Subgroups — C₂×#20242912 #26198814

Order 12032.

694 conjugacy classes of subgroups.

The subgroup lattice has 298 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×#15689414	C₂	1	2	yes	maximal
3	C₂×#15689202	C₂	1	1	yes	maximal, characteristic
4	#20242912	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688871	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10625324	—	2	1	no	—
8	C₂×#13094024	C₂²	1	2	yes	—
9	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
10	#15689528	—	2	2	no	—
11	#15689528	C₂²	1	4	yes	complemented
12	#15689414	—	2	4	no	—
13	#15689414	C₂²	1	8	yes	—
14	#15689202	—	2	2	no	—
15	#15689202	C₂²	1	4	yes	—
16	C₂×#13093958	—	2	1	no	—
17	C₂×#13093958	—	2	1	no	—
18	C₂×#13093956	—	2	2	no	—
19	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#13093946	—	2	2	no	—
21	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
22	#15688871	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×#334	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×#10625324	—	2	4	no	—
30	C₂×#10625324	—	4	4	no	—
31	C₂²×C₄₇⋊C₈	—	2	1	no	—
32	#13094024	—	2	2	no	—
33	#13094024	—	2	2	no	—
34	#13094024	—	2	2	no	—
35	#13094024	C₂³	1	4	yes	—
36	C₄×D₁₈₈	—	2	1	no	—
37	C₄×D₁₈₈	—	2	1	no	—
38	C₄×D₁₈₈	D₄	1	2	yes	—
39	C₄×D₁₈₈	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₄₇	—	2	2	no	—
41	C₂×C₄⋊Dic₄₇	—	2	2	no	—
42	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
43	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
44	#13093958	—	2	4	no	—
45	#13093958	—	2	4	no	—
46	#13093956	—	2	8	no	—
47	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
48	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
49	#13093946	—	2	8	no	—
50	(C₂×D₄₇)⋊C₈	—	2	4	no	—
51	C₂²×C₈×C₄₇	—	2	2	no	—
52	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₇	—	2	2	no	—
56	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
57	C₄×C₈×C₄₇	D₄	1	2	yes	—
58	C₄×C₈×C₄₇	C₂³	1	2	yes	—
59	C₂³×D₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	1	no	—
64	C₂²×Dic₄₇	—	4	2	no	—
65	C₂×D₁₈₈	—	2	4	no	—
66	C₂×D₁₈₈	—	2	1	no	—
67	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
68	C₂×C₄×D₄₇	—	2	2	no	—
69	C₂×C₄×D₄₇	—	2	2	no	—
70	C₂×C₄×D₄₇	—	4	2	no	—
71	C₂×C₄×D₄₇	—	4	2	no	—
72	C₂×Dic₉₄	—	2	4	no	—
73	C₂×Dic₉₄	—	4	2	no	—
74	C₂×Dic₉₄	—	8	4	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	4	4	no	—
78	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
79	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
81	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₄₇	#41	1	1	yes	characteristic
84	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
85	(C₂×D₄₇)⋊C₄	—	2	2	no	—
86	(C₂×D₄₇)⋊C₄	—	2	2	no	—
87	C₄⋊Dic₄₇	—	2	1	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	4	2	no	—
91	C₄⋊Dic₄₇	—	4	2	no	—
92	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	#10625324	—	4	8	no	—
96	C₂³×C₄₇	#121	1	1	yes	characteristic
97	C₂²×D₄₇	—	2	2	no	—
98	C₂²×D₄₇	—	4	2	no	—
99	C₂²×D₄₇	—	4	2	no	—
100	C₂²×D₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₄₇	#137	1	4	yes	—
107	C₂×C₄×C₄₇	#137	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₄₇	—	2	2	no	—
