Subgroups — C₂×#20244195 #26219904

Order 12032.

2414 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093945	C₂	1	4	yes	maximal, complemented
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688993	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244195	C₂	1	8	yes	maximal, direct product factor
5	C₂²×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂×#13093945	—	2	16	no	—
10	C₂×#13093945	C₂²	1	16	yes	complemented
11	C₂×C₄×C₄₇⋊C₈	—	2	4	no	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	complemented
13	#15688993	—	2	2	no	—
14	#15688993	C₂²	1	4	yes	complemented
15	C₂³×D₄×C₄₇	—	2	2	no	—
16	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
17	C₂×#127×C₄₇	—	2	2	no	—
18	C₂×#127×C₄₇	—	2	4	no	—
19	C₂×#127×C₄₇	C₄	1	8	yes	complemented
20	C₂×#127×C₄₇	C₂²	1	4	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂³×Dic₄₇	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	D₄	1	8	yes	complemented
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₂×C₄⋊Dic₄₇	—	2	8	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	8	no	—
30	C₂×C₄⋊Dic₄₇	—	4	8	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	#13093945	—	2	32	no	—
33	C₄×C₄₇⋊C₈	—	2	8	no	—
34	C₂⁵×C₄₇	—	2	2	no	—
35	C₂²×D₄×C₄₇	—	2	16	no	—
36	C₂²×D₄×C₄₇	—	2	8	no	—
37	C₂²×D₄×C₄₇	—	2	2	no	—
38	C₂²×D₄×C₄₇	—	2	4	no	—
39	C₂²×D₄×C₄₇	—	2	8	no	—
40	C₂²×D₄×C₄₇	—	2	8	no	—
41	C₂²×D₄×C₄₇	—	4	8	no	—
42	C₂²×D₄×C₄₇	—	4	8	no	—
43	C₂²×D₄×C₄₇	C₂×C₄	1	8	yes	—
44	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
45	C₂³×C₄×C₄₇	D₄	1	2	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	#127×C₄₇	—	2	16	no	—
48	#127×C₄₇	—	2	8	no	—
49	#127×C₄₇	C₂×C₄	1	16	yes	complemented
50	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₇	D₄	1	2	yes	—
52	C₂×C₄²×C₄₇	D₄	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
54	C₂⁴×C₄₇	—	2	8	no	—
55	C₂⁴×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	—	4	4	no	—
57	C₂⁴×C₄₇	—	4	4	no	—
58	C₂⁴×C₄₇	—	4	8	no	—
59	C₂⁴×C₄₇	—	4	8	no	—
60	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄×C₄₇	—	2	8	no	—
62	C₂×D₄×C₄₇	—	2	32	no	—
63	C₂×D₄×C₄₇	—	2	32	no	—
64	C₂×D₄×C₄₇	—	2	8	no	—
65	C₂×D₄×C₄₇	—	2	4	no	—
66	C₂×D₄×C₄₇	—	2	8	no	—
67	C₂×D₄×C₄₇	—	4	64	no	—
68	C₂×D₄×C₄₇	—	4	16	no	—
69	C₂×D₄×C₄₇	—	4	4	no	—
70	C₂×D₄×C₄₇	—	4	4	no	—
71	C₂×D₄×C₄₇	—	4	16	no	—
72	C₂×D₄×C₄₇	—	4	16	no	—
73	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
75	C₂²×C₄×C₄₇	D₈	1	8	yes	—
76	C₂²×C₄×C₄₇	SD₁₆	1	8	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₄₇	#41	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₄₇	—	2	4	no	—
85	C₂²×Dic₄₇	—	4	4	no	—
86	C₂²×Dic₄₇	—	4	8	no	—
