Subgroups — #26219556

Order 12032.

1038 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689150	C₂	1	2	yes	maximal
2	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#20244107	C₂	1	8	yes	maximal, complemented
5	#20243085	C₂	1	2	yes	maximal
6	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15689150	—	2	8	no	—
11	#15689147	—	2	4	no	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
14	C₂×C₄×C₄₇⋊C₈	—	2	4	no	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
16	#15688878	C₂²	1	8	yes	—
17	C₂×#117×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#384	C₄	1	2	yes	—
20	C₄₇×#382	C₄	1	2	yes	—
21	C₄₇×#380	C₄	1	4	yes	—
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
23	C₂×#10625329	—	2	4	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
31	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
37	C₄×C₄₇⋊C₈	—	2	8	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	#117×C₄₇	—	2	4	no	—
41	#117×C₄₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
44	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	4	yes	—
48	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
51	#95×C₄₇	C₂×C₄	1	8	yes	—
52	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	8	no	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₄₇	—	2	8	no	—
67	C₄⋊C₄×C₄₇	—	2	16	no	—
68	C₂²⋊C₄×C₄₇	—	2	8	no	—
69	C₄²×C₄₇	—	2	4	no	—
70	C₄²×C₄₇	—	2	4	no	—
71	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
72	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
73	C₄²×C₄₇	C₂×D₄	1	4	yes	—
74	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
75	#10625329	—	4	8	no	—
76	C₂×C₄₇⋊C₈	—	2	8	no	—
77	C₂×C₄₇⋊C₈	—	2	8	no	—
78	C₂×C₄₇⋊C₈	—	2	2	no	—
79	C₂×C₄₇⋊C₈	—	2	2	no	—
80	C₂×C₄₇⋊C₈	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	2	2	no	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
86	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₇	C₄×D₄	1	2	yes	—
90	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
91	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	16	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	2	16	no	—
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₄₇	#121	1	4	yes	—
102	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
103	C₂×C₄×C₄₇	#125	1	4	yes	—
104	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
105	C₂×C₄×C₄₇	#131	1	8	yes	—
106	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
112	C₄₇⋊C₈	—	2	4	no	—
113	C₄₇⋊C₈	—	2	4	no	—
114	C₄₇⋊C₈	—	4	4	no	—
115	#21143	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	#389	1	4	yes	—
120	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×#131	1	4	yes	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	—	4	8	no	—
126	C₄×C₄₇	#389	1	4	yes	—
127	C₄×C₄₇	#424	1	8	yes	complemented
128	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
129	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
131	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×#424	—	47	2	no	—
133	C₂×#423	—	47	1	no	—
134	C₂²×C₄×C₈	—	47	1	no	—
135	#2285	D₄₇	1	1	yes	characteristic
136	#1939	—	47	8	no	—
137	#1729	—	47	2	no	—
138	C₂×C₄₇	—	2	4	no	—
139	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
140	C₂×C₄₇	C₂×#424	1	2	yes	—
141	C₂×C₄₇	#2931	1	2	yes	—
142	C₂×C₄₇	#2960	1	2	yes	—
143	C₂²×M₁₆	—	47	2	no	—
144	C₂³×C₈	—	47	1	no	—
145	C₂³×C₈	—	47	1	no	—
146	C₂×#117	C₂×D₄₇	1	2	yes	—
147	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
148	#424	—	94	8	no	—
149	#423	—	94	4	no	—
150	C₂×C₄⋊C₈	—	47	4	no	—
151	C₂×C₄⋊C₈	—	47	8	no	—
152	C₂×C₄×C₈	—	47	4	no	—
153	C₂×C₄×C₈	—	94	4	no	—
154	#384	Dic₄₇	1	2	yes	—
155	#382	Dic₄₇	1	2	yes	—
156	#380	Dic₄₇	1	4	yes	—
157	C₄×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
158	#336	—	47	8	no	—
159	C₄₇	#21143	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
