Subgroups — C₂×#20241399 #26126500

Order 12032.

506 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689364	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#20241399	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#340	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₈₈	C₄	1	1	yes	characteristic, complemented
6	C₂×#13094003	—	2	1	no	—
7	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
8	#15689364	—	2	1	no	—
9	#15689364	—	2	1	no	—
10	#15689364	C₄	1	2	yes	complemented
11	#15689364	C₂²	1	2	yes	—
12	C₂×#13093958	C₄	1	1	yes	characteristic, complemented
13	C₂×#13093949	—	2	1	no	—
14	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
15	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₄₇×#340	—	2	1	no	—
19	C₄₇×#340	C₂²	1	2	yes	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂×Dic₁₈₈	—	2	1	no	—
22	C₂×Dic₁₈₈	—	4	2	no	—
23	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	complemented
24	#13094003	—	2	2	no	—
25	#13094003	—	2	2	no	—
26	#13094003	—	4	1	no	—
27	#13094003	—	4	1	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	C₂×Dic₄₇⋊C₄	—	4	1	no	—
30	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
31	C₂×C₄₇⋊C₁₆	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	—	4	2	no	—
33	#13093958	—	2	1	no	—
34	#13093958	C₂×C₄	1	2	yes	complemented
35	#13093949	—	2	2	no	—
36	#13093949	—	2	2	no	—
37	C₂³×C₄×C₄₇	—	4	1	no	—
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	—	4	2	no	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
50	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	—	8	2	no	—
56	C₂²×C₄×C₄₇	—	8	2	no	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	4	1	no	—
64	C₂²×Dic₄₇	—	4	1	no	—
65	C₂×Dic₉₄	—	2	2	no	—
66	C₂×Dic₉₄	—	4	2	no	—
67	C₂×Dic₉₄	—	4	2	no	—
68	C₂×Dic₉₄	—	4	1	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	4	1	no	—
72	C₂×C₈×C₄₇	—	4	1	no	—
73	C₂×C₈×C₄₇	—	4	1	no	—
74	C₂×C₈×C₄₇	—	4	1	no	—
75	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	—	4	4	no	—
81	C₄⋊C₄×C₄₇	—	4	2	no	—
82	C₂²⋊Dic₄₇	—	4	2	no	—
83	C₂²⋊Dic₄₇	—	4	2	no	—
84	C₄⋊Dic₄₇	—	2	2	no	—
85	C₄⋊Dic₄₇	—	4	1	no	—
86	Dic₄₇⋊C₄	—	4	2	no	—
87	Dic₄₇⋊C₄	—	4	2	no	—
88	Dic₁₈₈	—	4	2	no	—
89	Dic₁₈₈	—	4	2	no	—
90	C₄₇⋊C₁₆	—	4	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	#102	1	1	yes	characteristic
94	C₂³×C₄₇	#102	1	1	yes	characteristic
95	C₂³×C₄₇	SD₃₂	1	2	yes	—
96	C₂³×C₄₇	Q₃₂	1	2	yes	—
97	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	8	8	no	—
107	C₂×C₄×C₄₇	—	8	2	no	—
108	C₂×C₄×C₄₇	—	8	2	no	—
109	C₂×C₄×C₄₇	#99	1	2	yes	—
110	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#104	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×Dic₄₇	—	4	2	no	—
116	C₂×Dic₄₇	—	4	2	no	—
117	C₂×Dic₄₇	—	8	2	no	—
118	C₂×Dic₄₇	—	8	2	no	—
119	Dic₉₄	—	4	2	no	—
120	Dic₉₄	—	4	2	no	—
121	Dic₉₄	—	8	2	no	—
122	C₈×C₄₇	—	2	1	no	—
123	C₈×C₄₇	—	2	1	no	—
124	C₈×C₄₇	—	2	1	no	—
125	C₈×C₄₇	—	2	1	no	—
126	C₂×#1348	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	#327	1	2	yes	—
134	C₂²×C₄₇	#358	1	2	yes	—
135	C₂²×C₄₇	#358	1	2	yes	—
136	C₂²×C₄₇	#360	1	2	yes	—
137	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
140	C₄×C₄₇	—	2	1	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	8	4	no	—
143	C₄×C₄₇	#327	1	2	yes	—
144	C₄×C₄₇	C₂×#99	1	1	yes	characteristic
145	C₄×C₄₇	C₂×#104	1	1	yes	characteristic
146	Dic₄₇	—	8	2	no	—
147	Dic₄₇	—	8	2	no	—
148	C₂×#467	—	47	1	no	—
149	C₂×C₂²⋊C₁₆	—	47	1	no	—
150	C₂×#340	D₄₇	1	1	yes	characteristic
151	#1348	—	47	4	no	—
152	C₂×C₄₇	—	2	1	no	—
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	—	2	1	no	—
155	C₂×C₄₇	#1348	1	4	yes	complemented
156	C₂×C₄₇	C₂×#327	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#360	1	1	yes	characteristic
159	C₂²×Q₁₆	—	47	1	no	—
160	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
161	C₂×C₂²⋊Q₈	—	94	1	no	—
162	C₂²×C₄⋊C₄	—	2	1	no	—
163	C₂²×C₁₆	—	94	1	no	—
164	#467	—	47	2	no	—
165	#467	—	47	2	no	—
166	#467	—	94	1	no	—
167	#467	—	94	1	no	—
168	C₂×#107	—	47	1	no	—
