Subgroups — #26226608

Order 12032.

978 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20244106	C₂	1	4	yes	maximal, complemented
4	#20244104	C₂	1	4	yes	maximal, complemented
5	#20243366	C₂	1	4	yes	maximal
6	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
10	#15689176	—	2	4	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₈	C₂²	1	8	yes	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
16	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₂×C₄₇⋊#95	C₄	1	2	yes	—
20	#15688877	C₂²	1	8	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#431	—	2	4	no	—
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
26	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
27	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	C₈	1	8	yes	complemented
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₄₇	—	4	4	no	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
38	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
39	Dic₄₇⋊C₈	—	4	8	no	—
40	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
41	C₄×C₄₇⋊C₈	—	2	4	no	—
42	C₄₇⋊#95	C₂×C₄	1	4	yes	—
43	C₄₇⋊#95	C₂×C₄	1	4	yes	—
44	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	D₄	1	4	yes	—
51	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₇	—	2	4	no	—
55	C₂²⋊C₈×C₄₇	—	2	4	no	—
56	C₄×C₈×C₄₇	—	2	4	no	—
57	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	4	1	no	—
61	C₂²×C₄×C₄₇	—	4	1	no	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	4	2	no	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	C₂²×Dic₄₇	—	4	2	no	—
78	C₂²×Dic₄₇	—	4	2	no	—
79	C₂²×Dic₄₇	C₂×C₈	1	8	yes	—
80	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
81	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
87	C₄²×C₄₇	C₂×D₄	1	2	yes	—
88	C₄²×C₄₇	C₂×D₄	1	2	yes	—
89	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	C₂×C₈	1	8	yes	—
94	C₄⋊Dic₄₇	C₂×C₈	1	8	yes	—
95	C₄×Dic₄₇	—	2	8	no	—
96	C₄×Dic₄₇	—	4	4	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	2	4	no	—
100	C₂³×C₄₇	—	2	1	no	—
101	C₂³×C₄₇	—	2	1	no	—
102	C₂³×C₄₇	—	2	1	no	—
103	C₂³×C₄₇	—	2	1	no	—
104	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
106	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
107	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	C₄⋊C₈	1	16	yes	complemented
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
130	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	—	2	4	no	—
134	C₂×Dic₄₇	—	2	4	no	—
135	C₂×Dic₄₇	—	2	4	no	—
136	C₂×Dic₄₇	—	4	8	no	—
137	C₂×Dic₄₇	—	4	8	no	—
138	C₂×Dic₄₇	C₂²×C₈	1	8	yes	—
139	C₈×C₄₇	—	4	4	no	—
140	C₄₇⋊C₈	—	4	4	no	—
141	#21951	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
149	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₄₇	#518	1	1	yes	characteristic
151	C₂²×C₄₇	#519	1	1	yes	characteristic
152	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
153	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
154	C₂²×C₄₇	#568	1	1	yes	characteristic
155	C₂²×C₄₇	#568	1	1	yes	characteristic
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
160	C₄×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
161	C₄×C₄₇	#537	1	2	yes	—
162	C₄×C₄₇	#544	1	2	yes	—
163	Dic₄₇	—	2	8	no	—
164	Dic₄₇	—	4	4	no	—
165	C₂×#431	—	47	1	no	—
166	C₂×#431	D₄₇	1	1	yes	characteristic
167	C₂×C₄×C₄⋊C₄	—	47	1	no	—
168	#1939	—	47	4	no	—
169	#1939	—	47	4	no	—
170	#1797	—	47	4	no	—
171	C₂×C₄₇	—	2	4	no	—
172	C₂×C₄₇	#2316	1	1	yes	characteristic
173	C₂×C₄₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
174	C₂×C₄₇	#2905	1	1	yes	characteristic
175	C₂×C₄₇	#2972	1	1	yes	characteristic
176	C₂×C₄₇	#2975	1	1	yes	characteristic
177	C₂×C₄₇	#2982	1	1	yes	characteristic
178	C₂×C₄₇	#2983	1	1	yes	characteristic
179	C₂²×C₄⋊C₄	—	47	1	no	—
180	C₂²×C₄⋊C₄	—	47	1	no	—
181	C₂²×C₄²	—	47	2	no	—
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	#431	—	2	4	no	—
184	#431	—	94	4	no	—
185	C₂×C₄⋊C₈	—	47	2	no	—
186	C₂×C₄⋊C₈	—	47	8	no	—
187	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
188	C₂×C₂²⋊C₈	—	47	2	no	—
189	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
190	C₂×C₄×C₈	—	47	2	no	—
191	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
192	C₄×C₄⋊C₄	—	47	2	no	—
193	C₄×C₄⋊C₄	—	47	2	no	—
194	C₄×C₄⋊C₄	—	47	2	no	—
195	C₄×C₄⋊C₄	—	47	2	no	—
