Subgroups — C₂×#20242778 #26198536

Order 12032.

790 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689329	C₂	1	2	yes	maximal
3	C₂×C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
4	#20242778	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688870	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10625325	—	2	2	no	—
8	C₂²×C₈×D₄₇	—	2	1	no	—
9	C₂×#13094024	C₂²	1	2	yes	—
10	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#15689528	—	2	2	no	—
12	#15689528	C₂²	1	4	yes	complemented
13	#15689329	—	2	4	no	—
14	#15689329	C₂²	1	8	yes	—
15	C₈×D₁₈₈	—	2	2	no	—
16	C₈×D₁₈₈	C₂²	1	4	yes	—
17	C₂×#13093958	—	2	1	no	—
18	C₂×#13093958	—	2	1	no	—
19	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#13093949	—	2	2	no	—
21	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
22	#15688870	C₂²	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#335	C₂²	1	4	yes	complemented
26	C₂²×Dic₉₄	—	2	2	no	—
27	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	—	2	1	no	—
29	C₂×#10625325	—	2	8	no	—
30	C₂×#10625325	—	4	8	no	—
31	C₂²×C₄₇⋊C₈	—	2	1	no	—
32	C₂×C₈×D₄₇	—	2	4	no	—
33	C₂×C₈×D₄₇	—	4	4	no	—
34	#13094024	—	2	2	no	—
35	#13094024	—	2	2	no	—
36	#13094024	—	2	2	no	—
37	#13094024	C₂³	1	4	yes	—
38	C₄×D₁₈₈	—	2	1	no	—
39	C₄×D₁₈₈	—	2	1	no	—
40	C₄×D₁₈₈	D₄	1	2	yes	—
41	C₄×D₁₈₈	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	—	2	2	no	—
43	C₂×C₄⋊Dic₄₇	—	2	2	no	—
44	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
45	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
46	#13093958	—	2	4	no	—
47	#13093958	—	2	4	no	—
48	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
50	#13093949	—	2	8	no	—
51	(C₂×D₄₇)⋊C₈	—	2	4	no	—
52	C₂²×C₈×C₄₇	—	2	2	no	—
53	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	—	2	2	no	—
57	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
58	C₄×C₈×C₄₇	D₄	1	2	yes	—
59	C₄×C₈×C₄₇	C₂³	1	2	yes	—
60	C₂³×D₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	2	1	no	—
65	C₂²×Dic₄₇	—	4	2	no	—
66	C₂×D₁₈₈	—	2	1	no	—
67	C₂×D₁₈₈	—	2	2	no	—
68	C₂×D₁₈₈	D₈	1	4	yes	—
69	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
70	C₂×C₄×D₄₇	—	2	2	no	—
71	C₂×C₄×D₄₇	—	2	2	no	—
72	C₂×C₄×D₄₇	—	4	2	no	—
73	C₂×C₄×D₄₇	—	4	2	no	—
74	C₂×Dic₉₄	—	2	4	no	—
75	C₂×Dic₉₄	—	4	4	no	—
76	C₂×Dic₉₄	—	4	4	no	—
77	C₂×Dic₉₄	—	4	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	4	4	no	—
81	C₂×C₈×C₄₇	SD₁₆	1	4	yes	complemented
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₇	#41	1	1	yes	characteristic
87	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
88	(C₂×D₄₇)⋊C₄	—	2	2	no	—
89	(C₂×D₄₇)⋊C₄	—	2	2	no	—
90	C₄⋊Dic₄₇	—	2	1	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₄⋊Dic₄₇	—	4	2	no	—
95	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	#10625325	—	4	16	no	—
99	C₈×D₄₇	—	4	8	no	—
100	C₂³×C₄₇	#121	1	1	yes	characteristic
101	C₂²×D₄₇	—	2	2	no	—
102	C₂²×D₄₇	—	2	4	no	—
103	C₂²×D₄₇	—	4	2	no	—
104	C₂²×D₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
111	C₂×C₄×C₄₇	#135	1	2	yes	—
112	C₂×C₄×C₄₇	#137	1	2	yes	—
113	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	—	4	2	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	8	4	no	—
120	D₁₈₈	—	2	2	no	—
121	D₁₈₈	—	2	4	no	—
122	D₁₈₈	C₂×D₈	1	4	yes	—