169	C₂×#103	—	94	1	no	—
170	C₂²⋊C₁₆	—	47	2	no	—
171	C₂²⋊C₁₆	—	94	1	no	—
172	#340	—	2	1	no	—
173	#340	C₂×D₄₇	1	2	yes	—
174	C₄₇	C₂×#1348	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₈	—	94	1	no	—
176	C₂³×C₄	—	4	1	no	—
177	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
178	C₂×Q₁₆	—	47	2	no	—
179	C₂×Q₁₆	—	94	1	no	—
180	C₂×Q₁₆	—	188	2	no	—
181	C₂²×C₈	—	4	2	no	—
182	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
183	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
184	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
185	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
186	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₂²⋊Q₈	—	94	2	no	—
189	C₂²⋊Q₈	—	94	2	no	—
190	C₂²⋊Q₈	—	188	1	no	—
191	C₂²⋊Q₈	—	188	1	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	4	2	no	—
195	C₂×C₄⋊C₄	—	4	1	no	—
196	C₂×C₄⋊C₄	—	94	1	no	—
197	C₂×C₄⋊C₄	—	188	1	no	—
198	C₂×C₂²⋊C₄	—	188	1	no	—
199	C₂×C₁₆	—	94	2	no	—
200	C₂×C₁₆	—	188	2	no	—
201	#107	—	47	2	no	—
202	#107	—	94	1	no	—
203	#103	—	94	2	no	—
204	#103	—	94	2	no	—
205	C₂⁴	#10625334	1	1	yes	characteristic
206	C₂×Q₈	—	94	2	no	—
207	C₂×Q₈	—	188	2	no	—
208	C₂×Q₈	—	188	2	no	—
209	C₂×Q₈	—	188	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	8	2	no	—
215	C₂²×C₄	—	8	2	no	—
216	C₂²×C₄	—	188	1	no	—
217	C₂²×C₄	—	188	1	no	—
218	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	#10625334	1	1	yes	characteristic
221	C₂²×C₄	#10625336	1	1	yes	characteristic
222	C₂²×C₄	#10625336	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	Q₁₆	—	188	2	no	—
225	Q₁₆	—	188	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
233	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	94	2	no	—
240	C₄⋊C₄	—	188	2	no	—
241	C₄⋊C₄	—	188	1	no	—
242	C₄⋊C₄	—	188	2	no	—
243	C₂²⋊C₄	—	188	2	no	—
244	C₂²⋊C₄	—	188	2	no	—
245	C₁₆	—	188	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	#13093944	1	1	yes	characteristic
249	C₂³	#13093944	1	1	yes	characteristic
250	C₂³	#13093968	1	2	yes	—
251	C₂³	#13093970	1	2	yes	—
252	C₂³	C₂×#10625334	1	1	yes	characteristic
253	Q₈	—	376	2	no	—
254	Q₈	—	188	2	no	—
255	Q₈	—	188	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	2	no	—
265	C₂×C₄	—	8	8	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	376	2	no	—
268	C₂×C₄	—	376	2	no	—
269	C₂×C₄	—	188	2	no	—
270	C₂×C₄	—	188	2	no	—
271	C₂×C₄	#13093938	1	2	yes	—
272	C₂×C₄	#13093944	1	1	yes	characteristic
273	C₂×C₄	#13093944	1	1	yes	characteristic
274	C₂×C₄	#13093952	1	2	yes	—
275	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	#15688856	1	2	yes	—
288	C₂²	#15688926	1	2	yes	—
289	C₂²	#15688926	1	2	yes	—
290	C₂²	#15688932	1	2	yes	—
291	C₂²	C₂×#13093944	1	1	yes	characteristic
292	C₂²	C₂×#13093968	1	1	yes	characteristic
293	C₂²	C₂×#13093970	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	8	4	no	—
297	C₄	—	376	2	no	—
298	C₄	—	376	2	no	—
299	C₄	#15688856	1	2	yes	—
300	C₄	C₂×#13093938	1	1	yes	characteristic
301	C₄	C₂×#13093952	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	2	no	—
305	C₂	#20241399	1	4	yes	direct product factor
306	C₂	C₂×#15688856	1	1	yes	characteristic
307	C₂	C₂×#15688926	1	1	yes	characteristic
308	C₂	C₂×#15688932	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093944
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#340	C₂
socle	C₂³×C₄₇	#102
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#340	D₄₇
hypocenter	C₄₇	C₂×#1348
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
3. #20241399	C₂	direct (N × H)
5. C₂²×Dic₁₈₈	C₄	semidirect
10. #15689364	C₄	semidirect
12. C₂×#13093958	C₄	semidirect
23. C₂×Dic₁₈₈	C₂×C₄	semidirect
34. #13093958	C₂×C₄	semidirect
155. C₂×C₄₇	#1348	semidirect
174. C₄₇	C₂×#1348	semidirect
305. C₂	#20241399	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×#15689364	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20241399	2
C₂×C₄₇×#340	2
C₂×#1348	47

Subgroups — C2×#20241399 #26126500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241399 #26126500