Subgroups — #26246261

Order 12032.

648 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689172	C₂	1	1	yes	maximal, characteristic
2	#20244968	C₂	1	8	yes	maximal, complemented
3	C₂×#15688893	C₂	1	2	yes	maximal
4	#20243115	C₂	1	2	yes	maximal
5	C₂×C₄₇⋊#338	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#2905	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂²×#10625329	C₂²	1	2	yes	—
9	#15689172	—	2	2	no	—
10	#15689172	C₂²	1	4	yes	—
11	C₂×#13093960	—	2	2	no	—
12	#15688893	C₂²	1	8	yes	—
13	#15688888	C₂²	1	8	yes	—
14	C₄₇⋊#338	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#424	—	2	2	no	—
19	C₄₇×#424	C₂²	1	4	yes	—
20	C₄₇×#423	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625329	—	2	2	no	—
31	C₂×#10625329	C₂³	1	4	yes	—
32	C₂×#10625329	C₂³	1	4	yes	—
33	C₂²×C₄₇⋊C₈	—	2	2	no	—
34	C₂²×C₄₇⋊C₈	—	2	2	no	—
35	#13093960	—	2	8	no	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	2	1	no	—
40	C₂×M₁₆×C₄₇	—	2	1	no	—
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
44	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₄₇	D₄	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
47	#117×C₄₇	—	2	1	no	—
48	#117×C₄₇	—	2	1	no	—
49	#117×C₄₇	Q₈	1	2	yes	—
50	#117×C₄₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
52	C₂×C₂²⋊C₄×C₄₇	Q₈	1	2	yes	—
53	C₂×C₄²×C₄₇	—	2	1	no	—
54	C₄⋊C₈×C₄₇	—	2	8	no	—
55	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	—	2	4	no	—
69	M₁₆×C₄₇	—	4	1	no	—
70	M₁₆×C₄₇	—	4	1	no	—
71	M₁₆×C₄₇	—	4	2	no	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₂²⋊C₄×C₄₇	—	2	2	no	—
80	C₂²⋊C₄×C₄₇	C₂×Q₈	1	4	yes	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	2	4	no	—
85	#10625329	—	2	4	no	—
86	#10625329	—	2	4	no	—
87	#10625329	—	4	2	no	—
88	#10625329	—	4	2	no	—
89	#10625329	—	4	2	no	—
90	#10625329	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂³×C₄₇	—	8	1	no	—
98	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₄₇	#125	1	2	yes	—
101	C₂³×C₄₇	#125	1	2	yes	—
102	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	8	2	no	—
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#123	1	2	yes	—
110	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#124	1	2	yes	—
112	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#125	1	2	yes	—
114	C₂×C₄×C₄₇	#125	1	2	yes	—
115	C₂×C₄×C₄₇	#126	1	2	yes	—
116	C₂×C₄×C₄₇	#126	1	2	yes	—
117	C₂×C₄×C₄₇	#126	1	2	yes	—
118	C₂×C₄×C₄₇	#126	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
123	C₈×C₄₇	—	2	4	no	—
124	C₈×C₄₇	—	4	2	no	—
125	C₄₇⋊C₈	—	4	4	no	—
126	C₄₇⋊C₈	—	4	4	no	—
127	#23753	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	4	1	no	—
129	C₂²×C₄₇	—	4	1	no	—
130	C₂²×C₄₇	—	4	2	no	—
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	—	4	1	no	—
133	C₂²×C₄₇	#400	1	1	yes	characteristic
134	C₂²×C₄₇	#400	1	1	yes	characteristic
135	C₂²×C₄₇	#400	1	2	yes	—
136	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#125	1	2	yes	—
138	C₄×C₄₇	—	4	4	no	—
139	C₄×C₄₇	#400	1	2	yes	—
140	C₄×C₄₇	#400	1	1	yes	characteristic
141	C₄×C₄₇	#400	1	1	yes	characteristic
142	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
144	C₄×C₄₇	C₂×#126	1	2	yes	—
145	C₂×#430	—	47	1	no	—
146	#2905	D₄₇	1	1	yes	characteristic
147	#2103	—	47	8	no	—
148	C₂×#344	—	47	2	no	—
149	#1738	—	47	2	no	—
150	C₂×#338	—	47	1	no	—
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	—	2	1	no	—
156	C₂×C₄₇	C₂×#400	1	1	yes	characteristic
157	C₂²×M₁₆	—	47	2	no	—
158	C₂²×M₁₆	—	47	2	no	—
159	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
160	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
161	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
162	#430	—	47	4	no	—
163	#430	—	94	2	no	—
164	C₂×#108	—	94	2	no	—
165	#424	—	2	2	no	—
166	#424	C₂×D₄₇	1	4	yes	—
