Subgroups — #26122494

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251807	C₂	1	1	yes	maximal, characteristic
2	#20246008	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688894	C₂	1	2	yes	maximal
4	C₂×#15688891	C₂	1	2	yes	maximal
5	C₄₇×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
9	#15689079	—	2	4	no	—
10	C₄×#10625329	—	2	2	no	—
11	#15689021	—	2	2	no	—
12	#15688995	C₄	1	4	yes	—
13	#15688992	—	2	1	no	—
14	#15688988	—	2	1	no	—
15	#15688894	C₄	1	8	yes	complemented
16	#15688891	—	2	4	no	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#343	—	2	1	no	—
20	C₄₇×#343	—	2	1	no	—
21	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	C₂×C₄	1	4	yes	—
26	C₂×#10625329	D₄	1	4	yes	—
27	C₂×#10625329	Q₈	1	4	yes	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	#13093983	—	2	2	no	—
30	#13093983	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×C₄⋊Dic₄₇	—	2	1	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×Dic₄₇⋊C₄	—	2	4	no	—
37	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	—	2	1	no	—
39	#13093935	—	2	4	no	—
40	C₄×C₄₇⋊C₈	—	2	4	no	—
41	C₄₇⋊#95	—	2	4	no	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	2	no	—
48	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₄₇	—	2	2	no	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₇	D₄	1	2	yes	—
56	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×Dic₄₇	—	2	2	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	8	2	no	—
71	M₁₆×C₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	2	4	no	—
73	M₁₆×C₄₇	—	4	2	no	—
74	M₁₆×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₄⋊C₄×C₄₇	—	2	2	no	—
80	C₂²⋊C₄×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₂²⋊Dic₄₇	—	2	2	no	—
84	C₂²⋊Dic₄₇	C₄²	1	4	yes	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	4	1	no	—
87	C₄⋊Dic₄₇	—	4	2	no	—
88	C₄⋊Dic₄₇	—	4	1	no	—
89	Dic₄₇⋊C₄	—	8	4	no	—
90	C₄×Dic₄₇	—	2	4	no	—
91	#10625329	—	2	4	no	—
92	#10625329	—	2	4	no	—
93	#10625329	—	2	4	no	—
94	#10625329	C₄⋊C₄	1	8	yes	complemented
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂³×C₄₇	—	4	1	no	—
99	C₂³×C₄₇	—	4	1	no	—
100	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
101	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₄₇	#117	1	1	yes	characteristic
103	C₂³×C₄₇	C₄×D₄	1	2	yes	—
104	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₇	#117	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#121	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	#123	1	2	yes	—
120	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#125	1	2	yes	—
123	C₂×C₄×C₄₇	#126	1	2	yes	—
124	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	8	4	no	—
131	C₈×C₄₇	—	4	4	no	—
132	C₄₇⋊C₈	—	2	8	no	—
133	#10916	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	4	1	no	—
139	C₂²×C₄₇	—	4	1	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
143	C₂²×C₄₇	#386	1	1	yes	characteristic
144	C₂²×C₄₇	#387	1	2	yes	—
145	C₂²×C₄₇	#391	1	1	yes	characteristic
146	C₄×C₄₇	—	4	2	no	—
147	C₄×C₄₇	—	4	2	no	—
148	C₄×C₄₇	#385	1	2	yes	—
149	C₄×C₄₇	#387	1	2	yes	—
150	C₄×C₄₇	#388	1	2	yes	—
151	C₄×C₄₇	#389	1	2	yes	—
152	Dic₄₇	—	4	4	no	—
153	Dic₄₇	—	8	2	no	—
154	#2873	—	47	1	no	—
155	#2317	—	47	1	no	—
156	C₂×#344	—	47	2	no	—
157	#1742	D₄₇	1	1	yes	characteristic
158	C₂×#343	—	47	2	no	—
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	—	2	1	no	—
163	C₂×C₄₇	#1436	1	1	yes	characteristic
164	C₂×C₄₇	#1914	1	1	yes	characteristic
165	C₂×C₄₇	#1915	1	1	yes	characteristic
166	C₂²×M₁₆	—	47	2	no	—
167	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
168	C₂×#117	—	47	1	no	—
169	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
170	C₂²×C₄⋊C₄	—	47	1	no	—
171	#405	—	94	4	no	—
172	C₄×M₁₆	—	94	2	no	—
173	#389	—	94	2	no	—
174	#385	—	47	4	no	—
175	#384	—	94	1	no	—
176	#382	—	94	1	no	—
177	#344	—	47	8	no	—
178	#343	—	2	1	no	—
179	#343	—	2	1	no	—
180	#343	—	94	4	no	—
181	C₄₇	#10916	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	—	47	2	no	—
