Subgroups — #26104306

Order 12032.

424 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245865	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	#20241213	C₂	1	2	yes	maximal
4	#20241212	C₂	1	2	yes	maximal
5	C₄₇×#1839	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₄	1	1	yes	characteristic
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	#15689029	C₄	1	1	yes	characteristic
9	#15689029	C₂²	1	1	yes	characteristic
10	#15689021	C₄	1	1	yes	characteristic
11	#15689021	C₂²	1	1	yes	characteristic
12	#15689007	—	2	1	no	—
13	#15689007	—	2	1	no	—
14	C₄×Dic₄₇⋊C₄	—	2	1	no	—
15	C₄×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
16	#15688969	—	2	1	no	—
17	#15688878	—	2	1	no	—
18	#15688878	C₂²	1	2	yes	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#336	—	2	1	no	—
21	C₄₇×#336	—	2	1	no	—
22	C₄₇×#336	C₂²	1	2	yes	—
23	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₄×Dic₉₄	—	2	2	no	—
26	C₄×Dic₉₄	C₈	1	4	yes	complemented
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×Dic₄₇⋊C₄	—	2	1	no	—
31	C₂×Dic₄₇⋊C₄	—	2	1	no	—
32	C₂×Dic₄₇⋊C₄	—	2	1	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₂×C₄×Dic₄₇	—	2	1	no	—
35	C₄×C₄₇⋊C₈	—	2	4	no	—
36	C₄₇⋊#95	—	2	1	no	—
37	C₄₇⋊#95	—	2	1	no	—
38	C₄₇⋊#95	—	2	1	no	—
39	C₄₇⋊#95	—	2	1	no	—
40	C₄₇⋊#95	—	2	1	no	—
41	C₄₇⋊#95	—	2	1	no	—
42	C₄₇⋊#95	—	2	1	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	1	no	—
59	C₂²×Dic₄₇	—	2	1	no	—
60	C₂²×Dic₄₇	—	2	1	no	—
61	C₂²×Dic₄₇	—	2	1	no	—
62	C₂×Dic₉₄	—	2	1	no	—
63	C₂×Dic₉₄	—	4	1	no	—
64	C₂×Dic₉₄	—	4	1	no	—
65	C₂×Dic₉₄	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₄₇	—	4	2	no	—
67	C₂×C₈×C₄₇	—	4	2	no	—
68	C₂×C₈×C₄₇	—	4	2	no	—
69	C₄²×C₄₇	—	2	2	no	—
70	C₄²×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	4	1	no	—
73	C₄²×C₄₇	—	4	1	no	—
74	C₄²×C₄₇	—	4	1	no	—
75	C₄²×C₄₇	C₂×C₈	1	2	yes	—
76	C₄²×C₄₇	M₁₆	1	2	yes	—
77	C₄⋊Dic₄₇	—	2	1	no	—
78	C₄⋊Dic₄₇	—	4	1	no	—
79	C₄⋊Dic₄₇	—	4	1	no	—
80	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
81	Dic₄₇⋊C₄	—	2	4	no	—
82	Dic₄₇⋊C₄	—	4	2	no	—
83	Dic₄₇⋊C₄	—	4	2	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₄×Dic₄₇	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₄₇	#104	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#135	1	2	yes	—
111	C₂×C₄×C₄₇	#137	1	2	yes	—
112	C₂×Dic₄₇	—	2	2	no	—
113	C₂×Dic₄₇	—	2	2	no	—
114	C₂×Dic₄₇	—	4	1	no	—
115	C₂×Dic₄₇	—	4	1	no	—
116	C₂×Dic₄₇	—	4	1	no	—
117	C₂×Dic₄₇	—	4	1	no	—
118	C₂×Dic₄₇	—	4	1	no	—
119	C₂×Dic₄₇	—	4	1	no	—
120	C₂×Dic₄₇	—	4	1	no	—
121	C₂×Dic₄₇	—	4	1	no	—
122	C₂×Dic₄₇	—	4	1	no	—
123	C₂×Dic₄₇	—	4	1	no	—
124	Dic₉₄	—	4	2	no	—
125	Dic₉₄	—	4	2	no	—
126	C₈×C₄₇	—	8	2	no	—
127	C₄₇⋊C₈	—	4	4	no	—
128	#7581	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₄₇	#410	1	1	yes	characteristic
131	C₂²×C₄₇	#413	1	1	yes	characteristic
132	C₂²×C₄₇	#416	1	1	yes	characteristic
133	C₂²×C₄₇	#419	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
135	C₂²×C₄₇	#421	1	1	yes	characteristic
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	1	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	4	1	no	—
144	Dic₄₇	—	4	2	no	—
145	Dic₄₇	—	4	2	no	—
146	Dic₄₇	—	8	1	no	—
147	Dic₄₇	—	8	1	no	—
148	#2295	—	47	1	no	—
149	#1839	D₄₇	1	1	yes	characteristic
150	#1744	—	47	1	no	—
151	#1272	—	47	2	no	—
152	#1271	—	47	2	no	—
153	C₂×C₄₇	#1458	1	1	yes	characteristic
154	C₂×C₄₇	#1476	1	1	yes	characteristic
155	C₂×C₄₇	#1479	1	1	yes	characteristic
156	C₂×C₄₇	#1506	1	1	yes	characteristic
157	C₂×C₄₇	#1521	1	1	yes	characteristic
158	C₂×C₄₇	#1532	1	1	yes	characteristic
159	C₂×C₄₇	#1554	1	1	yes	characteristic
160	C₂×C₄×Q₈	—	47	1	no	—
161	C₂×C₄×C₈	—	94	1	no	—
162	#391	—	47	1	no	—
163	#391	—	47	1	no	—