112	C₂×Dic₄₇	—	4	2	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	8	4	no	—
115	D₁₈₈	—	2	4	no	—
116	D₁₈₈	—	4	2	no	—
117	C₄×D₄₇	—	4	2	no	—
118	C₄×D₄₇	—	4	2	no	—
119	C₄×D₄₇	—	4	4	no	—
120	Dic₉₄	—	8	4	no	—
121	Dic₉₄	—	8	4	no	—
122	C₈×C₄₇	—	2	2	no	—
123	C₈×C₄₇	—	4	4	no	—
124	C₈×C₄₇	C₂×SD₁₆	1	4	yes	—
125	C₄₇⋊C₈	—	4	2	no	—
126	C₂×#1695	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	#461	1	2	yes	—
128	C₂²×C₄₇	#470	1	2	yes	—
129	C₂²×C₄₇	#472	1	2	yes	—
130	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
131	C₂×D₄₇	—	4	2	no	—
132	C₂×D₄₇	—	4	2	no	—
133	C₂×D₄₇	—	4	2	no	—
134	C₂×D₄₇	—	8	4	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	#461	1	2	yes	—
138	C₄×C₄₇	#470	1	2	yes	—
139	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
140	C₄×C₄₇	C₂×#137	1	2	yes	—
141	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
142	Dic₄₇	—	4	2	no	—
143	Dic₄₇	—	8	4	no	—
144	C₂×#500	—	47	1	no	—
145	C₂×#476	—	47	2	no	—
146	C₂×#436	—	47	1	no	—
147	#1695	—	47	8	no	—
148	C₂×#334	D₄₇	1	1	yes	characteristic, complemented
149	C₂×#333	—	47	2	no	—
150	C₂×C₄₇	#1695	1	4	yes	complemented
151	C₂×C₄₇	C₂×#461	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#470	1	1	yes	characteristic
153	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
154	D₄₇	—	8	2	no	—
155	C₂²×M₁₆	—	94	1	no	—
156	C₂×C₄⋊Q₈	—	47	2	no	—
157	C₂×C₄×D₄	—	47	1	no	—
158	#500	—	47	4	no	—
159	#500	—	94	2	no	—
160	#476	—	47	8	no	—
161	#476	—	94	4	no	—
162	#436	—	47	4	no	—
163	#436	—	94	2	no	—
164	C₂×#107	—	94	1	no	—
165	C₂×#107	—	94	1	no	—
166	C₂×#106	—	94	2	no	—
167	C₂×C₄⋊C₈	—	47	1	no	—
168	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
169	C₂×#102	—	94	2	no	—
170	C₂×C₂²⋊C₈	—	94	1	no	—
171	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
172	#334	C₂×D₄₇	1	4	yes	complemented
173	#333	—	47	8	no	—
174	C₄₇	C₂×#1695	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₈	—	94	2	no	—
176	C₂²×D₄	—	47	1	no	—
177	C₂³×C₄	—	94	1	no	—
178	C₂×M₁₆	—	94	4	no	—
179	C₂×M₁₆	—	188	4	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	94	1	no	—
182	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
183	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
184	C₄⋊Q₈	—	47	4	no	—
185	C₄⋊Q₈	—	94	2	no	—
186	C₄⋊Q₈	—	94	2	no	—
187	C₄⋊Q₈	—	94	2	no	—
188	C₄×D₄	—	47	2	no	—
189	C₄×D₄	—	47	2	no	—
190	C₄×D₄	—	94	1	no	—
191	C₄×D₄	—	94	1	no	—
192	C₂×C₄⋊C₄	—	47	1	no	—
193	C₂×C₄⋊C₄	—	94	2	no	—
194	C₂×C₄⋊C₄	—	94	2	no	—
195	C₂×C₂²⋊C₄	—	94	1	no	—
196	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
197	#107	—	94	4	no	—
198	#107	—	94	4	no	—
199	#106	—	94	8	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	47	2	no	—
202	C₄⋊C₈	—	94	1	no	—
203	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
204	#102	—	94	8	no	—
205	C₂²⋊C₈	—	94	4	no	—
206	C₄×C₈	D₁₈₈	1	2	yes	—
207	C₄×C₈	C₂²×D₄₇	1	2	yes	—
208	C₂⁴	—	94	1	no	—
209	C₂×Q₈	—	94	4	no	—
210	C₂×Q₈	—	376	4	no	—
211	C₂×Q₈	—	188	2	no	—
212	C₂×D₄	—	47	2	no	—
213	C₂×D₄	—	94	1	no	—
214	C₂×D₄	—	94	4	no	—
215	C₂²×C₄	—	94	1	no	—
216	C₂²×C₄	—	94	2	no	—
217	C₂²×C₄	—	94	2	no	—
218	C₂²×C₄	—	188	2	no	—
