Subgroups — C₂×#20241495 #26131194

Order 12032.

630 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
3	#20241495	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#106×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	—	2	1	no	—
6	C₂×#13093964	—	2	1	no	—
7	C₂×#13093964	—	2	2	no	—
8	C₂×#13093964	C₄	1	4	yes	complemented
9	C₂×#13093964	C₂²	1	2	yes	—
10	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
11	#15689161	—	2	1	no	—
12	#15689161	—	2	1	no	—
13	#15689161	C₄	1	2	yes	complemented
14	#15689161	C₂²	1	2	yes	—
15	C₂×#13093958	C₄	1	2	yes	complemented
16	C₂×#13093956	—	2	1	no	—
17	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
19	C₂×#106×C₄₇	—	2	1	no	—
20	C₂×#106×C₄₇	—	2	2	no	—
21	C₂×#106×C₄₇	C₄	1	4	yes	complemented
22	C₂×#106×C₄₇	C₂²	1	2	yes	—
23	#13093983	—	2	2	no	—
24	#13093983	—	2	2	no	—
25	#13093983	—	4	1	no	—
26	#13093983	—	4	1	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
29	C₂×C₄×Dic₄₇	—	4	1	no	—
30	#13093964	—	2	4	no	—
31	#13093964	—	2	2	no	—
32	#13093964	C₂×C₄	1	4	yes	complemented
33	C₂×C₄₇⋊C₁₆	—	2	4	no	—
34	C₂×C₄₇⋊C₁₆	—	2	2	no	—
35	#13093958	—	2	2	no	—
36	#13093958	C₂×C₄	1	4	yes	complemented
37	#13093956	—	2	2	no	—
38	#13093956	—	2	2	no	—
39	C₂³×C₄×C₄₇	—	4	1	no	—
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	D₄	1	2	yes	—
49	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
52	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
54	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
55	#106×C₄₇	—	2	4	no	—
56	#106×C₄₇	—	2	2	no	—
57	#106×C₄₇	C₂×C₄	1	4	yes	complemented
58	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	—	8	4	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	4	2	no	—
73	C₂×C₈×C₄₇	C₄²	1	2	yes	—
74	C₂×C₈×C₄₇	C₄²	1	2	yes	—
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	2	yes	—
78	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
91	C₄⋊C₄×C₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	—	4	4	no	—
93	C₄⋊C₄×C₄₇	—	4	2	no	—
94	C₂²⋊Dic₄₇	—	4	2	no	—
95	C₂²⋊Dic₄₇	—	4	2	no	—
96	C₄⋊Dic₄₇	—	2	4	no	—
97	C₄⋊Dic₄₇	—	4	2	no	—
98	C₄×Dic₄₇	—	4	2	no	—
99	C₄×Dic₄₇	—	4	2	no	—
100	C₄₇⋊C₁₆	—	2	4	no	—
101	C₂³×C₄₇	—	2	4	no	—
102	C₂³×C₄₇	#102	1	1	yes	characteristic
103	C₂³×C₄₇	#102	1	1	yes	characteristic
104	C₂³×C₄₇	#102	1	2	yes	—
105	C₂³×C₄₇	#107	1	2	yes	—
106	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	8	4	no	—
109	C₂×C₄×C₄₇	—	8	8	no	—
110	C₂×C₄×C₄₇	#95	1	2	yes	—
111	C₂×C₄×C₄₇	#95	1	2	yes	—
112	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#102	1	2	yes	—
115	C₂×C₄×C₄₇	#103	1	2	yes	—
116	C₂×C₄×C₄₇	#103	1	2	yes	—
117	C₂×C₄×C₄₇	#103	1	2	yes	—
118	C₂×C₄×C₄₇	#103	1	2	yes	—
119	C₂×C₄×C₄₇	#106	1	2	yes	—
120	C₂×C₄×C₄₇	#106	1	2	yes	—
121	C₂×C₄×C₄₇	#107	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	8	4	no	—
130	C₈×C₄₇	#95	1	2	yes	—
131	C₈×C₄₇	#95	1	2	yes	—
132	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
133	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×#1386	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	#340	1	2	yes	—
142	C₂²×C₄₇	#340	1	2	yes	—
143	C₂²×C₄₇	#360	1	4	yes	—
144	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
147	C₄×C₄₇	—	8	4	no	—
148	C₄×C₄₇	#340	1	2	yes	—
149	C₄×C₄₇	#340	1	2	yes	—
150	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
151	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
152	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
153	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
154	Dic₄₇	—	8	4	no	—
155	C₂²×M₃₂	—	47	1	no	—
156	C₂×#427	—	47	1	no	—
157	C₂²×#106	D₄₇	1	1	yes	characteristic
158	#1386	—	47	4	no	—
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	1	no	—
162	C₂×C₄₇	#1386	1	4	yes	complemented
163	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
164	C₂×C₄₇	C₂×#360	1	2	yes	—
165	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
166	C₂×#117	—	94	1	no	—
167	C₂²×C₄⋊C₄	—	2	1	no	—
168	C₂×M₃₂	—	47	4	no	—
169	C₂×M₃₂	—	47	2	no	—
170	C₂×M₃₂	—	94	1	no	—
171	C₂×M₃₂	—	94	2	no	—
172	C₂²×C₁₆	—	94	1	no	—
173	#427	—	47	2	no	—
174	#427	—	47	2	no	—
175	#427	—	94	1	no	—
176	#427	—	94	1	no	—
177	C₂×#107	—	47	2	no	—
178	C₂×#106	—	2	1	no	—
179	C₂×#106	—	2	2	no	—
180	C₂×#106	—	94	1	no	—
181	C₂×#106	Dic₄₇	1	4	yes	complemented
182	C₂×#106	C₂×D₄₇	1	2	yes	—
183	C₄₇	C₂×#1386	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	4	1	no	—
185	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
186	C₂²×C₈	Dic₉₄	1	2	yes	—
187	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
188	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
189	C₂²×C₈	C₄×D₄₇	1	2	yes	—
190	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
191	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
193	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	#117	—	94	2	no	—
197	#117	—	94	2	no	—
