Subgroups — #26125043

Order 12032.

622 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	#20246006	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688895	C₂	1	2	yes	maximal
3	C₂×#15688894	C₂	1	2	yes	maximal
4	C₂×#15688892	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#343	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂×#13093983	C₂²	1	1	yes	characteristic
9	#15689022	—	2	2	no	—
10	#15689022	—	2	2	no	—
11	#15688983	C₄	1	4	yes	—
12	#15688895	—	2	4	no	—
13	#15688894	—	2	4	no	—
14	#15688892	—	2	2	no	—
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#343	—	2	2	no	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	D₄	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	#13093983	—	2	2	no	—
26	#13093983	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	—	2	1	no	—
35	C₂×C₄×Dic₄₇	—	2	1	no	—
36	C₄₇⋊#95	—	2	4	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
50	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	8	2	no	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₇	#41	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×Dic₄₇	—	2	2	no	—
64	C₂²×Dic₄₇	—	2	2	no	—
65	M₁₆×C₄₇	—	2	4	no	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	4	2	no	—
68	M₁₆×C₄₇	—	4	2	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	4	2	no	—
71	C₂×C₈×C₄₇	—	4	2	no	—
72	C₄⋊C₄×C₄₇	—	2	4	no	—
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₂²⋊Dic₄₇	—	2	4	no	—
75	C₂²⋊Dic₄₇	—	2	4	no	—
76	C₄⋊Dic₄₇	—	4	2	no	—
77	C₄⋊Dic₄₇	—	4	1	no	—
78	C₄⋊Dic₄₇	—	4	1	no	—
79	C₄⋊Dic₄₇	—	8	4	no	—
80	C₄×Dic₄₇	—	4	2	no	—
81	C₄×Dic₄₇	—	4	1	no	—
82	C₄×Dic₄₇	—	4	1	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	2	4	no	—
85	#10625329	—	4	2	no	—
86	#10625329	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂³×C₄₇	—	4	1	no	—
91	C₂³×C₄₇	—	4	1	no	—
92	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₇	#117	1	1	yes	characteristic
94	C₂³×C₄₇	#117	1	1	yes	characteristic
95	C₂³×C₄₇	C₄×D₄	1	2	yes	—
96	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	—	8	2	no	—
100	C₂×C₄×C₄₇	—	8	2	no	—
101	C₂×C₄×C₄₇	#121	1	2	yes	—
102	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₄₇	#123	1	2	yes	—
106	C₂×C₄×C₄₇	#123	1	2	yes	—
107	C₂×C₄×C₄₇	#123	1	2	yes	—
108	C₂×C₄×C₄₇	#123	1	2	yes	—
109	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#125	1	2	yes	—
112	C₂×C₄×C₄₇	#125	1	2	yes	—
113	C₂×C₄×C₄₇	#126	1	2	yes	—
114	C₂×C₄×C₄₇	#126	1	2	yes	—
115	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×Dic₄₇	—	4	2	no	—
118	C₂×Dic₄₇	—	4	2	no	—
119	C₂×Dic₄₇	—	4	2	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₈×C₄₇	—	4	4	no	—
124	C₄₇⋊C₈	—	4	4	no	—
125	#11352	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	1	no	—
127	C₂²×C₄₇	—	2	1	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	4	2	no	—
133	C₂²×C₄₇	—	4	2	no	—
134	C₂²×C₄₇	#381	1	1	yes	characteristic
135	C₂²×C₄₇	#387	1	2	yes	—
136	C₂²×C₄₇	#387	1	2	yes	—
137	C₂²×C₄₇	#390	1	1	yes	characteristic
138	C₂²×C₄₇	#391	1	1	yes	characteristic
139	C₄×C₄₇	—	4	4	no	—
140	C₄×C₄₇	#397	1	2	yes	—
141	C₄×C₄₇	#398	1	2	yes	—
142	C₄×C₄₇	#399	1	2	yes	—
143	C₄×C₄₇	#400	1	2	yes	—
144	Dic₄₇	—	8	2	no	—
145	Dic₄₇	—	8	2	no	—
146	#2316	—	47	1	no	—
147	C₂×#344	—	47	2	no	—
148	C₂×#344	—	47	2	no	—
149	C₂×#343	—	47	1	no	—
150	C₂×#343	D₄₇	1	1	yes	characteristic
151	C₂×C₄₇	—	2	1	no	—
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	—	2	1	no	—
155	C₂×C₄₇	#1445	1	1	yes	characteristic
156	C₂×C₄₇	#1829	1	1	yes	characteristic
157	C₂×C₄₇	#1830	1	1	yes	characteristic
158	C₂²×M₁₆	—	47	2	no	—
159	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
160	C₂×#117	—	47	1	no	—
161	C₂×#117	—	47	1	no	—
162	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
163	#389	—	94	2	no	—
164	#389	—	94	2	no	—
165	#380	—	47	4	no	—
166	#344	—	94	4	no	—
167	#344	—	94	4	no	—
168	#343	—	2	2	no	—
169	#343	—	94	2	no	—
