Subgroups — C₂×#20256793 #27123122

Order 12032.

7022 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256793	C₂	1	32	yes	maximal, direct product factor
2	C₂×#15689754	C₂	1	4	yes	maximal, complemented
3	C₂×#15689751	C₂	1	8	yes	maximal, complemented
4	C₂×#15689749	C₂	1	4	yes	maximal
5	C₂×#15689680	C₂	1	4	yes	maximal, complemented
6	C₂×#15689676	C₂	1	8	yes	maximal, complemented
7	C₂²×#13093983	C₂	1	2	yes	maximal
8	C₂²×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#15689754	C₂²	1	64	yes	complemented
10	#15689751	C₂²	1	128	yes	complemented
11	#15689749	C₂²	1	64	yes	—
12	#15689680	C₂²	1	64	yes	complemented
13	#15689676	C₂²	1	128	yes	complemented
14	C₂×#13094024	C₂²	1	8	yes	complemented
15	C₂×#13094023	C₂²	1	16	yes	complemented
16	C₂×#13094016	C₂²	1	8	yes	complemented
17	C₂×#13094015	C₂²	1	16	yes	complemented
18	C₂×#13094003	C₂²	1	16	yes	—
19	C₂×#13094001	C₂²	1	32	yes	—
20	C₂×C₄×Dic₉₄	C₂²	1	16	yes	complemented
21	C₂×#13093983	C₂²	1	4	yes	—
22	C₂×#13093983	C₂²	1	4	yes	—
23	C₂×#13093983	C₂²	1	32	yes	—
24	C₂×#13093983	C₂²	1	4	yes	—
25	C₂×#13093983	C₂²	1	4	yes	—
26	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
27	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
28	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
29	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#117×C₄₇	C₂²	1	4	yes	—
31	C₂×#117×C₄₇	C₂²	1	16	yes	—
32	C₂×#117×C₄₇	C₂²	1	4	yes	—
33	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
35	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
36	C₂²×Dic₉₄	C₂³	1	16	yes	—
37	C₂³×Dic₄₇	C₂³	1	8	yes	—
38	#13094024	C₂³	1	64	yes	complemented
39	#13094023	C₂³	1	128	yes	complemented
40	#13094016	C₂³	1	64	yes	complemented
41	#13094015	C₂³	1	128	yes	complemented
42	#13094003	C₂³	1	128	yes	—
43	#13094001	C₂³	1	256	yes	—
44	C₄×Dic₉₄	C₂³	1	128	yes	complemented
45	#13093983	C₂³	1	32	yes	—
46	#13093983	C₂³	1	32	yes	—
47	#13093983	C₂³	1	32	yes	—
48	#13093983	C₂³	1	32	yes	—
49	C₂×C₄⋊Dic₄₇	C₂³	1	32	yes	—
50	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
51	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
52	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
53	C₂×Dic₄₇⋊C₄	C₂³	1	32	yes	—
54	C₂×C₂²⋊Dic₄₇	C₂³	1	32	yes	—
55	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
56	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
57	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
58	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
59	C₂×C₄×Dic₄₇	C₂³	1	32	yes	—
60	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₄₇	Q₈	1	16	yes	complemented
62	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
63	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
66	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₄₇	C₂³	1	8	yes	—
68	#117×C₄₇	C₂³	1	32	yes	—
69	#117×C₄₇	C₂³	1	32	yes	—
70	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
71	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
72	C₂×C₂²⋊C₄×C₄₇	C₂³	1	16	yes	—
73	C₂×C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
74	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
75	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
76	C₂×C₄²×C₄₇	C₂³	1	16	yes	—
77	C₂⁴×C₄₇	—	4	4	no	—
