Subgroups — C₂×#20246453 #26375878

Order 12032.

5062 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×Dic₄₇	C₂	1	4	yes	maximal, complemented
2	#20246453	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689026	C₂	1	4	yes	maximal, complemented
4	C₂×#15689003	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₄⋊Dic₄₇	C₂	1	2	yes	maximal, complemented
6	C₂²×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₇	C₂²	1	4	yes	—
8	C₂×D₄×Dic₄₇	—	2	32	no	—
9	C₂×D₄×Dic₄₇	C₂²	1	32	yes	complemented
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂²×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
13	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
14	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
15	#15689026	C₂²	1	32	yes	complemented
16	#15689003	C₂²	1	32	yes	complemented
17	C₄×C₄⋊Dic₄₇	C₂²	1	16	yes	complemented
18	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
19	C₂³×D₄×C₄₇	C₂²	1	4	yes	—
20	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
21	C₂×#127×C₄₇	C₄	1	16	yes	complemented
22	C₂×#127×C₄₇	C₂²	1	8	yes	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₂³×Dic₄₇	—	2	32	no	—
25	C₂³×Dic₄₇	—	4	8	no	—
26	C₂³×Dic₄₇	C₂³	1	4	yes	—
27	C₂³×Dic₄₇	C₂³	1	16	yes	—
28	C₂³×Dic₄₇	C₂³	1	4	yes	—
29	D₄×Dic₄₇	—	2	128	no	—
30	C₂×C₄⋊Dic₄₇	—	2	8	no	—
31	C₂×C₄⋊Dic₄₇	D₄	1	16	yes	complemented
32	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
34	C₂×C₂²⋊Dic₄₇	—	2	32	no	—
35	C₂×C₂²⋊Dic₄₇	—	4	8	no	—
36	C₂×C₂²⋊Dic₄₇	C₂³	1	32	yes	—
37	C₂×C₂²⋊Dic₄₇	C₂³	1	16	yes	—
38	C₂×C₄×Dic₄₇	—	2	16	no	—
39	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
40	C₄₇⋊#95	C₂³	1	16	yes	—
41	C₂⁵×C₄₇	C₂³	1	4	yes	—
42	C₂²×D₄×C₄₇	—	2	16	no	—
43	C₂²×D₄×C₄₇	—	2	16	no	—
44	C₂²×D₄×C₄₇	—	4	4	no	—
45	C₂²×D₄×C₄₇	—	4	4	no	—
46	C₂²×D₄×C₄₇	C₂×C₄	1	32	yes	complemented
47	C₂²×D₄×C₄₇	C₂×C₄	1	16	yes	complemented
48	C₂²×D₄×C₄₇	C₂³	1	16	yes	—
49	C₂²×D₄×C₄₇	C₂³	1	8	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
52	#127×C₄₇	C₂×C₄	1	64	yes	complemented
53	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	complemented
54	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
55	C₂⁴×C₄₇	—	2	16	no	—
56	C₂⁴×C₄₇	—	2	16	no	—
57	C₂⁴×C₄₇	—	2	8	no	—
58	C₂⁴×C₄₇	—	2	8	no	—
59	C₂⁴×C₄₇	C₂²×C₄	1	16	yes	—
60	C₂⁴×C₄₇	C₂⁴	1	8	yes	—
61	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂×D₄×C₄₇	—	2	128	no	—
63	C₂×D₄×C₄₇	—	2	32	no	—
64	C₂×D₄×C₄₇	—	2	32	no	—
65	C₂×D₄×C₄₇	—	4	32	no	—
66	C₂×D₄×C₄₇	—	4	8	no	—
67	C₂×D₄×C₄₇	—	4	8	no	—
68	C₂×D₄×C₄₇	C₂²×C₄	1	64	yes	complemented
69	C₂×D₄×C₄₇	C₂²×C₄	1	32	yes	complemented
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₄₇	#41	1	8	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
78	C₂²×Dic₄₇	—	2	8	no	—
79	C₂²×Dic₄₇	—	2	8	no	—
80	C₂²×Dic₄₇	—	2	8	no	—
81	C₂²×Dic₄₇	—	2	8	no	—