198	#117	—	188	1	no	—
199	#117	—	188	1	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	4	1	no	—
203	C₂×C₄⋊C₄	—	4	2	no	—
204	C₂×C₄⋊C₄	—	94	2	no	—
205	C₂×C₂²⋊C₄	—	188	1	no	—
206	C₂×C₄²	—	188	1	no	—
207	M₃₂	—	47	4	no	—
208	M₃₂	—	94	4	no	—
209	M₃₂	—	94	2	no	—
210	C₂×C₁₆	—	94	4	no	—
211	C₂×C₁₆	—	94	2	no	—
212	#107	—	47	4	no	—
213	#107	—	94	2	no	—
214	#106	—	2	4	no	—
215	#106	—	2	2	no	—
216	#106	—	94	2	no	—
217	#106	—	94	2	no	—
218	#106	C₂×Dic₄₇	1	4	yes	complemented
219	C₂⁴	#10625334	1	1	yes	characteristic
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	8	4	no	—
223	C₂²×C₄	—	188	2	no	—
224	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
225	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	#10625334	1	1	yes	characteristic
228	C₂²×C₄	#10625335	1	2	yes	—
229	C₂²×C₄	#10625336	1	1	yes	characteristic
230	C₂²×C₄	#10625336	1	1	yes	characteristic
231	C₂²×C₄	#10625337	1	2	yes	—
232	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
233	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
235	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
236	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
237	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
238	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
239	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
240	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
241	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
242	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
243	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
244	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
245	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
246	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
249	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
250	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	94	4	no	—
256	C₄⋊C₄	—	188	2	no	—
257	C₂²⋊C₄	—	188	2	no	—
258	C₂²⋊C₄	—	188	2	no	—
259	C₄²	—	188	2	no	—
260	C₄²	—	188	2	no	—
261	C₁₆	—	94	4	no	—
262	C₂³	—	2	4	no	—
263	C₂³	#13093944	1	1	yes	characteristic
264	C₂³	#13093944	1	1	yes	characteristic
265	C₂³	#13093945	1	2	yes	—
266	C₂³	#13093957	1	2	yes	—
267	C₂³	C₂×#10625334	1	1	yes	characteristic
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	8	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	—	376	4	no	—
272	C₂×C₄	—	188	4	no	—
273	C₂×C₄	C₄₇⋊#95	1	2	yes	—
274	C₂×C₄	C₄₇⋊#95	1	2	yes	—
275	C₂×C₄	#13093944	1	1	yes	characteristic
276	C₂×C₄	#13093944	1	1	yes	characteristic
277	C₂×C₄	#13093945	1	2	yes	—
278	C₂×C₄	#13093947	1	2	yes	—
279	C₂×C₄	#13093947	1	2	yes	—
280	C₂×C₄	#13093948	1	2	yes	—
281	C₂×C₄	#13093948	1	2	yes	—
282	C₂×C₄	#13093955	1	2	yes	—
283	C₂×C₄	#13093955	1	2	yes	—
284	C₂×C₄	#13093957	1	2	yes	—
285	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
287	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
289	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
290	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
291	C₈	C₄₇⋊#95	1	2	yes	—
292	C₈	C₄₇⋊#95	1	2	yes	—
293	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
294	C₈	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
295	C₈	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
296	C₈	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#15688885	1	2	yes	—
302	C₂²	#15688885	1	2	yes	—
303	C₂²	#15688932	1	4	yes	—
304	C₂²	C₂×#13093944	1	1	yes	characteristic
305	C₂²	C₂×#13093945	1	1	yes	characteristic
306	C₂²	C₂×#13093957	1	1	yes	characteristic
307	C₄	—	8	4	no	—
308	C₄	—	376	4	no	—
309	C₄	#15688885	1	2	yes	—
310	C₄	#15688885	1	2	yes	—
311	C₄	C₂×C₄₇⋊#95	1	1	yes	characteristic
312	C₄	C₂×#13093947	1	1	yes	characteristic
313	C₄	C₂×#13093948	1	1	yes	characteristic
314	C₄	C₂×#13093955	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	#20241495	1	4	yes	direct product factor
319	C₂	C₂×#15688885	1	1	yes	characteristic
320	C₂	C₂×#15688932	1	2	yes	—

	subgroup	G/H
center	C₂³	#13093944
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂²×#106×C₄₇	C₂
socle	C₂³×C₄₇	#102
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×#106	D₄₇
hypocenter	C₄₇	C₂×#1386
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
3. #20241495	C₂	direct (N × H)
8. C₂×#13093964	C₄	semidirect
13. #15689161	C₄	semidirect
15. C₂×#13093958	C₄	semidirect
21. C₂×#106×C₄₇	C₄	semidirect
32. #13093964	C₂×C₄	semidirect
36. #13093958	C₂×C₄	semidirect
57. #106×C₄₇	C₂×C₄	semidirect
162. C₂×C₄₇	#1386	semidirect
181. C₂×#106	Dic₄₇	semidirect
183. C₄₇	C₂×#1386	semidirect
218. #106	C₂×Dic₄₇	semidirect
318. C₂	#20241495	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×#13093964	2
C₂×#15689161	2
#20241495	2
C₂²×#106×C₄₇	2
C₂×#1386	47

Subgroups — C2×#20241495 #26131194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241495 #26131194