Subgroups — C₂×#20252256 #27080424

Order 12032.

2654 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689601	C₂	1	2	yes	maximal
2	#20252256	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689230	C₂	1	4	yes	maximal, complemented
4	C₂×Q₁₆×Dic₄₇	C₂	1	4	yes	maximal, complemented
5	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
6	C₂×#15689132	C₂	1	2	yes	maximal
7	C₂×#15689078	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	#15689601	—	2	8	no	—
11	#15689601	C₂²	1	16	yes	—
12	C₂×Q₈×Dic₄₇	—	2	4	no	—
13	C₂×Q₈×Dic₄₇	C₂²	1	8	yes	—
14	C₂×#13093983	—	2	4	no	—
15	C₂×#13093983	C₂²	1	2	yes	—
16	C₂×#13093983	C₂²	1	1	yes	characteristic
17	#15689230	C₂²	1	32	yes	complemented
18	Q₁₆×Dic₄₇	C₂²	1	32	yes	complemented
19	#15689161	C₂²	1	8	yes	—
20	C₂×#13093958	C₂²	1	2	yes	—
21	C₂×#13093956	C₂²	1	2	yes	—
22	#15689132	C₂²	1	16	yes	—
23	C₂×#13093948	C₂²	1	8	yes	—
24	#15689078	C₂²	1	8	yes	—
25	C₂×#13093934	C₂²	1	2	yes	—
26	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
27	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
28	C₂²×Q₁₆×C₄₇	C₄	1	16	yes	complemented
29	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
30	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
31	C₂²×Q₁₆×C₄₇	C₂²	1	8	yes	—
32	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#10625329	—	2	4	no	—
34	C₂×#10625329	C₂³	1	4	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	Q₈×Dic₄₇	—	2	32	no	—
37	Q₈×Dic₄₇	—	2	16	no	—
38	Q₈×Dic₄₇	C₂³	1	32	yes	—
39	#13093983	—	2	4	no	—
40	#13093983	—	2	32	no	—
41	#13093983	D₄	1	4	yes	—
42	#13093983	C₂³	1	8	yes	—
43	#13093983	C₂³	1	4	yes	—
44	C₂×C₄⋊Dic₄₇	—	2	8	no	—
45	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
47	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
48	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
49	C₂×C₄×Dic₄₇	—	2	2	no	—
50	C₂×C₄×Dic₄₇	—	2	8	no	—
51	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
52	#13093958	C₂³	1	8	yes	—
53	#13093956	C₂³	1	8	yes	—
54	#13093948	C₂³	1	32	yes	—
55	#13093934	C₂³	1	8	yes	—
56	C₈×Dic₄₇	C₂³	1	8	yes	—
57	C₂²×Q₈×C₄₇	—	2	2	no	—
58	C₂²×Q₈×C₄₇	—	2	4	no	—
59	C₂²×Q₈×C₄₇	—	2	8	no	—
60	C₂²×Q₈×C₄₇	C₂×C₄	1	16	yes	—
61	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
62	C₂²×Q₈×C₄₇	C₂³	1	8	yes	—
63	C₂³×C₄×C₄₇	—	2	2	no	—
64	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₄₇	C₂×C₄	1	16	yes	—
66	C₂×Q₁₆×C₄₇	C₂×C₄	1	16	yes	—
67	C₂×Q₁₆×C₄₇	C₂×C₄	1	64	yes	complemented
68	C₂×Q₁₆×C₄₇	C₂³	1	8	yes	—
69	C₂×Q₁₆×C₄₇	C₂³	1	8	yes	—
70	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
74	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×Q₈×C₄₇	—	2	16	no	—
76	C₂×Q₈×C₄₇	—	2	32	no	—
77	C₂×Q₈×C₄₇	—	2	8	no	—
78	C₂×Q₈×C₄₇	—	2	4	no	—
79	C₂×Q₈×C₄₇	—	2	8	no	—
80	C₂×Q₈×C₄₇	—	2	16	no	—