78	C₂⁴×C₄₇	C₂×Q₈	1	8	yes	—
79	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
80	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂⁴×C₄₇	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₄₇	—	4	8	no	—
83	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
84	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
85	C₂²×C₄×C₄₇	C₂×Q₈	1	16	yes	—
86	C₂²×C₄×C₄₇	C₂×Q₈	1	16	yes	—
87	C₂²×C₄×C₄₇	C₂×Q₈	1	64	yes	complemented
88	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
89	C₂²×C₄×C₄₇	C₂⁴	1	16	yes	—
90	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
92	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
93	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
94	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
95	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
96	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₂²×Dic₄₇	—	4	16	no	—
98	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
99	C₂²×Dic₄₇	C₂⁴	1	32	yes	—
100	C₂×Dic₉₄	—	4	32	no	—
101	C₂×Dic₉₄	C₂⁴	1	64	yes	—
102	C₄⋊C₄×C₄₇	C₂⁴	1	32	yes	—
103	C₂²⋊C₄×C₄₇	C₂⁴	1	32	yes	—
104	C₄²×C₄₇	C₂⁴	1	16	yes	—
105	C₄²×C₄₇	C₂⁴	1	16	yes	—
106	C₂²⋊Dic₄₇	C₂⁴	1	32	yes	—
107	C₂²⋊Dic₄₇	C₂⁴	1	32	yes	—
108	C₄⋊Dic₄₇	C₂⁴	1	32	yes	—
109	C₄⋊Dic₄₇	C₂⁴	1	32	yes	—
110	C₄⋊Dic₄₇	C₂⁴	1	64	yes	—
111	Dic₄₇⋊C₄	C₂⁴	1	128	yes	—
112	C₄×Dic₄₇	C₂⁴	1	32	yes	—
113	C₄×Dic₄₇	C₂⁴	1	32	yes	—
114	C₂³×C₄₇	—	2	8	no	—
115	C₂³×C₄₇	—	2	4	no	—
116	C₂³×C₄₇	—	4	16	no	—
117	C₂³×C₄₇	—	4	8	no	—
118	C₂³×C₄₇	C₂²×Q₈	1	8	yes	—
119	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
120	C₂³×C₄₇	C₂²×Q₈	1	16	yes	—
121	C₂³×C₄₇	C₂⁵	1	2	yes	—
122	C₂³×C₄₇	C₂⁵	1	4	yes	—
123	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
124	C₂×C₄×C₄₇	—	2	16	no	—
125	C₂×C₄×C₄₇	—	4	16	no	—
126	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
127	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
128	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
129	C₂×C₄×C₄₇	C₂²×Q₈	1	16	yes	—
130	C₂×C₄×C₄₇	C₂²×Q₈	1	16	yes	—
131	C₂×C₄×C₄₇	C₂²×Q₈	1	32	yes	—
132	C₂×C₄×C₄₇	C₂²×Q₈	1	32	yes	—
133	C₂×C₄×C₄₇	C₂⁵	1	16	yes	—
134	C₂×C₄×C₄₇	C₂⁵	1	2	yes	—
135	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
136	C₂×C₄×C₄₇	C₂⁵	1	2	yes	—
137	C₂×Dic₄₇	—	2	32	no	—
138	C₂×Dic₄₇	—	4	32	no	—
139	C₂×Dic₄₇	C₂⁵	1	32	yes	—
140	Dic₉₄	—	4	64	no	—
141	C₂×#3458	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	8	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	16	no	—
145	C₂²×C₄₇	—	2	8	no	—
146	C₂²×C₄₇	—	4	16	no	—
147	C₂²×C₄₇	C₂³×Q₈	1	4	yes	—
148	C₂²×C₄₇	C₂³×Q₈	1	8	yes	—
149	C₂²×C₄₇	C₂³×Q₈	1	2	yes	—
150	C₂²×C₄₇	#585	1	4	yes	—
151	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
152	C₄×C₄₇	—	2	16	no	—
153	C₄×C₄₇	C₂³×Q₈	1	4	yes	—
154	C₄×C₄₇	C₂³×Q₈	1	8	yes	—
155	C₄×C₄₇	C₂³×Q₈	1	4	yes	—
156	Dic₄₇	—	2	32	no	—
157	#3458	—	47	32	no	—
158	C₂×#544	—	47	8	no	—
159	C₂×#544	—	47	4	no	—
160	C₂×#543	—	47	4	no	—
161	C₂×#533	—	47	4	no	—
162	C₂×#533	—	47	8	no	—
163	C₂²×#117	—	47	2	no	—
164	C₂²×#117	D₄₇	1	1	yes	characteristic
165	C₂×C₄₇	—	2	8	no	—
166	C₂×C₄₇	—	2	4	no	—
167	C₂×C₄₇	#3458	1	4	yes	complemented
168	C₂×C₄₇	C₂⁴×Q₈	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#585	1	2	yes	—