87	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₄₇	C₂×D₄	1	4	yes	—
90	C₄⋊Dic₄₇	—	2	8	no	—
91	C₄⋊Dic₄₇	—	4	4	no	—
92	C₄⋊Dic₄₇	—	4	16	no	—
93	C₂×C₄₇⋊C₈	—	2	16	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	complemented
97	C₂³×C₄₇	—	2	8	no	—
98	C₂³×C₄₇	—	4	4	no	—
99	C₂³×C₄₇	—	4	4	no	—
100	C₂³×C₄₇	—	4	8	no	—
101	C₂³×C₄₇	—	4	8	no	—
102	C₂³×C₄₇	—	4	16	no	—
103	C₂³×C₄₇	—	4	32	no	—
104	C₂³×C₄₇	—	8	16	no	—
105	C₂³×C₄₇	—	8	8	no	—
106	C₂³×C₄₇	—	8	8	no	—
107	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂³×C₄₇	#124	1	2	yes	—
109	C₂³×C₄₇	#127	1	2	yes	—
110	C₂³×C₄₇	C₂×D₈	1	4	yes	—
111	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	D₄×C₄₇	—	2	16	no	—
114	D₄×C₄₇	—	4	8	no	—
115	D₄×C₄₇	—	4	16	no	—
116	D₄×C₄₇	—	4	64	no	—
117	C₂×C₄×C₄₇	#102	1	32	yes	complemented
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
120	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
121	C₂×C₄×C₄₇	#121	1	8	yes	—
122	C₂×C₄×C₄₇	#124	1	2	yes	—
123	C₂×C₄×C₄₇	#127	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×D₈	1	8	yes	—
125	C₂×C₄×C₄₇	C₂×SD₁₆	1	8	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	8	no	—
132	C₂×Dic₄₇	—	8	8	no	—
133	C₄₇⋊C₈	—	2	8	no	—
134	C₂×#1957	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	4	16	no	—
136	C₂²×C₄₇	—	4	8	no	—
137	C₂²×C₄₇	—	8	8	no	—
138	C₂²×C₄₇	—	8	4	no	—
139	C₂²×C₄₇	—	8	4	no	—
140	C₂²×C₄₇	—	8	32	no	—
141	C₂²×C₄₇	#390	1	4	yes	—
142	C₂²×C₄₇	C₂×#102	1	8	yes	—
143	C₂²×C₄₇	#486	1	8	yes	—
144	C₂²×C₄₇	#492	1	4	yes	—
145	C₂²×C₄₇	#493	1	4	yes	—
146	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
149	C₂²×C₄₇	C₂²×D₈	1	2	yes	—
150	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
151	C₄×C₄₇	#390	1	4	yes	—
152	C₄×C₄₇	C₂×#102	1	16	yes	complemented
153	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
154	C₄×C₄₇	C₂×#121	1	2	yes	—
155	Dic₄₇	—	8	4	no	—
156	C₂²×#127	D₄₇	1	1	yes	characteristic
157	C₂²×#102	—	47	4	no	—
158	C₂²×C₄×C₈	—	47	1	no	—
159	C₂×#384	—	47	1	no	—
160	#1957	—	47	8	no	—
161	C₂×C₄₇	—	8	8	no	—
162	C₂×C₄₇	#1957	1	8	yes	complemented
163	C₂×C₄₇	C₂×#390	1	1	yes	characteristic
164	C₂×C₄₇	C₂²×#102	1	2	yes	—
165	C₂×C₄₇	C₂×#486	1	2	yes	—
166	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
168	C₂³×D₄	—	2	2	no	—
169	C₂³×D₄	C₂×D₄₇	1	2	yes	—
170	C₂³×C₈	—	47	2	no	—
171	C₂×#127	—	2	2	no	—
172	C₂×#127	—	2	4	no	—
173	C₂×#127	Dic₄₇	1	8	yes	complemented
174	C₂×#127	C₂×D₄₇	1	4	yes	—
175	C₂²×C₄⋊C₄	—	47	2	no	—
176	C₂²×C₄⋊C₄	—	94	2	no	—
177	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
178	C₂×#102	—	47	16	no	—
179	C₂×#102	—	94	16	no	—
180	C₂×C₄×C₈	—	47	4	no	—
181	C₂×C₄×C₈	—	94	4	no	—
182	#384	—	47	4	no	—
183	#384	—	94	2	no	—