183	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	47	4	no	—
190	C₂×M₁₆	—	47	4	no	—
191	C₂×M₁₆	—	47	4	no	—
192	C₂×M₁₆	—	94	2	no	—
193	C₂×M₁₆	—	94	2	no	—
194	C₂×M₁₆	—	94	2	no	—
195	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	94	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	47	4	no	—
201	#117	—	94	2	no	—
202	C₂×C₄⋊C₄	—	94	1	no	—
203	C₂×C₄⋊C₄	—	94	4	no	—
204	C₂×C₄⋊C₄	—	94	1	no	—
205	C₂×C₄⋊C₄	—	94	2	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	—	94	1	no	—
208	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
209	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	47	2	no	—
211	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
212	C₂×C₄²	—	94	1	no	—
213	C₂×C₄²	D₁₈₈	1	2	yes	—
214	#97	—	94	4	no	—
215	C₄×C₈	—	94	4	no	—
216	#95	—	94	4	no	—
217	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	—	94	2	no	—
219	C₂²×C₄	—	94	4	no	—
220	C₂²×C₄	—	94	2	no	—
221	C₂²×C₄	—	376	2	no	—
222	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
223	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
224	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
225	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
226	C₂²×C₄	#10625350	1	2	yes	—
227	C₂²×C₄	D₄×D₄₇	1	2	yes	—
228	C₂²×C₄	#10625352	1	2	yes	—
229	C₂²×C₄	#10625352	1	2	yes	—
230	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	47	8	no	—
237	M₁₆	—	94	4	no	—
238	M₁₆	—	94	4	no	—
239	M₁₆	—	94	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	94	4	no	—
245	C₂×C₈	—	94	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	94	4	no	—
249	C₄⋊C₄	—	376	4	no	—
250	C₄⋊C₄	—	188	1	no	—
251	C₄⋊C₄	—	188	2	no	—
252	C₄⋊C₄	—	188	1	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	47	4	no	—
255	C₂²⋊C₄	—	94	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	94	4	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	4	1	no	—
261	C₂³	C₄²×D₄₇	1	1	yes	characteristic
262	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
263	C₂³	#13093981	1	1	yes	characteristic
264	C₂³	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
265	C₂³	#13093989	1	2	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	376	4	no	—
272	C₂×C₄	—	188	4	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	—	188	2	no	—
276	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
277	C₂×C₄	#13093980	1	2	yes	—
278	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
279	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
280	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
281	C₂×C₄	#13093989	1	2	yes	—
282	C₂×C₄	#13093994	1	1	yes	characteristic
283	C₂×C₄	#13093996	1	2	yes	—
284	C₂×C₄	#13094002	1	2	yes	—
285	C₂×C₄	#13094004	1	2	yes	—
286	C₂×C₄	#13094006	1	2	yes	—
287	C₂×C₄	#13094008	1	1	yes	characteristic
288	C₂×C₄	#13094011	1	1	yes	characteristic
289	C₂×C₄	#13094014	1	2	yes	—
290	C₂×C₄	#13094017	1	2	yes	—
291	C₂×C₄	#13094022	1	1	yes	characteristic
292	C₈	—	4	4	no	—
293	C₈	—	94	8	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
302	C₂²	#15688974	1	2	yes	—
303	C₂²	#15689000	1	1	yes	characteristic
304	C₂²	#15689005	1	2	yes	—
305	C₂²	#15689028	1	1	yes	characteristic
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	376	2	no	—
309	C₄	—	188	4	no	—
310	C₄	#15688997	1	2	yes	—
311	C₄	#15689008	1	2	yes	—
312	C₄	#15689017	1	2	yes	—
313	C₄	#15689018	1	2	yes	—
314	C₂	—	2	1	no	—
315	C₂	—	2	2	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	#20241623	1	1	yes	characteristic
319	C₂	#20243933	1	1	yes	characteristic
320	C₂	#20243938	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689000
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1742	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1742	D₄₇
hypocenter	C₄₇	#10916
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
15. #15688894	C₄	semidirect
94. #10625329	C₄⋊C₄	semidirect
181. C₄₇	#10916	semidirect

Maximal subgroups

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

subgroup	index
#20251807	2
#20246008	2
C₂×#15688894	2
C₂×#15688891	2
C₄₇×#1742	2
#10916	47