123	C₄×D₄₇	—	4	2	no	—
124	C₄×D₄₇	—	4	2	no	—
125	C₄×D₄₇	—	4	4	no	—
126	Dic₉₄	—	4	4	no	—
127	Dic₉₄	—	4	4	no	—
128	Dic₉₄	—	8	4	no	—
129	C₈×C₄₇	—	2	2	no	—
130	C₈×C₄₇	—	4	4	no	—
131	C₈×C₄₇	C₂×SD₁₆	1	4	yes	complemented
132	C₄₇⋊C₈	—	4	2	no	—
133	C₂×#1673	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	#461	1	2	yes	—
135	C₂²×C₄₇	#466	1	2	yes	—
136	C₂²×C₄₇	#473	1	2	yes	—
137	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
138	C₂×D₄₇	—	2	4	no	—
139	C₂×D₄₇	—	4	8	no	—
140	C₂×D₄₇	—	4	2	no	—
141	C₂×D₄₇	—	4	2	no	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	#461	1	2	yes	—
145	C₄×C₄₇	#466	1	2	yes	—
146	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
148	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
149	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
150	Dic₄₇	—	4	2	no	—
151	Dic₄₇	—	8	4	no	—
152	C₂×#500	—	47	1	no	—
153	C₂×#461	—	47	2	no	—
154	C₂×C₈×D₄	—	47	1	no	—
155	#1673	—	47	8	no	—
156	C₂×#335	D₄₇	1	1	yes	characteristic, complemented
157	C₂×#332	—	47	2	no	—
158	C₂×C₄₇	#1673	1	4	yes	complemented
159	C₂×C₄₇	C₂×#461	1	1	yes	characteristic
160	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
161	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
162	D₄₇	—	4	4	no	—
163	C₂²×SD₁₆	—	94	2	no	—
164	C₂³×C₈	—	94	1	no	—
165	C₂×C₄⋊Q₈	—	47	2	no	—
166	C₂×C₄×D₄	—	47	1	no	—
167	#500	—	47	4	no	—
168	#500	—	94	2	no	—
169	#461	—	47	8	no	—
170	#461	—	94	4	no	—
171	C₈×D₄	—	47	4	no	—
172	C₈×D₄	—	94	2	no	—
173	C₂×#107	—	94	1	no	—
174	C₂×#107	—	94	1	no	—
175	C₂×C₄⋊C₈	—	47	1	no	—
176	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
177	C₂×#103	—	94	2	no	—
178	C₂×C₂²⋊C₈	—	94	1	no	—
179	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
180	#335	C₂×D₄₇	1	4	yes	complemented
181	#332	—	47	8	no	—
182	C₄₇	C₂×#1673	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	94	2	no	—
184	C₂²×D₄	—	47	1	no	—
185	C₂³×C₄	—	94	1	no	—
186	C₂×SD₁₆	—	94	8	no	—
187	C₂×SD₁₆	—	188	8	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	94	1	no	—
190	C₂²×C₈	—	94	4	no	—
191	C₂²×C₈	—	188	4	no	—
192	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
193	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
194	C₄⋊Q₈	—	47	4	no	—
195	C₄⋊Q₈	—	94	2	no	—
196	C₄⋊Q₈	—	94	2	no	—
197	C₄⋊Q₈	—	94	2	no	—
198	C₄×D₄	—	47	2	no	—
199	C₄×D₄	—	47	2	no	—
200	C₄×D₄	—	94	1	no	—
201	C₄×D₄	—	94	1	no	—
202	C₂×C₄⋊C₄	—	47	1	no	—
203	C₂×C₄⋊C₄	—	94	2	no	—
204	C₂×C₄⋊C₄	—	94	2	no	—
205	C₂×C₂²⋊C₄	—	94	1	no	—
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	#107	—	94	4	no	—
208	#107	—	94	4	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	47	2	no	—
211	C₄⋊C₈	—	94	1	no	—
212	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
213	#103	—	94	8	no	—
214	C₂²⋊C₈	—	94	4	no	—
215	C₄×C₈	D₁₈₈	1	2	yes	—
216	C₄×C₈	C₂²×D₄₇	1	2	yes	—
217	C₂⁴	—	94	1	no	—
218	C₂×Q₈	—	94	4	no	—
219	C₂×Q₈	—	188	4	no	—
220	C₂×Q₈	—	188	4	no	—
221	C₂×Q₈	—	188	2	no	—
222	C₂×D₄	—	47	4	no	—
223	C₂×D₄	—	47	2	no	—
224	C₂×D₄	—	94	1	no	—
225	C₂×D₄	—	94	2	no	—
226	C₂²×C₄	—	94	1	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	—	188	2	no	—
230	C₂²×C₄	—	188	2	no	—
231	C₂²×C₄	—	188	2	no	—
232	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	#10625354	1	1	yes	characteristic