184	C₄₇	C₂×#1957	1	1	yes	characteristic, complemented, Sylow
185	C₂⁵	—	2	2	no	—
186	C₂²×D₄	—	2	16	no	—
187	C₂²×D₄	—	2	8	no	—
188	C₂²×D₄	—	2	8	no	—
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	—	2	4	no	—
191	C₂²×D₄	—	2	8	no	—
192	C₂²×D₄	—	4	8	no	—
193	C₂²×D₄	—	4	8	no	—
194	C₂²×D₄	C₂×Dic₄₇	1	8	yes	—
195	C₂²×D₄	C₂²×D₄₇	1	4	yes	—
196	C₂³×C₄	—	94	2	no	—
197	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
198	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×C₈	—	47	8	no	—
200	C₂²×C₈	—	47	4	no	—
201	C₂²×C₈	—	94	4	no	—
202	C₂²×C₈	—	94	4	no	—
203	#127	—	2	16	no	—
204	#127	—	2	8	no	—
205	#127	C₂×Dic₄₇	1	16	yes	complemented
206	C₂×C₄⋊C₄	—	47	4	no	—
207	C₂×C₄⋊C₄	—	94	8	no	—
208	C₂×C₄⋊C₄	—	94	2	no	—
209	C₂×C₄⋊C₄	—	94	8	no	—
210	C₂×C₄⋊C₄	—	188	8	no	—
211	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
212	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
213	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
214	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
215	#102	—	94	32	no	—
216	C₄×C₈	—	94	8	no	—
217	C₂⁴	—	2	8	no	—
218	C₂⁴	—	2	4	no	—
219	C₂⁴	—	4	4	no	—
220	C₂⁴	—	4	4	no	—
221	C₂⁴	—	4	8	no	—
222	C₂⁴	—	4	8	no	—
223	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂×D₄	—	2	8	no	—
225	C₂×D₄	—	2	32	no	—
226	C₂×D₄	—	2	32	no	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	8	no	—
230	C₂×D₄	—	4	64	no	—
231	C₂×D₄	—	4	16	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	—	4	16	no	—
235	C₂×D₄	—	4	16	no	—
236	C₂×D₄	C₂²×Dic₄₇	1	8	yes	—
237	C₂²×C₄	—	94	4	no	—
238	C₂²×C₄	—	188	4	no	—
239	C₂²×C₄	—	188	8	no	—
240	C₂²×C₄	#10625334	1	8	yes	—
241	C₂²×C₄	#10625335	1	8	yes	—
242	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
243	C₂²×C₄	D₄×D₄₇	1	4	yes	—
244	C₂²×C₄	#10625352	1	4	yes	—
245	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
248	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
250	C₂×C₈	—	47	8	no	—
251	C₂×C₈	—	94	16	no	—
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	—	94	4	no	—
254	C₄⋊C₄	—	94	8	no	—
255	C₄⋊C₄	—	188	4	no	—
256	C₄⋊C₄	—	188	16	no	—
257	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
258	C₄²	C₂²×Dic₄₇	1	4	yes	—
259	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
260	C₂³	—	2	8	no	—
261	C₂³	—	4	16	no	—
262	C₂³	—	4	32	no	—
263	C₂³	—	4	8	no	—
264	C₂³	—	4	4	no	—
265	C₂³	—	4	8	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	8	8	no	—
268	C₂³	—	8	8	no	—
269	C₂³	—	8	16	no	—
270	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
271	C₂³	#13094011	1	2	yes	—
272	C₂³	#13094020	1	2	yes	—
273	C₂³	C₂×#10625334	1	4	yes	—
274	C₂³	C₂×#10625335	1	4	yes	—
275	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