167	#423	—	2	2	no	—
168	#344	—	47	8	no	—
169	#341	—	47	8	no	—
170	#338	—	47	4	no	—
171	C₄₇	#23753	1	1	yes	characteristic, complemented, Sylow
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	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	47	4	no	—
180	C₂×M₁₆	—	47	4	no	—
181	C₂×M₁₆	—	94	2	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	—	94	2	no	—
184	C₂×M₁₆	—	94	2	no	—
185	C₂×M₁₆	—	94	2	no	—
186	C₂×M₁₆	—	94	2	no	—
187	C₂×M₁₆	—	94	2	no	—
188	C₂×M₁₆	—	94	2	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
192	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
193	C₂²×C₈	—	94	2	no	—
194	C₂²×C₈	—	94	2	no	—
195	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
196	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
197	#117	—	2	1	no	—
198	#117	—	2	1	no	—
199	#117	Dic₉₄	1	2	yes	—
200	#117	C₂²×D₄₇	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₂²⋊C₄	Dic₉₄	1	2	yes	—
203	C₂×C₄²	—	2	1	no	—
204	#108	—	94	8	no	—
205	C₄⋊C₈	—	2	8	no	—
206	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
209	C₂²×C₄	#10625350	1	1	yes	characteristic
210	C₂²×C₄	#10625350	1	1	yes	characteristic
211	C₂²×C₄	#10625352	1	2	yes	—
212	C₂²×C₄	#10625352	1	2	yes	—
213	C₂²×C₄	#10625354	1	2	yes	—
214	C₂²×C₄	#10625354	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	94	4	no	—
224	M₁₆	—	94	4	no	—
225	M₁₆	—	94	4	no	—
226	M₁₆	—	94	4	no	—
227	M₁₆	—	188	2	no	—
228	M₁₆	—	188	2	no	—
229	M₁₆	—	188	2	no	—
230	M₁₆	—	188	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	94	4	no	—
237	C₂×C₈	—	188	2	no	—
238	C₂×C₈	—	188	2	no	—
239	C₂×C₈	—	188	2	no	—
240	C₂×C₈	—	188	2	no	—
241	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	C₂×Dic₉₄	1	4	yes	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₂³	—	8	1	no	—
249	C₂³	#13094003	1	1	yes	characteristic
250	C₂³	#13094003	1	1	yes	characteristic
251	C₂³	#13094015	1	2	yes	—
252	C₂³	#13094015	1	2	yes	—
253	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	#13094003	1	1	yes	characteristic
257	C₂×C₄	#13094003	1	1	yes	characteristic
258	C₂×C₄	#13094005	1	1	yes	characteristic
259	C₂×C₄	#13094005	1	1	yes	characteristic
260	C₂×C₄	#13094005	1	2	yes	—
261	C₂×C₄	#13094013	1	1	yes	characteristic
262	C₂×C₄	#13094013	1	2	yes	—
263	C₂×C₄	#13094013	1	1	yes	characteristic
264	C₂×C₄	#13094015	1	2	yes	—
265	C₂×C₄	#13094015	1	2	yes	—
266	C₂×C₄	#13094019	1	2	yes	—
267	C₂×C₄	#13094019	1	2	yes	—
268	C₂×C₄	#13094019	1	2	yes	—
269	C₂×C₄	#13094019	1	2	yes	—
270	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#10625352	1	2	yes	—
272	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
273	C₂×C₄	C₂×#10625354	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	188	4	no	—
277	C₈	—	188	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	#15689062	1	1	yes	characteristic
284	C₂²	#15689062	1	1	yes	characteristic
285	C₂²	#15689062	1	2	yes	—
286	C₂²	C₂×#13094003	1	1	yes	characteristic
287	C₂²	C₂×#13094015	1	2	yes	—
288	C₄	—	4	4	no	—
289	C₄	#15689062	1	2	yes	—
290	C₄	#15689062	1	1	yes	characteristic
291	C₄	#15689062	1	1	yes	characteristic
292	C₄	C₂×#13094005	1	1	yes	characteristic
293	C₄	C₂×#13094013	1	1	yes	characteristic
294	C₄	C₂×#13094019	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	C₂×#15689062	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13094013
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2905	C₂
socle	C₂×C₄₇	C₂×#400
prime radical	C₂⁴×C₄₇	C₂×Q₈
hypercenter	#2905	D₄₇
hypocenter	C₄₇	#23753
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
2. #20244968	C₂	semidirect
171. C₄₇	#23753	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689172	2
#20244968	2
C₂×#15688893	2
#20243115	2
C₂×C₄₇⋊#338	2
C₄₇×#2905	2
#23753	47