161	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
162	C₂×M₁₆	—	94	4	no	—
163	C₂×M₁₆	—	94	4	no	—
164	C₂×M₁₆	—	94	4	no	—
165	C₂²×C₈	—	47	4	no	—
166	C₂²×C₈	—	47	4	no	—
167	C₂²×C₈	—	47	2	no	—
168	C₂²×C₈	—	47	2	no	—
169	C₂²×C₈	—	47	4	no	—
170	C₂²×C₈	—	94	2	no	—
171	C₂²×C₈	—	94	2	no	—
172	C₂²×C₈	—	94	2	no	—
173	C₂²×C₈	—	94	2	no	—
174	#117	—	2	4	no	—
175	#117	—	2	4	no	—
176	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
177	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
178	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
179	C₂×C₄²	Dic₉₄	1	4	yes	—
180	C₂×C₄²	D₁₈₈	1	4	yes	—
181	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
184	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
185	C₄⋊C₈	—	94	16	no	—
186	C₄⋊C₈	—	94	8	no	—
187	C₄×C₈	—	94	8	no	—
188	#95	C₂×Dic₄₇	1	8	yes	—
189	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
192	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
193	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
194	C₂²×C₄	D₄×D₄₇	1	2	yes	—
195	C₂²×C₄	#10625352	1	2	yes	—
196	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
197	C₂²×C₄	#10625354	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
203	M₁₆	—	188	8	no	—
204	C₂×C₈	—	47	4	no	—
205	C₂×C₈	—	47	4	no	—
206	C₂×C₈	—	94	8	no	—
207	C₂×C₈	—	94	8	no	—
208	C₂×C₈	—	94	2	no	—
209	C₂×C₈	—	94	2	no	—
210	C₂×C₈	—	94	2	no	—
211	C₂×C₈	—	94	2	no	—
212	C₂×C₈	—	94	4	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	94	4	no	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	16	no	—
217	C₂²⋊C₄	—	2	8	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	C₄⋊Dic₄₇	1	4	yes	—
221	C₄²	C₄⋊Dic₄₇	1	4	yes	—
222	C₄²	C₂×Dic₉₄	1	4	yes	—
223	C₄²	C₂×D₁₈₈	1	4	yes	—
224	C₂³	—	2	4	no	—
225	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
226	C₂³	D₄×Dic₄₇	1	2	yes	—
227	C₂³	Q₈×Dic₄₇	1	2	yes	—
228	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	16	no	—
233	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
235	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
236	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
237	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
238	C₂×C₄	#13093999	1	4	yes	—
239	C₂×C₄	#13094004	1	4	yes	—
240	C₂×C₄	#13094015	1	4	yes	—
241	C₂×C₄	#13094023	1	4	yes	—
242	C₂×C₄	#13094031	1	8	yes	—
243	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
245	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
247	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
248	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
249	C₈	—	94	4	no	—
250	C₈	—	94	4	no	—
251	C₈	—	188	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	#15689022	1	4	yes	—
256	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
257	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
258	C₂²	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
259	C₂²	C₂×#13094031	1	4	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	#15689022	1	4	yes	—
263	C₄	#15689150	1	8	yes	complemented
264	C₄	C₂×#13093999	1	1	yes	characteristic
265	C₄	C₂×#13094004	1	1	yes	characteristic
266	C₄	C₂×#13094015	1	1	yes	characteristic
267	C₄	C₂×#13094023	1	1	yes	characteristic
268	C₂	—	2	4	no	—
269	C₂	C₂×#15689022	1	1	yes	characteristic
270	C₂	C₂×#15689150	1	2	yes	—
271	C₂	#20252100	1	2	yes	—
272	C₂	#20252323	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#2285	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#21143
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
4. #20244107	C₂	semidirect
127. C₄×C₄₇	#424	semidirect
159. C₄₇	#21143	semidirect
263. C₄	#15689150	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689150	2
C₂×#15689147	2
C₂²×C₄×C₄₇⋊C₈	2
#20244107	2
#20243085	2
C₄₇×#2285	2
#21143	47