276	D₄	—	2	16	no	—
277	D₄	—	4	16	no	—
278	D₄	—	4	8	no	—
279	D₄	—	4	64	no	—
280	C₂×C₄	—	376	8	no	—
281	C₂×C₄	—	188	8	no	—
282	C₂×C₄	—	188	4	no	—
283	C₂×C₄	#13093945	1	32	yes	complemented
284	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
285	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
286	C₂×C₄	D₄×Dic₄₇	1	8	yes	—
287	C₂×C₄	#13093995	1	8	yes	—
288	C₂×C₄	#13094011	1	2	yes	—
289	C₂×C₄	#13094020	1	2	yes	—
290	C₂×C₄	C₂×#10625334	1	8	yes	—
291	C₂×C₄	C₂×#10625335	1	8	yes	—
292	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
294	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂×#10625352	1	2	yes	—
296	C₈	—	94	8	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	16	no	—
299	C₂²	—	8	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	32	no	—
302	C₂²	—	8	8	no	—
303	C₂²	#15689026	1	4	yes	—
304	C₂²	C₂×#13093945	1	8	yes	—
305	C₂²	#15689469	1	8	yes	—
306	C₂²	#15689495	1	4	yes	—
307	C₂²	#15689499	1	4	yes	—
308	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
309	C₂²	C₂×#13094011	1	1	yes	characteristic
310	C₂²	C₂×#13094020	1	1	yes	characteristic
311	C₂²	C₂²×#10625334	1	2	yes	—
312	C₂²	C₂²×#10625335	1	2	yes	—
313	C₄	—	376	4	no	—
314	C₄	#15689026	1	4	yes	—
315	C₄	C₂×#13093945	1	16	yes	complemented
316	C₄	C₂×D₄×Dic₄₇	1	2	yes	—
317	C₄	C₂×#13093995	1	2	yes	—
318	C₂	—	8	8	no	—
319	C₂	#20244195	1	8	yes	direct product factor
320	C₂	C₂×#15689026	1	1	yes	characteristic
321	C₂	C₂²×#13093945	1	2	yes	—
322	C₂	C₂×#15689469	1	2	yes	—
323	C₂	C₂×#15689495	1	1	yes	characteristic
324	C₂	C₂×#15689499	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting, prime radical	C₂²×#127×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#127	D₄₇
hypocenter	C₄₇	C₂×#1957
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093945	C₂	semidirect
2. C₂²×C₄×C₄₇⋊C₈	C₂	semidirect
3. C₂×#15688993	C₂	semidirect
4. #20244195	C₂	direct (N × H)
10. C₂×#13093945	C₂²	semidirect
12. C₂×C₄×C₄₇⋊C₈	C₂²	semidirect
14. #15688993	C₂²	semidirect
19. C₂×#127×C₄₇	C₄	semidirect
25. C₂²×C₄₇⋊C₈	D₄	semidirect
49. #127×C₄₇	C₂×C₄	semidirect
96. C₂×C₄₇⋊C₈	C₂×D₄	semidirect
117. C₂×C₄×C₄₇	#102	semidirect
152. C₄×C₄₇	C₂×#102	semidirect
162. C₂×C₄₇	#1957	semidirect
173. C₂×#127	Dic₄₇	semidirect
184. C₄₇	C₂×#1957	semidirect
205. #127	C₂×Dic₄₇	semidirect
283. C₂×C₄	#13093945	semidirect
315. C₄	C₂×#13093945	semidirect
319. C₂	#20244195	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×#13093945	2
C₂²×C₄×C₄₇⋊C₈	2
C₂×#15688993	2
#20244195	2
C₂²×#127×C₄₇	2
C₂×#1957	47

Subgroups — C2×#20244195 #26219904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244195 #26219904