219	C₂²×C₄	—	188	2	no	—
220	C₂²×C₄	—	188	2	no	—
221	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
222	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
223	C₂²×C₄	#10625354	1	1	yes	characteristic
224	M₁₆	—	188	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	4	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	—	188	2	no	—
230	C₂×C₈	#10625325	1	4	yes	—
231	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
232	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
233	C₄⋊C₄	—	47	2	no	—
234	C₄⋊C₄	—	94	1	no	—
235	C₄⋊C₄	—	94	4	no	—
236	C₄⋊C₄	—	94	4	no	—
237	C₄⋊C₄	—	188	2	no	—
238	C₄⋊C₄	—	188	2	no	—
239	C₂²⋊C₄	—	94	2	no	—
240	C₂²⋊C₄	—	94	2	no	—
241	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
242	C₄²	D₄×D₄₇	1	1	yes	characteristic
243	C₄²	#10625354	1	1	yes	characteristic
244	C₄²	C₂³×D₄₇	1	1	yes	characteristic
245	C₂³	—	94	2	no	—
246	C₂³	—	188	2	no	—
247	C₂³	—	188	2	no	—
248	C₂³	—	188	2	no	—
249	C₂³	#13093999	1	1	yes	characteristic
250	Q₈	—	376	4	no	—
251	Q₈	—	376	4	no	—
252	D₄	—	94	4	no	—
253	D₄	—	188	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	94	2	no	—
256	C₂×C₄	—	376	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	—	188	2	no	—
259	C₂×C₄	—	188	4	no	—
260	C₂×C₄	—	188	2	no	—
261	C₂×C₄	—	188	2	no	—
262	C₂×C₄	#13093999	1	1	yes	characteristic
263	C₂×C₄	#13093999	1	1	yes	characteristic
264	C₂×C₄	#13093999	1	1	yes	characteristic
265	C₂×C₄	C₂×#10625325	1	2	yes	—
266	C₂×C₄	#13094053	1	4	yes	—
267	C₂×C₄	#13094057	1	2	yes	—
268	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
270	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	4	4	no	—
273	C₈	—	188	2	no	—
274	C₈	C₂×#10625325	1	4	yes	—
275	C₂²	—	376	4	no	—
276	C₂²	—	188	2	no	—
277	C₂²	—	188	2	no	—
278	C₂²	—	188	2	no	—
279	C₂²	#15689329	1	2	yes	—
280	C₂²	#15689382	1	2	yes	—
281	C₂²	#15689394	1	2	yes	—
282	C₂²	C₂×#13093999	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	376	4	no	—
286	C₄	—	188	2	no	—
287	C₄	#15689329	1	2	yes	—
288	C₄	#15689382	1	2	yes	—
289	C₄	C₂²×#10625325	1	1	yes	characteristic
290	C₄	C₂×#13094053	1	2	yes	—
291	C₄	C₂×#13094057	1	1	yes	characteristic
292	C₂	—	376	2	no	—
293	C₂	#20242912	1	4	yes	direct product factor
294	C₂	C₂×#15689329	1	1	yes	characteristic
295	C₂	C₂×#15689382	1	1	yes	characteristic
296	C₂	C₂×#15689394	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093999
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	C₂×#1695
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
1. C₂×#15689528	C₂	semidirect
4. #20242912	C₂	direct (N × H)
5. C₂×#15688871	C₂	semidirect
6. C₂×C₄₇×#334	C₂	semidirect
11. #15689528	C₂²	semidirect
22. #15688871	C₂²	semidirect
25. C₄₇×#334	C₂²	semidirect
148. C₂×#334	D₄₇	semidirect
150. C₂×C₄₇	#1695	semidirect
172. #334	C₂×D₄₇	semidirect
174. C₄₇	C₂×#1695	semidirect
293. C₂	#20242912	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×#15689414	2
C₂×#15689202	2
#20242912	2
C₂×#15688871	2
C₂×C₄₇×#334	2
C₂×#1695	47

Subgroups — C2×#20242912 #26198814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242912 #26198814