82	C₂²×Dic₄₇	—	2	8	no	—
83	C₂²×Dic₄₇	—	2	64	no	—
84	C₂²×Dic₄₇	—	4	16	no	—
85	C₂²×Dic₄₇	—	4	64	no	—
86	C₂²×Dic₄₇	C₂×D₄	1	16	yes	—
87	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
88	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	16	yes	complemented
90	C₂²⋊Dic₄₇	—	2	64	no	—
91	C₂²⋊Dic₄₇	—	4	16	no	—
92	C₄⋊Dic₄₇	—	2	16	no	—
93	C₄⋊Dic₄₇	C₂×D₄	1	32	yes	complemented
94	C₄×Dic₄₇	—	2	32	no	—
95	C₂³×C₄₇	—	2	64	no	—
96	C₂³×C₄₇	—	2	32	no	—
97	C₂³×C₄₇	—	2	16	no	—
98	C₂³×C₄₇	—	2	16	no	—
99	C₂³×C₄₇	—	2	8	no	—
100	C₂³×C₄₇	—	2	8	no	—
101	C₂³×C₄₇	—	4	16	no	—
102	C₂³×C₄₇	—	4	32	no	—
103	C₂³×C₄₇	—	4	16	no	—
104	C₂³×C₄₇	C₂³×C₄	1	16	yes	—
105	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
106	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
107	C₂³×C₄₇	C₂×#41	1	4	yes	—
108	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
109	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
110	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
111	D₄×C₄₇	—	2	128	no	—
112	D₄×C₄₇	—	4	32	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	C₄×D₄	1	32	yes	complemented
117	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
120	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
121	C₂×Dic₄₇	—	2	32	no	—
122	C₂×Dic₄₇	—	2	8	no	—
123	C₂×Dic₄₇	—	2	8	no	—
124	C₂×Dic₄₇	—	2	8	no	—
125	C₂×Dic₄₇	—	2	8	no	—
126	C₂×Dic₄₇	—	2	8	no	—
127	C₂×Dic₄₇	—	4	64	no	—
128	C₂×Dic₄₇	—	4	16	no	—
129	C₂×Dic₄₇	C₂²×D₄	1	16	yes	—
130	C₂×#2366	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	32	no	—
132	C₂²×C₄₇	—	2	16	no	—
133	C₂²×C₄₇	—	4	64	no	—
134	C₂²×C₄₇	—	4	8	no	—
135	C₂²×C₄₇	—	4	16	no	—
136	C₂²×C₄₇	—	4	8	no	—
137	C₂²×C₄₇	C₂×C₄×D₄	1	8	yes	—
138	C₂²×C₄₇	#518	1	4	yes	—
139	C₂²×C₄₇	D₄²	1	4	yes	—
140	C₂²×C₄₇	#550	1	8	yes	—
141	C₂²×C₄₇	#559	1	4	yes	—
142	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
143	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
144	C₂²×C₄₇	C₂²×#41	1	2	yes	—
145	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
147	C₄×C₄₇	—	2	4	no	—
148	C₄×C₄₇	C₂×C₄×D₄	1	16	yes	complemented
149	Dic₄₇	—	2	16	no	—
150	Dic₄₇	—	4	8	no	—
151	C₂²×#127	D₄₇	1	1	yes	characteristic
152	C₂²×C₄×D₄	—	47	4	no	—
153	#2366	—	47	16	no	—
154	C₂×#390	—	47	4	no	—
155	C₂×#386	—	47	4	no	—
156	C₂×C₄×C₄⋊C₄	—	47	2	no	—
157	C₂×C₄₇	—	4	16	no	—
158	C₂×C₄₇	#2366	1	8	yes	complemented
159	C₂×C₄₇	C₂²×C₄×D₄	1	2	yes	—
160	C₂×C₄₇	C₂×#518	1	1	yes	characteristic
161	C₂×C₄₇	C₂×D₄²	1	1	yes	characteristic
162	C₂×C₄₇	C₂×#550	1	2	yes	—
163	C₂×C₄₇	C₂×#559	1	1	yes	characteristic
164	C₂³×D₄	C₂×D₄₇	1	4	yes	—
165	C₂³×D₄	C₂×D₄₇	1	2	yes	—
166	C₂⁴×C₄	—	47	4	no	—
167	C₂×#127	Dic₄₇	1	16	yes	complemented
168	C₂×#127	C₂×D₄₇	1	8	yes	—
169	C₂×C₄×D₄	—	47	32	no	—
170	C₂×C₄×D₄	—	94	32	no	—
171	C₂²×C₄⋊C₄	—	47	2	no	—
172	C₂²×C₄⋊C₄	—	47	2	no	—