235	SD₁₆	—	188	16	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	4	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	—	188	2	no	—
241	C₂×C₈	—	188	8	no	—
242	C₂×C₈	#10625325	1	4	yes	complemented
243	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
244	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
245	C₄⋊C₄	—	47	2	no	—
246	C₄⋊C₄	—	94	1	no	—
247	C₄⋊C₄	—	94	4	no	—
248	C₄⋊C₄	—	94	4	no	—
249	C₄⋊C₄	—	188	2	no	—
250	C₄⋊C₄	—	188	2	no	—
251	C₂²⋊C₄	—	94	2	no	—
252	C₂²⋊C₄	—	94	2	no	—
253	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
254	C₄²	D₄×D₄₇	1	1	yes	characteristic
255	C₄²	#10625354	1	1	yes	characteristic
256	C₄²	C₂³×D₄₇	1	1	yes	characteristic
257	C₂³	—	94	4	no	—
258	C₂³	—	94	2	no	—
259	C₂³	—	188	2	no	—
260	C₂³	—	188	2	no	—
261	C₂³	#13093999	1	1	yes	characteristic
262	Q₈	—	376	4	no	—
263	Q₈	—	188	4	no	—
264	Q₈	—	188	4	no	—
265	D₄	—	47	4	no	—
266	D₄	—	94	4	no	—
267	D₄	—	94	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	—	376	4	no	—
271	C₂×C₄	—	188	4	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	—	188	4	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	—	188	2	no	—
276	C₂×C₄	#13093999	1	1	yes	characteristic
277	C₂×C₄	#13093999	1	1	yes	characteristic
278	C₂×C₄	#13093999	1	1	yes	characteristic
279	C₂×C₄	D₈×D₄₇	1	2	yes	—
280	C₂×C₄	C₂×#10625325	1	2	yes	—
281	C₂×C₄	#13094047	1	2	yes	—
282	C₂×C₄	#13094057	1	2	yes	—
283	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
285	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	4	4	no	—
288	C₈	—	188	2	no	—
289	C₈	C₂×#10625325	1	4	yes	complemented
290	C₂²	—	94	4	no	—
291	C₂²	—	188	8	no	—
292	C₂²	—	188	2	no	—
293	C₂²	—	188	2	no	—
294	C₂²	#15689329	1	2	yes	—
295	C₂²	#15689360	1	2	yes	—
296	C₂²	#15689399	1	2	yes	—
297	C₂²	C₂×#13093999	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	376	4	no	—
301	C₄	—	188	2	no	—
302	C₄	#15689329	1	2	yes	—
303	C₄	#15689360	1	2	yes	—
304	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
305	C₄	C₂²×#10625325	1	1	yes	characteristic
306	C₄	C₂×#13094047	1	1	yes	characteristic
307	C₄	C₂×#13094057	1	1	yes	characteristic
308	C₂	—	188	4	no	—
309	C₂	#20242778	1	4	yes	direct product factor
310	C₂	C₂×#15689329	1	1	yes	characteristic
311	C₂	C₂×#15689360	1	1	yes	characteristic
312	C₂	C₂×#15689399	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093999
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	C₂×#1673
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
1. C₂×#15689528	C₂	semidirect
4. #20242778	C₂	direct (N × H)
5. C₂×#15688870	C₂	semidirect
6. C₂×C₄₇×#335	C₂	semidirect
12. #15689528	C₂²	semidirect
22. #15688870	C₂²	semidirect
25. C₄₇×#335	C₂²	semidirect
81. C₂×C₈×C₄₇	SD₁₆	semidirect
131. C₈×C₄₇	C₂×SD₁₆	semidirect
156. C₂×#335	D₄₇	semidirect
158. C₂×C₄₇	#1673	semidirect
180. #335	C₂×D₄₇	semidirect
182. C₄₇	C₂×#1673	semidirect
242. C₂×C₈	#10625325	semidirect
289. C₈	C₂×#10625325	semidirect
309. C₂	#20242778	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×#15689528	2
C₂×#15689329	2
C₂×C₈×D₁₈₈	2
#20242778	2
C₂×#15688870	2
C₂×C₄₇×#335	2
C₂×#1673	47

Subgroups — C2×#20242778 #26198536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242778 #26198536