170	C₄₇	#11352	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	C₄×D₄₇	1	2	yes	—
172	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	47	4	no	—
179	C₂×M₁₆	—	94	2	no	—
180	C₂×M₁₆	—	94	2	no	—
181	C₂×M₁₆	—	94	2	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	—	94	2	no	—
184	C₂×M₁₆	Dic₉₄	1	4	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	94	2	no	—
187	#117	—	94	2	no	—
188	#117	—	94	2	no	—
189	#117	—	94	2	no	—
190	#117	—	94	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	94	4	no	—
194	C₂×C₄⋊C₄	—	94	1	no	—
195	C₂×C₄⋊C₄	—	94	1	no	—
196	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
197	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
198	C₂×C₂²⋊C₄	—	47	2	no	—
199	C₂×C₂²⋊C₄	—	47	2	no	—
200	C₂×C₄²	—	94	1	no	—
201	C₂×C₄²	—	94	1	no	—
202	#95	—	94	4	no	—
203	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	8	2	no	—
207	C₂²×C₄	—	94	2	no	—
208	C₂²×C₄	—	94	2	no	—
209	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
210	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
211	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
212	C₂²×C₄	#10625350	1	2	yes	—
213	C₂²×C₄	D₄×D₄₇	1	2	yes	—
214	C₂²×C₄	#10625352	1	2	yes	—
215	C₂²×C₄	#10625352	1	2	yes	—
216	C₂²×C₄	#10625354	1	2	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	94	4	no	—
223	M₁₆	—	94	4	no	—
224	M₁₆	—	188	2	no	—
225	M₁₆	—	188	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	188	2	no	—
231	C₂×C₈	—	188	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	376	4	no	—
235	C₄⋊C₄	—	188	2	no	—
236	C₄⋊C₄	—	188	1	no	—
237	C₄⋊C₄	—	188	1	no	—
238	C₂²⋊C₄	—	94	4	no	—
239	C₂²⋊C₄	—	94	4	no	—
240	C₄²	—	188	2	no	—
241	C₄²	—	188	1	no	—
242	C₄²	—	188	1	no	—
243	C₂³	—	4	1	no	—
244	C₂³	—	4	1	no	—
245	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
246	C₂³	#13093980	1	1	yes	characteristic
247	C₂³	#13093980	1	1	yes	characteristic
248	C₂³	#13093986	1	2	yes	—
249	C₂³	C₄×Dic₉₄	1	2	yes	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	188	2	no	—
255	C₂×C₄	—	188	2	no	—
256	C₂×C₄	—	188	2	no	—
257	C₂×C₄	—	188	2	no	—
258	C₂×C₄	—	188	2	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	#13093999	1	2	yes	—
261	C₂×C₄	#13094001	1	2	yes	—
262	C₂×C₄	#13094001	1	2	yes	—
263	C₂×C₄	#13094003	1	2	yes	—
264	C₂×C₄	#13094005	1	2	yes	—
265	C₂×C₄	#13094006	1	2	yes	—
266	C₂×C₄	#13094006	1	2	yes	—
267	C₂×C₄	#13094008	1	2	yes	—
268	C₂×C₄	#13094012	1	1	yes	characteristic
269	C₂×C₄	#13094013	1	1	yes	characteristic
270	C₂×C₄	#13094015	1	2	yes	—
271	C₂×C₄	#13094015	1	2	yes	—
272	C₂×C₄	#13094019	1	2	yes	—
273	C₂×C₄	#13094019	1	2	yes	—
274	C₂×C₄	#13094020	1	1	yes	characteristic
275	C₂×C₄	#13094024	1	1	yes	characteristic
276	C₈	—	4	4	no	—
277	C₈	—	188	4	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	#15688984	1	1	yes	characteristic
287	C₂²	#15689006	1	2	yes	—
288	C₂²	#15689006	1	2	yes	—
289	C₂²	#15689024	1	1	yes	characteristic
290	C₂²	#15689029	1	1	yes	characteristic
291	C₄	—	4	4	no	—
292	C₄	—	376	2	no	—
293	C₄	—	376	2	no	—
294	C₄	#15689050	1	2	yes	—
295	C₄	#15689055	1	2	yes	—
296	C₄	#15689059	1	2	yes	—
297	C₄	#15689062	1	2	yes	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	1	no	—
302	C₂	#20241657	1	1	yes	characteristic
303	C₂	#20243497	1	1	yes	characteristic
304	C₂	#20243500	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688984
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#343	C₂
socle	C₂²×C₄₇	#381
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#343	D₄₇
hypocenter	C₄₇	#11352
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#343

Complements

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

N	H ≅ G/N	type
170. C₄₇	#11352	semidirect

Maximal subgroups

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

subgroup	index
#20246006	2
C₂×#15688895	2
C₂×#15688894	2
C₂×#15688892	2
C₂×C₄₇×#343	2
#11352	47