173	C₂²×C₂²⋊C₄	—	47	8	no	—
174	C₂²×C₂²⋊C₄	—	47	4	no	—
175	C₂²×C₄²	—	47	4	no	—
176	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
177	#390	—	47	32	no	—
178	#386	—	47	32	no	—
179	C₄×C₄⋊C₄	—	47	16	no	—
180	C₂×#95	—	47	4	no	—
181	C₄₇	C₂×#2366	1	1	yes	characteristic, complemented, Sylow
182	C₂⁵	C₂²×D₄₇	1	4	yes	—
183	C₂²×D₄	—	2	16	no	—
184	C₂²×D₄	—	2	16	no	—
185	C₂²×D₄	—	4	4	no	—
186	C₂²×D₄	—	4	4	no	—
187	C₂²×D₄	C₂×Dic₄₇	1	32	yes	complemented
188	C₂²×D₄	C₂×Dic₄₇	1	16	yes	complemented
189	C₂²×D₄	C₂²×D₄₇	1	16	yes	—
190	C₂²×D₄	C₂²×D₄₇	1	8	yes	—
191	C₂³×C₄	—	47	4	no	—
192	C₂³×C₄	—	47	4	no	—
193	C₂³×C₄	—	47	16	no	—
194	C₂³×C₄	—	94	32	no	—
195	C₂³×C₄	—	188	8	no	—
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
198	#127	C₂×Dic₄₇	1	64	yes	complemented
199	C₄×D₄	—	94	128	no	—
200	C₂×C₄⋊C₄	—	47	16	no	—
201	C₂×C₄⋊C₄	—	47	8	no	—
202	C₂×C₄⋊C₄	—	47	8	no	—
203	C₂×C₄⋊C₄	—	94	8	no	—
204	C₂×C₂²⋊C₄	—	47	32	no	—
205	C₂×C₂²⋊C₄	—	47	16	no	—
206	C₂×C₂²⋊C₄	—	94	32	no	—
207	C₂×C₂²⋊C₄	—	188	8	no	—
208	C₂×C₄²	—	47	16	no	—
209	C₂×C₄²	—	94	16	no	—
210	C₂×C₄²	C₂×Dic₄₇	1	8	yes	complemented
211	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
212	#95	—	47	16	no	—
213	C₂⁴	—	2	16	no	—
214	C₂⁴	—	2	16	no	—
215	C₂⁴	—	2	8	no	—
216	C₂⁴	—	2	8	no	—
217	C₂⁴	C₂²×Dic₄₇	1	16	yes	—
218	C₂⁴	C₂³×D₄₇	1	8	yes	—
219	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
220	C₂×D₄	—	2	128	no	—
221	C₂×D₄	—	2	32	no	—
222	C₂×D₄	—	2	32	no	—
223	C₂×D₄	—	4	32	no	—
224	C₂×D₄	—	4	8	no	—
225	C₂×D₄	—	4	8	no	—
226	C₂×D₄	C₂²×Dic₄₇	1	64	yes	complemented
227	C₂×D₄	C₂²×Dic₄₇	1	32	yes	complemented
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	47	16	no	—
231	C₂²×C₄	—	47	8	no	—
232	C₂²×C₄	—	47	8	no	—
233	C₂²×C₄	—	94	64	no	—
234	C₂²×C₄	—	94	8	no	—
235	C₂²×C₄	—	94	8	no	—
236	C₂²×C₄	—	94	8	no	—
237	C₂²×C₄	—	94	8	no	—
238	C₂²×C₄	—	94	8	no	—
239	C₂²×C₄	—	188	16	no	—
240	C₂²×C₄	—	188	64	no	—
241	C₂²×C₄	D₄×D₄₇	1	8	yes	—
242	C₂²×C₄	#10625352	1	8	yes	—
243	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
244	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
245	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
247	C₄⋊C₄	—	47	32	no	—
248	C₄⋊C₄	—	94	16	no	—
249	C₂²⋊C₄	—	94	64	no	—
250	C₂²⋊C₄	—	188	16	no	—
251	C₄²	—	94	32	no	—
252	C₄²	C₂²×Dic₄₇	1	16	yes	complemented
253	C₂³	—	2	64	no	—
254	C₂³	—	2	32	no	—
255	C₂³	—	2	16	no	—
256	C₂³	—	2	16	no	—
257	C₂³	—	2	8	no	—
258	C₂³	—	2	8	no	—
259	C₂³	—	4	16	no	—
260	C₂³	—	4	32	no	—
261	C₂³	—	4	16	no	—
262	C₂³	C₂³×Dic₄₇	1	16	yes	—
263	C₂³	C₂³×Dic₄₇	1	2	yes	—
264	C₂³	C₂×D₄×D₄₇	1	4	yes	—
265	C₂³	C₂×#10625352	1	4	yes	—
266	C₂³	#13094069	1	2	yes	—
267	C₂³	#13094069	1	2	yes	—
268	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
269	D₄	—	2	128	no	—
270	D₄	—	4	32	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	47	16	no	—