81	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
82	C₂×Q₈×C₄₇	C₂²×C₄	1	32	yes	—
83	C₂×Q₈×C₄₇	C₂⁴	1	8	yes	—
84	C₂²×C₄×C₄₇	—	2	16	no	—
85	C₂²×C₄×C₄₇	—	2	4	no	—
86	C₂²×C₄×C₄₇	—	2	8	no	—
87	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	#41	1	4	yes	—
91	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
93	C₂²×Dic₄₇	—	2	4	no	—
94	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
95	Q₁₆×C₄₇	C₂²×C₄	1	32	yes	—
96	Q₁₆×C₄₇	C₂²×C₄	1	32	yes	—
97	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₄₇	C₂²×C₄	1	16	yes	—
100	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
101	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
102	C₂²⋊Dic₄₇	—	2	8	no	—
103	C₂²⋊Dic₄₇	C₂×D₄	1	8	yes	—
104	C₄⋊Dic₄₇	—	2	4	no	—
105	C₄⋊Dic₄₇	—	2	32	no	—
106	C₄⋊Dic₄₇	C₂×D₄	1	8	yes	—
107	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
108	C₄×Dic₄₇	—	2	8	no	—
109	C₄×Dic₄₇	—	2	32	no	—
110	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
111	C₄×Dic₄₇	C₂⁴	1	4	yes	—
112	#10625329	—	2	8	no	—
113	C₂×C₄₇⋊C₈	—	2	4	no	—
114	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
115	C₂³×C₄₇	—	2	4	no	—
116	C₂³×C₄₇	C₄×D₄	1	4	yes	—
117	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
118	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	Q₈×C₄₇	—	2	16	no	—
120	Q₈×C₄₇	—	2	32	no	—
121	Q₈×C₄₇	—	2	8	no	—
122	Q₈×C₄₇	C₂³×C₄	1	16	yes	—
123	C₂×C₄×C₄₇	—	2	32	no	—
124	C₂×C₄×C₄₇	—	2	16	no	—
125	C₂×C₄×C₄₇	—	2	8	no	—
126	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
127	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
128	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
129	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
134	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
135	C₂×Dic₄₇	—	2	8	no	—
136	C₂×Dic₄₇	—	2	8	no	—
137	C₂×Dic₄₇	—	4	8	no	—
138	C₂×Dic₄₇	C₂²×D₄	1	8	yes	—
139	C₈×C₄₇	C₂³×C₄	1	4	yes	—
140	C₈×C₄₇	C₂³×C₄	1	4	yes	—
141	C₄₇⋊C₈	—	2	4	no	—
142	C₂×#2950	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	8	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	C₂×C₄×D₄	1	4	yes	—
146	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
147	C₂²×C₄₇	#578	1	4	yes	—
148	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
149	C₄×C₄₇	—	2	16	no	—
150	C₄×C₄₇	C₂×C₄×D₄	1	4	yes	—
151	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
152	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
153	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
154	Dic₄₇	—	2	8	no	—
155	Dic₄₇	—	4	8	no	—
156	C₂³×Q₁₆	D₄₇	1	1	yes	characteristic
157	C₂×#519	—	47	2	no	—
158	#2950	—	47	16	no	—
159	C₂×#441	—	47	4	no	—
160	C₂×C₄×Q₁₆	—	47	4	no	—
161	C₂×#427	—	47	1	no	—
162	C₂×#419	—	47	2	no	—
163	C₂×#405	—	47	1	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	#2950	1	4	yes	complemented