170	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
171	#544	—	47	128	no	—
172	#544	—	47	64	no	—
173	#543	—	47	64	no	—
174	#533	—	47	128	no	—
175	#533	—	47	64	no	—
176	C₂×C₄⋊Q₈	—	47	16	no	—
177	C₂×C₄⋊Q₈	—	47	8	no	—
178	C₂×#125	—	47	16	no	—
179	C₂×#125	—	47	8	no	—
180	C₂×C₂²⋊Q₈	—	47	16	no	—
181	C₂×C₂²⋊Q₈	—	47	32	no	—
182	C₂×C₄×Q₈	—	47	16	no	—
183	C₂×#117	—	47	4	no	—
184	C₂×#117	—	47	4	no	—
185	C₂×#117	—	47	4	no	—
186	C₂×#117	—	47	4	no	—
187	C₂×#117	—	47	32	no	—
188	C₂×#117	C₂×D₄₇	1	4	yes	—
189	C₂×#117	C₂×D₄₇	1	4	yes	—
190	C₂×#117	C₂×D₄₇	1	16	yes	—
191	C₂²×C₄⋊C₄	—	47	4	no	—
192	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
193	C₂²×C₂²⋊C₄	—	47	4	no	—
194	C₂²×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
195	C₂²×C₄²	—	47	4	no	—
196	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
197	C₄₇	C₂×#3458	1	1	yes	characteristic, complemented, Sylow
198	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×Q₈	—	47	16	no	—
200	C₂³×C₄	—	47	8	no	—
201	C₂³×C₄	Dic₉₄	1	16	yes	complemented
202	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
203	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
204	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
205	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
206	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
207	C₂³×C₄	C₂²×D₄₇	1	8	yes	—
208	C₄⋊Q₈	—	47	128	no	—
209	C₄⋊Q₈	—	47	64	no	—
210	#125	—	47	128	no	—
211	#125	—	47	64	no	—
212	C₂²⋊Q₈	—	47	128	no	—
213	C₂²⋊Q₈	—	47	256	no	—
214	C₄×Q₈	—	47	128	no	—
215	#117	—	47	32	no	—
216	#117	—	47	32	no	—
217	#117	—	47	32	no	—
218	#117	—	47	32	no	—
219	#117	C₂²×D₄₇	1	32	yes	—
220	#117	C₂²×D₄₇	1	32	yes	—
221	C₂×C₄⋊C₄	—	47	8	no	—
222	C₂×C₄⋊C₄	—	47	8	no	—
223	C₂×C₄⋊C₄	—	47	32	no	—
224	C₂×C₄⋊C₄	—	47	32	no	—
225	C₂×C₄⋊C₄	—	47	16	no	—
226	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
227	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
228	C₂×C₂²⋊C₄	—	47	8	no	—
229	C₂×C₂²⋊C₄	—	47	8	no	—
230	C₂×C₂²⋊C₄	—	47	32	no	—
231	C₂×C₂²⋊C₄	C₂²×D₄₇	1	8	yes	—
232	C₂×C₂²⋊C₄	C₂²×D₄₇	1	16	yes	—
233	C₂×C₄²	—	47	8	no	—
234	C₂×C₄²	—	47	32	no	—
235	C₂×C₄²	—	47	8	no	—
236	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
237	C₂×C₄²	C₂²×D₄₇	1	16	yes	—
238	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
239	C₂⁴	—	4	4	no	—
240	C₂⁴	C₂×Dic₉₄	1	8	yes	—
241	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
242	C₂⁴	C₂³×D₄₇	1	2	yes	—
243	C₂⁴	C₂³×D₄₇	1	4	yes	—
244	C₂×Q₈	—	47	64	no	—
245	C₂×Q₈	—	188	32	no	—
246	C₂²×C₄	—	4	8	no	—
247	C₂²×C₄	—	47	16	no	—
248	C₂²×C₄	—	47	32	no	—
249	C₂²×C₄	—	188	16	no	—
250	C₂²×C₄	C₂×Dic₉₄	1	16	yes	—
251	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
252	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
253	C₂²×C₄	C₂×Dic₉₄	1	16	yes	—
254	C₂²×C₄	C₂×Dic₉₄	1	64	yes	complemented
255	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
256	C₂²×C₄	C₂³×D₄₇	1	16	yes	—
257	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
258	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
259	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
260	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
261	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
