Subgroups — #26106900

Order 12032.

698 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243241	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
4	#20241229	C₂	1	2	yes	maximal, complemented
5	C₄²×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13093935	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	2	no	—
11	D₄⋊(C₄₇⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₇	—	2	1	no	—
15	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
18	C₄×C₂²⋊C₄×C₄₇	—	2	1	no	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	complemented
24	#13093935	—	4	2	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
27	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	—	2	2	no	—
29	C₂³×C₄×C₄₇	—	2	1	no	—
30	C₄×D₄×C₄₇	—	2	4	no	—
31	C₄×D₄×C₄₇	—	2	4	no	—
32	C₄×D₄×C₄₇	—	2	2	no	—
33	C₄×D₄×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄²×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	—	2	1	no	—
40	C₂×C₄²×C₄₇	—	4	2	no	—
41	C₂×C₄²×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	#95×C₄₇	—	2	2	no	—
48	#95×C₄₇	—	2	1	no	—
49	C₂⁴×C₄₇	—	2	1	no	—
50	C₂×D₄×C₄₇	—	2	1	no	—
51	C₂×D₄×C₄₇	—	2	2	no	—
52	C₂×D₄×C₄₇	C₄²	1	2	yes	—
53	C₂×D₄×C₄₇	M₁₆	1	4	yes	—
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	4	4	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	1	no	—
61	C₂²×C₄×C₄₇	—	4	1	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	—	2	1	no	—
69	C₄⋊C₄×C₄₇	—	2	2	no	—
70	C₄⋊C₄×C₄₇	C₄²	1	2	yes	—
71	C₄⋊C₄×C₄₇	M₁₆	1	4	yes	—
72	C₂²⋊C₄×C₄₇	—	2	8	no	—
73	C₂²⋊C₄×C₄₇	—	2	4	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	2	2	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	C₄²×C₄₇	—	4	1	no	—
81	C₄²×C₄₇	D₈	1	2	yes	—
82	C₄²×C₄₇	SD₁₆	1	2	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₄₇	#41	1	2	yes	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	4	4	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	4	no	—
92	C₂³×C₄₇	—	4	1	no	—
93	C₂³×C₄₇	—	4	1	no	—
94	C₂³×C₄₇	—	4	2	no	—
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	D₄×C₄₇	—	2	2	no	—
97	D₄×C₄₇	—	2	4	no	—
98	D₄×C₄₇	#97	1	4	yes	complemented
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	4	16	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	#102	1	4	yes	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#135	1	2	yes	—
124	C₄₇⋊C₈	—	4	4	no	—
125	C₄₇⋊C₈	—	4	4	no	—
126	#8174	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	4	4	no	—
129	C₂²×C₄₇	—	4	4	no	—
130	C₂²×C₄₇	—	4	2	no	—
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄₇	#407	1	2	yes	—
134	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
135	C₂²×C₄₇	#416	1	1	yes	characteristic
136	C₂²×C₄₇	#421	1	2	yes	—
137	C₄×C₄₇	—	2	4	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	4	4	no	—
140	C₄×C₄₇	—	4	2	no	—
141	C₄×C₄₇	—	4	2	no	—
142	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
143	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
144	C₄×C₄₇	#432	1	2	yes	—
145	C₄×C₄₇	C₄×D₈	1	2	yes	—
146	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
147	C₄²×D₄	D₄₇	1	1	yes	characteristic
148	#1767	—	47	1	no	—
149	#1744	—	47	1	no	—
150	C₂×D₄⋊C₈	—	47	2	no	—
151	#1279	—	47	2	no	—
152	C₂×C₄₇	—	4	4	no	—
153	C₂×C₄₇	#1486	1	2	yes	—
154	C₂×C₄₇	#1488	1	2	yes	—
155	C₂×C₄₇	#1749	1	1	yes	characteristic
156	C₂×C₄₇	#1759	1	1	yes	characteristic
157	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
158	C₂×C₄×D₄	Dic₄₇	1	2	yes	—
159	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
160	C₂²×C₄²	—	2	1	no	—
161	C₂×C₄⋊C₈	—	47	2	no	—
162	C₂×#97	—	94	1	no	—
163	C₂×C₄×C₈	—	47	2	no	—
164	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