164	#389	—	47	1	no	—
165	#389	—	47	1	no	—
166	#387	—	94	1	no	—
167	#387	—	94	1	no	—
168	C₄×C₄⋊C₄	—	47	1	no	—
169	C₄×C₄⋊C₄	—	94	1	no	—
170	#376	—	94	1	no	—
171	C₄³	C₂×D₄₇	1	1	yes	characteristic
172	#336	—	2	1	no	—
173	#336	—	2	1	no	—
174	#336	—	47	2	no	—
175	#336	—	94	1	no	—
176	#336	C₂×D₄₇	1	2	yes	—
177	C₄₇	#7581	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	—	47	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	94	2	no	—
181	C₄×Q₈	—	47	4	no	—
182	C₄×Q₈	—	94	2	no	—
183	C₂×C₄⋊C₄	—	47	1	no	—
184	C₂×C₄⋊C₄	—	47	1	no	—
185	C₂×C₄⋊C₄	—	47	1	no	—
186	C₂×C₄⋊C₄	—	94	1	no	—
187	C₂×C₄⋊C₄	—	94	1	no	—
188	C₂×C₄⋊C₄	—	94	1	no	—
189	C₂×C₄²	—	94	1	no	—
190	C₂×C₄²	—	94	1	no	—
191	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
192	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
193	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
194	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
198	C₄×C₈	—	94	4	no	—
199	#95	—	94	1	no	—
200	#95	—	94	1	no	—
201	#95	—	94	1	no	—
202	#95	—	94	1	no	—
203	#95	—	94	1	no	—
204	#95	—	94	1	no	—
205	#95	—	94	1	no	—
206	C₂×Q₈	—	47	2	no	—
207	C₂×Q₈	—	94	1	no	—
208	C₂×Q₈	—	188	1	no	—
209	C₂×Q₈	—	188	1	no	—
210	C₂²×C₄	—	94	1	no	—
211	C₂²×C₄	—	94	1	no	—
212	C₂²×C₄	—	94	1	no	—
213	C₂²×C₄	—	94	1	no	—
214	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	94	4	no	—
225	C₂×C₈	—	188	2	no	—
226	C₂×C₈	—	188	2	no	—
227	C₄⋊C₄	—	47	2	no	—
228	C₄⋊C₄	—	94	4	no	—
229	C₄⋊C₄	—	94	1	no	—
230	C₄⋊C₄	—	188	1	no	—
231	C₄⋊C₄	—	188	1	no	—
232	C₄⋊C₄	—	188	2	no	—
233	C₄⋊C₄	—	188	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	94	4	no	—
241	C₄²	—	188	2	no	—
242	C₄²	C₈×D₄₇	1	2	yes	—
243	C₄²	#10625324	1	2	yes	—
244	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
245	Q₈	—	188	2	no	—
246	Q₈	—	188	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	94	2	no	—
263	C₂×C₄	—	94	2	no	—
264	C₂×C₄	—	188	1	no	—
265	C₂×C₄	—	188	1	no	—
266	C₂×C₄	—	188	1	no	—
267	C₂×C₄	—	188	1	no	—
268	C₂×C₄	—	188	1	no	—
269	C₂×C₄	—	188	1	no	—
270	C₂×C₄	—	188	1	no	—
271	C₂×C₄	—	188	1	no	—
272	C₂×C₄	—	188	1	no	—
273	C₂×C₄	—	188	1	no	—
274	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
275	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
276	C₂×C₄	#13093950	1	2	yes	—
277	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
278	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
279	C₂×C₄	#13094044	1	2	yes	—
280	C₂×C₄	#13094057	1	2	yes	—
281	C₈	—	8	2	no	—
282	C₈	—	188	4	no	—
283	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
284	C₂²	#15689096	1	1	yes	characteristic
285	C₂²	#15689105	1	1	yes	characteristic
286	C₂²	#15689117	1	1	yes	characteristic
287	C₂²	#15689133	1	1	yes	characteristic
288	C₂²	C₂×#13093950	1	1	yes	characteristic
289	C₂²	#15689141	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	1	no	—
298	C₄	—	376	1	no	—
299	C₄	—	376	1	no	—
300	C₄	—	188	2	no	—
301	C₄	—	188	2	no	—
302	C₂	#20241700	1	1	yes	characteristic
303	C₂	#20241768	1	1	yes	characteristic
304	C₂	#20241784	1	1	yes	characteristic
305	C₂	#20241901	1	1	yes	characteristic
306	C₂	#20241963	1	1	yes	characteristic
307	C₂	#20242009	1	1	yes	characteristic
308	C₂	#20242099	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1839	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1839	D₄₇
hypocenter	C₄₇	#7581
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1839

Complements

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

N	H ≅ G/N	type
26. C₄×Dic₉₄	C₈	semidirect
177. C₄₇	#7581	semidirect

Maximal subgroups

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

subgroup	index
#20245865	2
#20243134	2
#20241213	2
#20241212	2
C₄₇×#1839	2
#7581	47