×#10621885	1	1	yes	characteristic
310	C₂	C₂×#14583707	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#557	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₂×#557	D₄₃
hypocenter	C₄₃	C₂×#3363
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#557

Complements

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

N	H ≅ G/N	type
2. #19147269	C₂	direct (N × H)
76. C₂×Q₈×C₄₃	Q₁₆	semidirect
115. Q₈×C₄₃	C₂×Q₁₆	semidirect
160. C₂×C₄₃	#3363	semidirect
189. C₄₃	C₂×#3363	semidirect
238. C₂×Q₈	#10621885	semidirect
275. Q₈	C₂×#10621885	semidirect
307. C₂	#19147269	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×Q₈×Dic₈₆	2
#19147269	2
C₂×#14583249	2
C₂×#14583194	2
C₂×#14583106	2
C₂×Q₈×C₄₃⋊C₈	2
C₂×C₄×#10621885	2
C₂×C₄₃×#557	2
C₂×#3363	43

Subgroups — C2×#19147269 #25976197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19147269 #25976197