275	C₂×C₄	—	94	32	no	—
276	C₂×C₄	—	94	8	no	—
277	C₂×C₄	—	94	8	no	—
278	C₂×C₄	—	94	8	no	—
279	C₂×C₄	—	94	8	no	—
280	C₂×C₄	—	94	8	no	—
281	C₂×C₄	—	188	64	no	—
282	C₂×C₄	—	188	16	no	—
283	C₂×C₄	D₄×Dic₄₇	1	32	yes	complemented
284	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
285	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
286	C₂×C₄	C₂×D₄×D₄₇	1	8	yes	—
287	C₂×C₄	C₂×#10625352	1	8	yes	—
288	C₂²	—	2	32	no	—
289	C₂²	—	2	16	no	—
290	C₂²	—	4	8	no	—
291	C₂²	—	4	16	no	—
292	C₂²	—	4	64	no	—
293	C₂²	—	4	8	no	—
294	C₂²	C₂×D₄×Dic₄₇	1	8	yes	—
295	C₂²	#15689598	1	4	yes	—
296	C₂²	#15689758	1	4	yes	—
297	C₂²	#15689793	1	8	yes	—
298	C₂²	#15689859	1	4	yes	—
299	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
300	C₂²	C₂²×D₄×D₄₇	1	2	yes	—
301	C₂²	C₂²×#10625352	1	2	yes	—
302	C₂²	C₂×#13094069	1	1	yes	characteristic
303	C₂²	C₂×#13094069	1	1	yes	characteristic
304	C₄	—	2	4	no	—
305	C₄	—	94	16	no	—
306	C₄	—	188	8	no	—
307	C₄	C₂×D₄×Dic₄₇	1	16	yes	complemented
308	C₂	—	4	16	no	—
309	C₂	#20246453	1	8	yes	direct product factor
310	C₂	C₂²×D₄×Dic₄₇	1	2	yes	—
311	C₂	C₂×#15689598	1	1	yes	characteristic
312	C₂	C₂×#15689758	1	1	yes	characteristic
313	C₂	C₂×#15689793	1	2	yes	—
314	C₂	C₂×#15689859	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting, prime radical	C₂²×#127×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×#127	D₄₇
hypocenter	C₄₇	C₂×#2366
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×Dic₄₇	C₂	semidirect
2. #20246453	C₂	direct (N × H)
3. C₂×#15689026	C₂	semidirect
4. C₂×#15689003	C₂	semidirect
5. C₂×C₄×C₄⋊Dic₄₇	C₂	semidirect
9. C₂×D₄×Dic₄₇	C₂²	semidirect
15. #15689026	C₂²	semidirect
16. #15689003	C₂²	semidirect
17. C₄×C₄⋊Dic₄₇	C₂²	semidirect
21. C₂×#127×C₄₇	C₄	semidirect
31. C₂×C₄⋊Dic₄₇	D₄	semidirect
46. C₂²×D₄×C₄₇	C₂×C₄	semidirect
47. C₂²×D₄×C₄₇	C₂×C₄	semidirect
52. #127×C₄₇	C₂×C₄	semidirect
53. C₂×C₄²×C₄₇	C₂×C₄	semidirect
68. C₂×D₄×C₄₇	C₂²×C₄	semidirect
69. C₂×D₄×C₄₇	C₂²×C₄	semidirect
89. C₄²×C₄₇	C₂²×C₄	semidirect
93. C₄⋊Dic₄₇	C₂×D₄	semidirect
116. C₂×C₄×C₄₇	C₄×D₄	semidirect
148. C₄×C₄₇	C₂×C₄×D₄	semidirect
158. C₂×C₄₇	#2366	semidirect
167. C₂×#127	Dic₄₇	semidirect
181. C₄₇	C₂×#2366	semidirect
187. C₂²×D₄	C₂×Dic₄₇	semidirect
188. C₂²×D₄	C₂×Dic₄₇	semidirect
198. #127	C₂×Dic₄₇	semidirect
210. C₂×C₄²	C₂×Dic₄₇	semidirect
226. C₂×D₄	C₂²×Dic₄₇	semidirect
227. C₂×D₄	C₂²×Dic₄₇	semidirect
252. C₄²	C₂²×Dic₄₇	semidirect
283. C₂×C₄	D₄×Dic₄₇	semidirect
307. C₄	C₂×D₄×Dic₄₇	semidirect
309. C₂	#20246453	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×D₄×Dic₄₇	2
#20246453	2
C₂×#15689026	2
C₂×#15689003	2
C₂×C₄×C₄⋊Dic₄₇	2
C₂²×#127×C₄₇	2
C₂×#2366	47

Subgroups — C2×#20246453 #26375878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20246453 #26375878