166	C₂×C₄₇	C₂²×C₄×D₄	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#578	1	2	yes	—
168	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
169	C₂²×Q₁₆	Dic₄₇	1	16	yes	complemented
170	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
171	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
172	C₂²×Q₁₆	C₂×D₄₇	1	8	yes	—
173	C₂²×M₁₆	—	47	1	no	—
174	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
175	#519	—	47	16	no	—
176	#519	—	94	8	no	—
177	C₂×C₄×Q₈	—	47	8	no	—
178	C₂×C₄×Q₈	—	94	4	no	—
179	C₂×#117	—	47	1	no	—
180	C₂×#117	—	47	2	no	—
181	C₂×#117	—	94	4	no	—
182	#441	—	47	32	no	—
183	C₄×Q₁₆	—	47	32	no	—
184	#427	—	47	8	no	—
185	C₂×#107	—	47	2	no	—
186	C₂×#106	—	47	2	no	—
187	#419	—	47	16	no	—
188	C₂×#103	—	47	8	no	—
189	#405	—	47	8	no	—
190	C₂×#97	—	47	2	no	—
191	C₂×C₄×C₈	—	47	2	no	—
192	C₄₇	C₂×#2950	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₈	—	2	2	no	—
194	C₂²×Q₈	—	2	4	no	—
195	C₂²×Q₈	—	2	8	no	—
196	C₂²×Q₈	C₂×Dic₄₇	1	16	yes	—
197	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
198	C₂²×Q₈	C₂²×D₄₇	1	8	yes	—
199	C₂³×C₄	—	2	2	no	—
200	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×Q₁₆	C₂×Dic₄₇	1	16	yes	—
202	C₂×Q₁₆	C₂×Dic₄₇	1	16	yes	—
203	C₂×Q₁₆	C₂×Dic₄₇	1	64	yes	complemented
204	C₂×Q₁₆	C₂²×D₄₇	1	8	yes	—
205	C₂×Q₁₆	C₂²×D₄₇	1	8	yes	—
206	C₂×M₁₆	—	47	4	no	—
207	C₂×M₁₆	—	94	4	no	—
208	C₂²×C₈	—	47	2	no	—
209	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
210	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
213	C₄×Q₈	—	47	32	no	—
214	C₄×Q₈	—	94	16	no	—
215	C₄×Q₈	—	94	32	no	—
216	#117	—	47	4	no	—
217	#117	—	47	4	no	—
218	#117	—	47	8	no	—
219	#117	—	94	32	no	—
220	#117	—	94	4	no	—
221	C₂×C₄⋊C₄	—	47	2	no	—
222	C₂×C₄⋊C₄	—	47	4	no	—
223	C₂×C₄⋊C₄	—	94	8	no	—
224	C₂×C₂²⋊C₄	—	47	2	no	—
225	C₂×C₂²⋊C₄	—	94	2	no	—
226	C₂×C₄²	—	47	2	no	—
227	C₂×C₄²	—	94	8	no	—
228	C₂×C₄²	—	94	2	no	—
229	#107	—	47	8	no	—
230	#106	—	47	8	no	—
231	#103	—	47	32	no	—
232	#97	—	47	8	no	—
233	C₄×C₈	—	47	8	no	—
234	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
235	C₂×Q₈	—	2	16	no	—
236	C₂×Q₈	—	2	32	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	2	8	no	—
239	C₂×Q₈	—	2	8	no	—
240	C₂×Q₈	—	2	16	no	—
241	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
242	C₂×Q₈	C₂²×Dic₄₇	1	32	yes	—
243	C₂×Q₈	C₂³×D₄₇	1	8	yes	—
244	C₂²×C₄	—	2	16	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	47	4	no	—
248	C₂²×C₄	—	94	4	no	—
249	C₂²×C₄	D₄×D₄₇	1	2	yes	—
250	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
251	C₂²×C₄	#10625352	1	4	yes	—
252	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
253	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
254	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
255	Q₁₆	C₂²×Dic₄₇	1	32	yes	—
