Subgroups — C₂×#20246353 #26375645

Order 12032.

2550 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246353	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689030	C₂	1	2	yes	maximal
3	C₂×#15689009	C₂	1	4	yes	maximal
4	C₂×#15688991	C₂	1	2	yes	maximal
5	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×Dic₄₇	C₂	1	2	yes	maximal
7	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#15688969	C₂	1	2	yes	maximal
9	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
13	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
14	#15689030	C₂²	1	16	yes	—
15	#15689009	C₂²	1	32	yes	—
16	#15688991	C₂²	1	16	yes	—
17	#15688989	C₂²	1	8	yes	—
18	C₄⋊C₄×Dic₄₇	C₂²	1	16	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	8	yes	—
20	#15688969	C₂²	1	16	yes	—
21	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
22	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
23	C₂×C₄₇⋊#95	C₂²	1	1	yes	characteristic
24	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
25	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
26	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
27	C₂×C₄₇⋊#95	C₂²	1	1	yes	characteristic
28	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₄₇	C₄	1	16	yes	complemented
30	C₂×C₄×Q₈×C₄₇	C₂²	1	8	yes	—
31	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
35	C₂³×Dic₄₇	C₂³	1	4	yes	—
36	C₂³×Dic₄₇	C₂³	1	2	yes	—
37	C₂³×Dic₄₇	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	—	2	8	no	—
39	C₂×C₄⋊Dic₄₇	—	2	8	no	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
42	C₂×C₄×Dic₄₇	—	2	8	no	—
43	C₂×C₄×Dic₄₇	—	2	16	no	—
44	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
46	C₄₇⋊#95	C₂³	1	4	yes	—
47	C₄₇⋊#95	C₂³	1	8	yes	—
48	C₄₇⋊#95	C₂³	1	16	yes	—
49	C₄₇⋊#95	C₂³	1	4	yes	—
50	C₄₇⋊#95	C₂³	1	16	yes	—
51	C₄₇⋊#95	C₂³	1	8	yes	—
52	C₄₇⋊#95	C₂³	1	16	yes	—
53	C₂²×Q₈×C₄₇	—	2	4	no	—
54	C₂²×Q₈×C₄₇	—	2	4	no	—
55	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
56	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
57	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
60	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
62	C₄×Q₈×C₄₇	C₂×C₄	1	64	yes	complemented
63	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
64	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
65	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
67	C₂×C₄²×C₄₇	C₂×C₄	1	16	yes	—
68	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
69	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
71	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂×Q₈×C₄₇	—	2	32	no	—
73	C₂×Q₈×C₄₇	—	2	8	no	—
74	C₂×Q₈×C₄₇	—	2	8	no	—
75	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
76	C₂²×C₄×C₄₇	—	2	4	no	—
77	C₂²×C₄×C₄₇	—	2	4	no	—
78	C₂²×C₄×C₄₇	—	2	8	no	—
79	C₂²×C₄×C₄₇	—	2	4	no	—
80	C₂²×C₄×C₄₇	—	2	4	no	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
83	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
85	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂²×C₄×C₄₇	#41	1	4	yes	—
87	C₂²×C₄×C₄₇	#41	1	4	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
90	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
91	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
92	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
93	C₂²×Dic₄₇	—	2	4	no	—
94	C₂²×Dic₄₇	—	2	4	no	—
95	C₂²×Dic₄₇	—	2	4	no	—
96	C₂²×Dic₄₇	—	2	8	no	—
97	C₂²×Dic₄₇	—	2	4	no	—
98	C₂²×Dic₄₇	—	2	8	no	—
99	C₂²×Dic₄₇	—	2	4	no	—
100	C₂²×Dic₄₇	—	2	8	no	—
101	C₂²×Dic₄₇	#41	1	8	yes	—
102	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
103	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
104	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
105	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
106	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
107	C₄²×C₄₇	C₂²×C₄	1	32	yes	—
108	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
109	C₄⋊Dic₄₇	—	2	16	no	—
110	C₄⋊Dic₄₇	—	2	16	no	—
111	C₄×Dic₄₇	—	2	16	no	—
112	C₄×Dic₄₇	—	2	32	no	—
113	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
114	C₂³×C₄₇	C₂×#41	1	4	yes	—
115	C₂³×C₄₇	C₂×#41	1	2	yes	—
116	C₂³×C₄₇	C₂×#41	1	2	yes	—
