Subgroups — #26226601

Order 12032.

1010 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689087	C₂	1	2	yes	maximal
2	#20243595	C₂	1	8	yes	maximal, complemented
3	#20243430	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#407	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	4	yes	—
6	C₂×#13093960	C₂²	1	8	yes	complemented
7	#15689087	—	2	2	no	—
8	#15689087	—	2	2	no	—
9	#15689087	C₂²	1	4	yes	—
10	#15689087	C₂²	1	4	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
13	#15688888	C₂²	1	8	yes	complemented
14	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₄₇×#407	C₄	1	2	yes	—
17	C₄₇×#407	C₄	1	4	yes	—
18	C₄₇×#407	C₄	1	2	yes	—
19	C₂×C₂²⋊C₈×C₄₇	C₂²	1	4	yes	—
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	—	4	8	no	—
22	C₂×#10625329	C₂³	1	8	yes	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	#13093960	—	4	8	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
27	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
29	C₂⁵×C₄₇	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	—	4	4	no	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	D₄	1	2	yes	—
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
38	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
41	C₂×M₁₆×C₄₇	—	4	4	no	—
42	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
45	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
46	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
47	C₂⁴×C₄₇	—	4	2	no	—
48	C₂⁴×C₄₇	—	4	4	no	—
49	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
51	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂⁴×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	8	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	4	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₄₇	—	4	4	no	—
83	M₁₆×C₄₇	—	4	4	no	—
84	C₂×C₈×C₄₇	—	2	8	no	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
86	#10625329	—	2	16	no	—
87	#10625329	—	4	8	no	—
88	C₂×C₄₇⋊C₈	—	2	8	no	—
89	C₂×C₄₇⋊C₈	—	4	8	no	—
90	C₂³×C₄₇	—	2	4	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	—	2	4	no	—
95	C₂³×C₄₇	—	4	1	no	—
96	C₂³×C₄₇	—	4	2	no	—
97	C₂³×C₄₇	—	4	1	no	—
98	C₂³×C₄₇	—	4	2	no	—
99	C₂³×C₄₇	—	4	2	no	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	—	4	2	no	—
102	C₂³×C₄₇	—	4	16	no	—
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	1	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	1	no	—
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
132	C₈×C₄₇	—	2	8	no	—
133	C₄₇⋊C₈	—	4	8	no	—
134	#21949	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	4	8	no	—
146	C₂²×C₄₇	—	4	4	no	—
147	C₂²×C₄₇	—	4	4	no	—
148	C₂²×C₄₇	—	4	8	no	—
149	C₂²×C₄₇	—	4	4	no	—
150	C₂²×C₄₇	#430	1	4	yes	—
151	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄₇	#543	1	2	yes	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	#518	1	2	yes	—
158	C₄×C₄₇	#534	1	2	yes	—
159	C₂×#407	—	47	2	no	—
160	C₂×#407	D₄₇	1	1	yes	characteristic
161	#1855	—	47	8	no	—
162	#1811	—	47	4	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	—	4	1	no	—
166	C₂×C₄₇	—	4	2	no	—
167	C₂×C₄₇	—	4	1	no	—
168	C₂×C₄₇	#2315	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#430	1	2	yes	—
170	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
171	C₂²×M₁₆	—	47	4	no	—
172	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
173	C₂×#108	—	47	8	no	—
174	#407	—	47	4	no	—
175	#407	—	47	4	no	—
176	#407	—	94	2	no	—
177	#407	—	94	2	no	—
178	#407	Dic₄₇	1	2	yes	—
179	#407	Dic₄₇	1	4	yes	—
180	#407	Dic₄₇	1	2	yes	—
181	C₂×C₂²⋊C₈	—	94	2	no	—
182	C₂×C₂²⋊C₈	—	94	2	no	—
183	C₂×C₂²⋊C₈	C₂×D₄₇	1	4	yes	—
184	#341	—	47	8	no	—
185	C₄₇	#21949	1	1	yes	characteristic, complemented, Sylow
186	C₂⁵	Dic₉₄	1	1	yes	characteristic
187	C₂³×C₄	—	4	4	no	—
188	C₂³×C₄	Dic₉₄	1	2	yes	—
189	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
190	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
191	C₂³×C₄	D₁₈₈	1	2	yes	—
192	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
193	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
194	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
195	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
197	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂×M₁₆	—	4	4	no	—
199	C₂×M₁₆	—	47	8	no	—
200	C₂×M₁₆	—	94	4	no	—
201	C₂×M₁₆	—	188	8	no	—
202	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
203	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
204	C₂²×C₈	—	94	4	no	—
205	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
206	#108	—	188	8	no	—
207	C₂²⋊C₈	—	94	4	no	—
208	C₂²⋊C₈	—	94	4	no	—
209	C₂²⋊C₈	—	94	4	no	—
210	C₂²⋊C₈	—	94	4	no	—
211	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
212	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
213	C₂⁴	—	4	2	no	—
214	C₂⁴	—	4	4	no	—
215	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
216	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
217	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
218	C₂⁴	C₂×Dic₉₄	1	2	yes	—
219	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
232	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
233	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
234	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
236	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
237	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
238	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
242	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
243	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
248	M₁₆	—	4	4	no	—
249	M₁₆	—	4	4	no	—
250	M₁₆	—	94	16	no	—
251	M₁₆	—	188	8	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	—	94	8	no	—
254	C₂×C₈	—	188	8	no	—
255	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	1	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	16	no	—
269	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
270	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
271	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
272	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
273	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
294	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
295	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
297	C₂×C₄	#13094069	1	4	yes	—
298	C₈	—	2	8	no	—
299	C₈	—	188	8	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	8	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	4	8	no	—
314	C₂²	—	4	4	no	—
315	C₂²	#15689172	1	4	yes	—
316	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
317	C₂²	#15689749	1	2	yes	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	#15689598	1	2	yes	—
323	C₄	#15689685	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	2	4	no	—
326	C₂	—	4	1	no	—
327	C₂	—	4	2	no	—
328	C₂	—	4	1	no	—
329	C₂	#20246002	1	1	yes	characteristic
330	C₂	C₂×#15689172	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#407	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁵×C₄₇	Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#407	D₄₇
hypocenter	C₄₇	#21949
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂×#407

Complements

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

N	H ≅ G/N	type
2. #20243595	C₂	semidirect
3. #20243430	C₂	semidirect
6. C₂×#13093960	C₂²	semidirect
13. #15688888	C₂²	semidirect
185. C₄₇	#21949	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689087	2
#20243595	2
#20243430	2
C₂×C₄₇×#407	2
#21949	47