196	C₂×#95	—	47	2	no	—
197	#336	—	47	8	no	—
198	C₄₇	#21951	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₄	—	47	2	no	—
200	C₂³×C₄	—	47	2	no	—
201	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
202	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
203	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
204	C₂²×C₈	—	47	4	no	—
205	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
206	C₂×C₄⋊C₄	—	47	8	no	—
207	C₂×C₄⋊C₄	—	47	2	no	—
208	C₂×C₄⋊C₄	—	47	2	no	—
209	C₂×C₄⋊C₄	—	47	2	no	—
210	C₂×C₄⋊C₄	—	47	2	no	—
211	C₂×C₄⋊C₄	—	94	4	no	—
212	C₂×C₄²	—	47	4	no	—
213	C₂×C₄²	—	47	4	no	—
214	C₂×C₄²	—	188	4	no	—
215	C₂×C₄²	Dic₉₄	1	4	yes	—
216	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
219	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
221	C₄⋊C₈	—	2	4	no	—
222	C₄⋊C₈	—	94	4	no	—
223	C₄⋊C₈	—	188	8	no	—
224	C₂²⋊C₈	—	2	4	no	—
225	C₂²⋊C₈	—	94	4	no	—
226	C₄×C₈	—	2	4	no	—
227	C₄×C₈	—	94	4	no	—
228	#95	—	47	4	no	—
229	#95	—	47	4	no	—
230	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	47	4	no	—
236	C₂²×C₄	—	47	8	no	—
237	C₂²×C₄	—	47	4	no	—
238	C₂²×C₄	—	94	4	no	—
239	C₂²×C₄	—	188	2	no	—
240	C₂²×C₄	—	188	2	no	—
241	C₂²×C₄	—	188	2	no	—
242	C₂²×C₄	—	188	2	no	—
243	C₂²×C₄	C₈×D₄₇	1	4	yes	—
244	C₂²×C₄	#10625324	1	4	yes	—
245	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
246	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
247	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
248	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
254	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	94	4	no	—
259	C₂×C₈	—	94	4	no	—
260	C₂×C₈	—	94	4	no	—
261	C₄⋊C₄	—	47	8	no	—
262	C₄⋊C₄	—	47	8	no	—
263	C₄⋊C₄	—	94	4	no	—
264	C₄⋊C₄	—	94	4	no	—
265	C₄²	—	94	8	no	—
266	C₄²	—	188	4	no	—
267	C₄²	Dic₄₇⋊C₄	1	4	yes	—
268	C₄²	Dic₄₇⋊C₄	1	4	yes	—
269	C₄²	C₂×Dic₉₄	1	2	yes	—
270	C₄²	C₂×Dic₉₄	1	2	yes	—
271	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
272	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
278	C₂³	C₂×C₈×D₄₇	1	2	yes	—
279	C₂³	C₂×#10625324	1	2	yes	—
280	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	47	8	no	—
291	C₂×C₄	—	94	4	no	—
292	C₂×C₄	—	94	4	no	—
293	C₂×C₄	—	94	4	no	—
294	C₂×C₄	—	94	4	no	—
295	C₂×C₄	—	94	4	no	—
296	C₂×C₄	—	188	8	no	—
297	C₂×C₄	—	188	8	no	—
298	C₂×C₄	Dic₄₇⋊C₈	1	16	yes	complemented
299	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
300	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
301	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
302	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
303	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
304	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
305	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
306	C₂×C₄	C₂×#10625324	1	2	yes	—
307	C₂×C₄	C₂×#10625324	1	2	yes	—
308	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
310	C₂×C₄	#13094070	1	2	yes	—
311	C₂×C₄	#13094071	1	2	yes	—
312	C₈	—	4	4	no	—
313	C₈	—	188	4	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	C₂×Dic₄₇⋊C₈	1	4	yes	—
321	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
322	C₂²	#15689597	1	1	yes	characteristic
323	C₂²	#15689600	1	1	yes	characteristic
324	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
325	C₂²	C₂²×#10625324	1	1	yes	characteristic
326	C₂²	#15689889	1	1	yes	characteristic
327	C₂²	#15689889	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	94	8	no	—
332	C₄	—	188	4	no	—
333	C₄	C₂×Dic₄₇⋊C₈	1	4	yes	—
334	C₄	C₂×Dic₄₇⋊C₈	1	4	yes	—
335	C₄	#15689701	1	2	yes	—
336	C₄	#15689751	1	2	yes	—
337	C₂	—	2	4	no	—
338	C₂	#20246004	1	1	yes	characteristic
339	C₂	C₂²×Dic₄₇⋊C₈	1	1	yes	characteristic
340	C₂	#20251951	1	1	yes	characteristic
341	C₂	#20252411	1	1	yes	characteristic
342	C₂	#20252428	1	1	yes	characteristic
343	C₂	#20252468	1	1	yes	characteristic
344	C₂	#20252471	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#21951
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20244106	C₂	semidirect
4. #20244104	C₂	semidirect
30. C₂×C₄⋊Dic₄₇	C₈	semidirect
117. C₂×C₄×C₄₇	C₄⋊C₈	semidirect
198. C₄₇	#21951	semidirect
298. C₂×C₄	Dic₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₂×C₄×C₄⋊Dic₄₇	2
#20244106	2
#20244104	2
#20243366	2
C₂×C₄₇×#431	2
#21951	47