117	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
118	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
119	Q₈×C₄₇	—	2	32	no	—
120	C₂×C₄×C₄₇	—	2	16	no	—
121	C₂×C₄×C₄₇	—	2	16	no	—
122	C₂×C₄×C₄₇	—	2	8	no	—
123	C₂×C₄×C₄₇	—	2	8	no	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	2	4	no	—
128	C₂×C₄×C₄₇	—	2	4	no	—
129	C₂×C₄×C₄₇	#117	1	16	yes	—
130	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
131	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
132	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
133	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
134	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
135	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
136	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
137	C₂×Dic₄₇	—	2	16	no	—
138	C₂×Dic₄₇	—	2	4	no	—
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	2	8	no	—
141	C₂×Dic₄₇	—	2	4	no	—
142	C₂×Dic₄₇	—	2	4	no	—
143	C₂×Dic₄₇	—	2	8	no	—
144	C₂×Dic₄₇	—	2	4	no	—
145	C₂×Dic₄₇	—	2	8	no	—
146	C₂×Dic₄₇	—	4	8	no	—
147	C₂×Dic₄₇	—	4	16	no	—
148	C₂×Dic₄₇	C₂×#41	1	8	yes	—
149	C₂×#2356	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	C₂×#117	1	4	yes	—
151	C₂²×C₄₇	C₄×#41	1	4	yes	—
152	C₂²×C₄₇	#519	1	4	yes	—
153	C₂²×C₄₇	#552	1	8	yes	—
154	C₂²×C₄₇	#556	1	8	yes	—
155	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
156	C₂²×C₄₇	C₂²×#41	1	2	yes	—
157	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
158	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
160	C₄×C₄₇	—	2	8	no	—
161	C₄×C₄₇	—	2	8	no	—
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	—	2	4	no	—
164	C₄×C₄₇	C₂×#117	1	8	yes	—
165	Dic₄₇	—	2	8	no	—
166	Dic₄₇	—	4	4	no	—
167	Dic₄₇	—	4	8	no	—
168	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
169	#2356	—	47	16	no	—
170	C₂×#391	—	47	2	no	—
171	C₂×#387	—	47	4	no	—
172	C₂×#383	—	47	2	no	—
173	C₂×#382	—	47	1	no	—
174	C₂×C₄×C₄⋊C₄	—	47	1	no	—
175	C₂×C₄×C₄⋊C₄	—	47	2	no	—
176	C₂×#376	—	47	2	no	—
177	C₂×C₄₇	#2356	1	8	yes	complemented
178	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
179	C₂×C₄₇	C₂×C₄×#41	1	1	yes	characteristic
180	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
181	C₂×C₄₇	C₂×#552	1	2	yes	—
182	C₂×C₄₇	C₂×#556	1	2	yes	—
183	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
184	C₂×C₄×Q₈	Dic₄₇	1	16	yes	complemented
185	C₂×C₄×Q₈	C₂×D₄₇	1	8	yes	—
186	C₂²×C₄⋊C₄	—	47	2	no	—
187	C₂²×C₄⋊C₄	—	47	2	no	—
188	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
189	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
190	C₂²×C₄²	—	47	4	no	—
191	C₂²×C₄²	—	47	2	no	—
192	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
193	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
194	#391	—	47	16	no	—
195	#387	—	47	32	no	—
196	#383	—	47	16	no	—
197	#382	—	47	8	no	—
198	C₄×C₄⋊C₄	—	47	16	no	—
199	C₄×C₄⋊C₄	—	47	8	no	—
200	#376	—	47	16	no	—
201	C₂×#95	—	47	1	no	—
202	C₂×#95	—	47	2	no	—
203	C₂×#95	—	47	4	no	—
204	C₂×#95	—	47	1	no	—
205	C₂×#95	—	47	4	no	—
206	C₂×#95	—	47	2	no	—
207	C₂×#95	—	47	4	no	—
208	C₄₇	C₂×#2356	1	1	yes	characteristic, complemented, Sylow
209	C₂²×Q₈	—	2	4	no	—
210	C₂²×Q₈	—	2	4	no	—
211	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
212	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
213	C₂³×C₄	—	47	2	no	—
214	C₂³×C₄	—	47	4	no	—
215	C₂³×C₄	—	47	2	no	—
216	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
217	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
218	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
219	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
220	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
221	C₄×Q₈	C₂×Dic₄₇	1	64	yes	complemented
222	C₂×C₄⋊C₄	—	47	8	no	—
223	C₂×C₄⋊C₄	—	47	8	no	—
224	C₂×C₄⋊C₄	—	94	8	no	—
225	C₂×C₄⋊C₄	—	94	8	no	—
226	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
227	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
228	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
229	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
230	C₂×C₄²	—	47	16	no	—
231	C₂×C₄²	—	47	8	no	—
232	C₂×C₄²	—	94	16	no	—
233	C₂×C₄²	—	94	8	no	—
234	C₂×C₄²	C₂×Dic₄₇	1	16	yes	—