165	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
166	C₄×C₂²⋊C₄	—	2	1	no	—
167	C₄×C₂²⋊C₄	—	2	2	no	—
168	C₄³	C₂×D₄₇	1	1	yes	characteristic
169	#336	—	94	2	no	—
170	#336	—	94	1	no	—
171	D₄⋊C₈	—	47	8	no	—
172	C₄₇	#8174	1	1	yes	characteristic, complemented, Sylow
173	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	—	2	1	no	—
176	C₂²×C₈	—	94	2	no	—
177	C₂²×C₈	—	94	2	no	—
178	C₄×D₄	—	2	4	no	—
179	C₄×D₄	—	2	4	no	—
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	C₄×D₄₇	1	4	yes	—
182	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	—	4	2	no	—
189	C₂×C₄²	—	4	2	no	—
190	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
193	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
195	C₄⋊C₈	—	47	4	no	—
196	C₄⋊C₈	—	94	2	no	—
197	#97	—	188	2	no	—
198	C₄×C₈	—	47	4	no	—
199	C₄×C₈	—	94	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	1	no	—
202	C₂⁴	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	#10625329	1	4	yes	—
206	C₂×D₄	C₄×Dic₄₇	1	2	yes	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×C₈	—	94	4	no	—
222	C₂×C₈	—	94	4	no	—
223	C₂×C₈	—	188	4	no	—
224	C₂×C₈	—	188	2	no	—
225	C₂×C₈	—	188	2	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	#10625329	1	4	yes	—
229	C₄⋊C₄	C₄×Dic₄₇	1	2	yes	—
230	C₂²⋊C₄	—	2	8	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	4	4	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	#10625334	1	2	yes	—
240	C₄²	#10625335	1	2	yes	—
241	C₄²	C₂×C₄×D₄₇	1	2	yes	—
242	C₄²	#10625350	1	2	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	4	1	no	—
246	C₂³	—	4	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
249	D₄	—	2	2	no	—
250	D₄	—	2	4	no	—
251	D₄	#13093935	1	4	yes	complemented
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	16	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	#13093945	1	4	yes	—
268	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
269	C₂×C₄	#13093983	1	1	yes	characteristic
270	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
271	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
272	C₂×C₄	C₂×#10625329	1	2	yes	—
273	C₂×C₄	C₂×#10625329	1	2	yes	—
274	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
275	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
276	C₂×C₄	#13094044	1	2	yes	—
277	C₈	—	188	4	no	—
278	C₈	—	188	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
285	C₂²	#15689087	1	2	yes	—
286	C₂²	C₂×#13093945	1	1	yes	characteristic
287	C₂²	#15689119	1	1	yes	characteristic
288	C₂²	#15689140	1	2	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	4	no	—
293	C₄	—	4	2	no	—
294	C₄	C₂×#13093935	1	1	yes	characteristic
295	C₄	C₄×#10625329	1	1	yes	characteristic
296	C₄	#15689180	1	2	yes	—
297	C₄	C₄×#10625334	1	2	yes	—
298	C₄	C₄×#10625335	1	2	yes	—
299	C₂	—	4	4	no	—
300	C₂	#20241813	1	2	yes	—
301	C₂	#20241822	1	2	yes	—
302	C₂	#20243150	1	1	yes	characteristic
303	C₂	#20243198	1	1	yes	characteristic
304	C₂	C₄×#13093945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄²×D₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄²×D₄	D₄₇
hypocenter	C₄₇	#8174
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
1. #20243241	C₂	semidirect
2. #20243134	C₂	semidirect
4. #20241229	C₂	semidirect
11. D₄⋊(C₄₇⋊C₈)	C₄	semidirect
23. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
26. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
98. D₄×C₄₇	#97	semidirect
172. C₄₇	#8174	semidirect
251. D₄	#13093935	semidirect

Maximal subgroups

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

subgroup	index
#20243241	2
#20243134	2
C₂×D₄⋊(C₄₇⋊C₈)	2
#20241229	2
C₄²×D₄×C₄₇	2
#8174	47