262	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
263	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
264	C₄⋊C₄	—	47	128	no	—
265	C₄⋊C₄	—	47	32	no	—
266	C₄⋊C₄	—	47	32	no	—
267	C₄⋊C₄	—	47	64	no	—
268	C₄⋊C₄	C₂³×D₄₇	1	32	yes	—
269	C₂²⋊C₄	—	47	32	no	—
270	C₂²⋊C₄	—	47	32	no	—
271	C₂²⋊C₄	C₂³×D₄₇	1	32	yes	—
272	C₄²	—	47	32	no	—
273	C₄²	—	47	32	no	—
274	C₄²	C₂³×D₄₇	1	16	yes	—
275	C₄²	C₂³×D₄₇	1	16	yes	—
276	C₂³	—	2	4	no	—
277	C₂³	—	2	8	no	—
278	C₂³	—	4	16	no	—
279	C₂³	—	4	8	no	—
280	C₂³	C₂²×Dic₉₄	1	4	yes	—
281	C₂³	C₂²×Dic₉₄	1	8	yes	—
282	C₂³	C₂²×Dic₉₄	1	16	yes	—
283	C₂³	C₂⁴×D₄₇	1	2	yes	—
284	C₂³	C₂⁴×D₄₇	1	4	yes	—
285	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
286	Q₈	—	188	64	no	—
287	C₂×C₄	—	2	16	no	—
288	C₂×C₄	—	4	16	no	—
289	C₂×C₄	—	47	32	no	—
290	C₂×C₄	—	94	32	no	—
291	C₂×C₄	—	188	32	no	—
292	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
293	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
294	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
295	C₂×C₄	C₂²×Dic₉₄	1	32	yes	—
296	C₂×C₄	C₂²×Dic₉₄	1	16	yes	—
297	C₂×C₄	C₂²×Dic₉₄	1	32	yes	—
298	C₂×C₄	C₂²×Dic₉₄	1	16	yes	—
299	C₂×C₄	C₂⁴×D₄₇	1	16	yes	—
300	C₂×C₄	C₂⁴×D₄₇	1	2	yes	—
301	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
302	C₂×C₄	C₂⁴×D₄₇	1	2	yes	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	16	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	4	16	no	—
308	C₂²	C₂³×Dic₉₄	1	4	yes	—
309	C₂²	C₂³×Dic₉₄	1	8	yes	—
310	C₂²	C₂³×Dic₉₄	1	2	yes	—
311	C₂²	#15689959	1	4	yes	—
312	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
313	C₄	—	2	16	no	—
314	C₄	—	94	32	no	—
315	C₄	C₂³×Dic₉₄	1	8	yes	—
316	C₄	C₂³×Dic₉₄	1	4	yes	—
317	C₄	C₂³×Dic₉₄	1	4	yes	—
318	C₂	—	2	4	no	—
319	C₂	—	2	8	no	—
320	C₂	#20256793	1	4	yes	direct product factor
321	C₂	C₂⁴×Dic₉₄	1	1	yes	characteristic
322	C₂	C₂×#15689959	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂²×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁵×C₄₇	C₂³
hypercenter	C₂²×#117	D₄₇
hypocenter	C₄₇	C₂×#3458
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. #20256793	C₂	direct (N × H)
2. C₂×#15689754	C₂	semidirect
3. C₂×#15689751	C₂	semidirect
5. C₂×#15689680	C₂	semidirect
6. C₂×#15689676	C₂	semidirect
9. #15689754	C₂²	semidirect
10. #15689751	C₂²	semidirect
12. #15689680	C₂²	semidirect
13. #15689676	C₂²	semidirect
14. C₂×#13094024	C₂²	semidirect
15. C₂×#13094023	C₂²	semidirect
16. C₂×#13094016	C₂²	semidirect
17. C₂×#13094015	C₂²	semidirect
20. C₂×C₄×Dic₉₄	C₂²	semidirect
38. #13094024	C₂³	semidirect
39. #13094023	C₂³	semidirect
40. #13094016	C₂³	semidirect
41. #13094015	C₂³	semidirect
44. C₄×Dic₉₄	C₂³	semidirect
61. C₂³×C₄×C₄₇	Q₈	semidirect
87. C₂²×C₄×C₄₇	C₂×Q₈	semidirect
167. C₂×C₄₇	#3458	semidirect
197. C₄₇	C₂×#3458	semidirect
201. C₂³×C₄	Dic₉₄	semidirect
254. C₂²×C₄	C₂×Dic₉₄	semidirect
320. C₂	#20256793	direct (N × H)

Maximal subgroups

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

subgroup	index
#20256793	2
C₂×#15689754	2
C₂×#15689751	2
C₂×#15689749	2
C₂×#15689680	2
C₂×#15689676	2
C₂²×#13093983	2
C₂²×#117×C₄₇	2
C₂×#3458	47

Subgroups — C2×#20256793 #27123122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20256793 #27123122