235	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
236	C₂×C₄²	C₂²×D₄₇	1	8	yes	—
237	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
238	#95	—	47	4	no	—
239	#95	—	47	8	no	—
240	#95	—	47	16	no	—
241	#95	—	47	4	no	—
242	#95	—	47	16	no	—
243	#95	—	47	8	no	—
244	#95	—	47	16	no	—
245	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
246	C₂×Q₈	—	2	32	no	—
247	C₂×Q₈	—	2	8	no	—
248	C₂×Q₈	—	2	8	no	—
249	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
250	C₂²×C₄	—	2	8	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	47	8	no	—
256	C₂²×C₄	—	47	4	no	—
257	C₂²×C₄	—	47	8	no	—
258	C₂²×C₄	—	47	4	no	—
259	C₂²×C₄	—	94	4	no	—
260	C₂²×C₄	—	94	4	no	—
261	C₂²×C₄	—	94	8	no	—
262	C₂²×C₄	—	94	4	no	—
263	C₂²×C₄	—	94	4	no	—
264	C₂²×C₄	—	94	8	no	—
265	C₂²×C₄	—	94	4	no	—
266	C₂²×C₄	—	94	8	no	—
267	C₂²×C₄	#10625350	1	4	yes	—
268	C₂²×C₄	#10625350	1	4	yes	—
269	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
270	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
271	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
272	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
273	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
274	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
275	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
276	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
277	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
278	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
279	C₄⋊C₄	—	94	16	no	—
280	C₄⋊C₄	—	94	16	no	—
281	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
282	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
283	C₄²	—	94	32	no	—
284	C₄²	—	94	16	no	—
285	C₄²	C₂²×Dic₄₇	1	32	yes	—
286	C₄²	C₂²×Dic₄₇	1	16	yes	—
287	C₂³	C₂³×Dic₄₇	1	2	yes	—
288	C₂³	#41×D₄₇	1	4	yes	—
289	C₂³	C₂×#10625350	1	2	yes	—
290	C₂³	C₂×#10625350	1	2	yes	—
291	C₂³	#13094072	1	4	yes	—
292	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
293	Q₈	—	2	32	no	—
294	C₂×C₄	—	2	16	no	—
295	C₂×C₄	—	2	16	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	47	8	no	—
304	C₂×C₄	—	94	16	no	—
305	C₂×C₄	—	94	4	no	—
306	C₂×C₄	—	94	4	no	—
307	C₂×C₄	—	94	8	no	—
308	C₂×C₄	—	94	4	no	—
309	C₂×C₄	—	94	4	no	—
310	C₂×C₄	—	94	8	no	—
311	C₂×C₄	—	94	4	no	—
312	C₂×C₄	—	94	8	no	—
313	C₂×C₄	—	188	8	no	—
314	C₂×C₄	—	188	16	no	—
315	C₂×C₄	#13093983	1	16	yes	—
316	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
317	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
318	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
319	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
320	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
321	C₂×C₄	C₂×#10625350	1	4	yes	—
322	C₂×C₄	C₂×#10625350	1	4	yes	—
323	C₂²	C₂×#13093983	1	4	yes	—
324	C₂²	#41×Dic₄₇	1	4	yes	—
325	C₂²	#15689601	1	4	yes	—
326	C₂²	#15689814	1	8	yes	—
327	C₂²	#15689845	1	8	yes	—
328	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
329	C₂²	C₂×#41×D₄₇	1	2	yes	—
330	C₂²	C₂²×#10625350	1	1	yes	characteristic
331	C₂²	C₂²×#10625350	1	1	yes	characteristic
332	C₂²	C₂×#13094072	1	2	yes	—
333	C₄	—	2	8	no	—
334	C₄	—	2	8	no	—
335	C₄	—	2	4	no	—
336	C₄	—	2	4	no	—
337	C₄	—	94	8	no	—
338	C₄	—	188	4	no	—
339	C₄	—	188	8	no	—
340	C₄	C₂×#13093983	1	8	yes	—
341	C₂	#20246353	1	8	yes	direct product factor
342	C₂	C₂²×#13093983	1	1	yes	characteristic
343	C₂	C₂×#41×Dic₄₇	1	1	yes	characteristic
344	C₂	C₂×#15689601	1	1	yes	characteristic
345	C₂	C₂×#15689814	1	2	yes	—
346	C₂	C₂×#15689845	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×#2356
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20246353	C₂	direct (N × H)
29. C₂×C₄×Q₈×C₄₇	C₄	semidirect
62. C₄×Q₈×C₄₇	C₂×C₄	semidirect
177. C₂×C₄₇	#2356	semidirect
184. C₂×C₄×Q₈	Dic₄₇	semidirect
208. C₄₇	C₂×#2356	semidirect
221. C₄×Q₈	C₂×Dic₄₇	semidirect
341. C₂	#20246353	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20246353	2
C₂×#15689030	2
C₂×#15689009	2
C₂×#15688991	2
C₂×#15688989	2
C₂×C₄⋊C₄×Dic₄₇	2
C₂×C₄×C₄⋊Dic₄₇	2
C₂×#15688969	2
C₂²×C₄×Q₈×C₄₇	2
C₂×#2356	47

Subgroups — C2×#20246353 #26375645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20246353 #26375645