256	Q₁₆	C₂²×Dic₄₇	1	32	yes	—
257	M₁₆	—	94	8	no	—
258	C₂×C₈	—	47	4	no	—
259	C₂×C₈	—	94	4	no	—
260	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
261	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
262	C₂×C₈	C₂²×Dic₄₇	1	16	yes	—
263	C₂×C₈	C₂³×D₄₇	1	2	yes	—
264	C₂×C₈	C₂³×D₄₇	1	2	yes	—
265	C₄⋊C₄	—	47	8	no	—
266	C₄⋊C₄	—	47	8	no	—
267	C₄⋊C₄	—	94	32	no	—
268	C₄⋊C₄	—	94	4	no	—
269	C₂²⋊C₄	—	47	8	no	—
270	C₂²⋊C₄	—	94	8	no	—
271	C₄²	—	47	4	no	—
272	C₄²	—	47	4	no	—
273	C₄²	—	94	32	no	—
274	C₄²	—	94	8	no	—
275	C₂³	—	2	4	no	—
276	C₂³	D₄×Dic₄₇	1	4	yes	—
277	C₂³	C₂×D₄×D₄₇	1	2	yes	—
278	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
279	Q₈	—	2	16	no	—
280	Q₈	—	2	32	no	—
281	Q₈	—	2	8	no	—
282	Q₈	C₂³×Dic₄₇	1	16	yes	—
283	C₂×C₄	—	2	32	no	—
284	C₂×C₄	—	2	16	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	47	8	no	—
287	C₂×C₄	—	94	8	no	—
288	C₂×C₄	—	94	8	no	—
289	C₂×C₄	—	188	8	no	—
290	C₂×C₄	D₄×Dic₄₇	1	8	yes	—
291	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
292	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
293	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
294	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
296	C₂×C₄	C₂×#10625352	1	2	yes	—
297	C₂×C₄	C₂×#10625352	1	4	yes	—
298	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
299	C₈	—	94	4	no	—
300	C₈	C₂³×Dic₄₇	1	4	yes	—
301	C₈	C₂³×Dic₄₇	1	4	yes	—
302	C₂²	—	2	8	no	—
303	C₂²	—	2	4	no	—
304	C₂²	C₂×D₄×Dic₄₇	1	4	yes	—
305	C₂²	C₂×D₄×Dic₄₇	1	2	yes	—
306	C₂²	#15689936	1	4	yes	—
307	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
308	C₄	—	2	16	no	—
309	C₄	—	94	8	no	—
310	C₄	—	188	8	no	—
311	C₄	C₂×D₄×Dic₄₇	1	2	yes	—
312	C₄	C₂×D₄×Dic₄₇	1	4	yes	—
313	C₄	C₂⁴×Dic₄₇	1	1	yes	characteristic
314	C₄	C₂²×#10625352	1	1	yes	characteristic
315	C₂	—	2	4	no	—
316	C₂	#20252256	1	4	yes	direct product factor
317	C₂	C₂²×D₄×Dic₄₇	1	1	yes	characteristic
318	C₂	C₂×#15689936	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂³×Q₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂³×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#2950
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20252256	C₂	direct (N × H)
3. C₂×#15689230	C₂	semidirect
4. C₂×Q₁₆×Dic₄₇	C₂	semidirect
17. #15689230	C₂²	semidirect
18. Q₁₆×Dic₄₇	C₂²	semidirect
28. C₂²×Q₁₆×C₄₇	C₄	semidirect
67. C₂×Q₁₆×C₄₇	C₂×C₄	semidirect
165. C₂×C₄₇	#2950	semidirect
169. C₂²×Q₁₆	Dic₄₇	semidirect
192. C₄₇	C₂×#2950	semidirect
203. C₂×Q₁₆	C₂×Dic₄₇	semidirect
316. C₂	#20252256	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×#15689601	2
#20252256	2
C₂×#15689230	2
C₂×Q₁₆×Dic₄₇	2
C₂×#15689161	2
C₂×#15689132	2
C₂×#15689078	2
C₂³×Q₁₆×C₄₇	2
C₂×#2950	47

Subgroups — C2×#20252256 #27080424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20252256 #27080424