Subgroups — #26122116

Order 12032.

898 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	2	yes	maximal
2	C₂×#15688892	C₂	1	4	yes	maximal
3	C₂×C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	C₂²	1	4	yes	—
5	#15689147	—	2	4	no	—
6	#15689147	C₄	1	8	yes	complemented
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
9	#15688892	—	2	8	no	—
10	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
14	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
15	C₂×#95×C₄₇	C₄	1	2	yes	—
16	C₂×#10625329	—	2	4	no	—
17	C₂×#10625329	—	2	8	no	—
18	C₂×#10625329	—	2	8	no	—
19	C₂×#10625329	C₂×C₄	1	8	yes	—
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₇	D₄	1	4	yes	—
30	C₂×C₄⋊C₄×C₄₇	Q₈	1	4	yes	—
31	C₂×C₄²×C₄₇	—	2	2	no	—
32	C₂×C₄²×C₄₇	—	2	2	no	—
33	C₂×C₄²×C₄₇	—	4	4	no	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₄₇	D₄	1	2	yes	—
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
39	#95×C₄₇	—	2	4	no	—
40	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	2	4	no	—
45	C₂²×C₄×C₄₇	—	4	2	no	—
46	C₂²×C₄×C₄₇	—	4	4	no	—
47	C₂²×C₄×C₄₇	—	4	4	no	—
48	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
49	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	4	yes	—
63	C₄⋊C₄×C₄₇	—	2	8	no	—
64	C₄⋊C₄×C₄₇	—	2	8	no	—
65	C₄²×C₄₇	—	4	4	no	—
66	C₄²×C₄₇	—	4	4	no	—
67	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
68	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
69	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
70	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
71	#10625329	—	2	16	no	—
72	#10625329	—	4	4	no	—
73	#10625329	—	4	4	no	—
74	C₂×C₄₇⋊C₈	—	2	8	no	—
75	C₂×C₄₇⋊C₈	—	4	8	no	—
76	C₂³×C₄₇	—	2	1	no	—
77	C₂³×C₄₇	—	2	1	no	—
78	C₂³×C₄₇	—	2	2	no	—
79	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
80	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
83	C₂³×C₄₇	#117	1	2	yes	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	4	2	no	—
91	C₂×C₄×C₄₇	—	4	2	no	—
92	C₂×C₄×C₄₇	—	4	8	no	—
93	C₂×C₄×C₄₇	—	4	4	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	4	8	no	—
96	C₂×C₄×C₄₇	#95	1	2	yes	—
97	C₂×C₄×C₄₇	#95	1	4	yes	—
98	C₂×C₄×C₄₇	#95	1	2	yes	—
99	C₂×C₄×C₄₇	#100	1	2	yes	—
100	C₂×C₄×C₄₇	#101	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
105	C₂×C₄×C₄₇	#121	1	4	yes	—
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₄₇	#124	1	2	yes	—
108	C₂×C₄×C₄₇	#125	1	2	yes	—
109	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
112	C₄₇⋊C₈	—	4	8	no	—
113	#10847	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	1	no	—
119	C₂²×C₄₇	—	2	1	no	—
120	C₂²×C₄₇	—	2	1	no	—
121	C₂²×C₄₇	—	2	1	no	—
122	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
123	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
124	C₂²×C₄₇	#380	1	2	yes	—
125	C₂²×C₄₇	#382	1	1	yes	characteristic
126	C₂²×C₄₇	#384	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
129	C₂²×C₄₇	#413	1	2	yes	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	4	no	—
132	C₄×C₄₇	—	4	8	no	—
133	C₄×C₄₇	#343	1	4	yes	complemented
134	C₄×C₄₇	#389	1	4	yes	—
135	C₄×C₄₇	#390	1	2	yes	—
136	C₄×C₄₇	#391	1	2	yes	—
137	C₂×#423	—	47	2	no	—
138	C₂×C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic
139	C₂×#343	—	47	4	no	—
140	C₂×C₄₇	—	2	2	no	—
141	C₂×C₄₇	—	2	1	no	—
142	C₂×C₄₇	—	2	1	no	—
143	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
145	C₂×C₄₇	#1742	1	1	yes	characteristic
146	C₂×C₄₇	#1800	1	2	yes	—
147	C₂×C₄₇	#1801	1	2	yes	—
148	C₂²×M₁₆	—	47	4	no	—
149	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
150	C₂²×C₄²	Dic₄₇	1	2	yes	—
151	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
152	#423	—	47	8	no	—
153	#423	—	94	4	no	—
154	C₂×C₄⋊C₈	—	94	2	no	—
155	C₂×C₄⋊C₈	—	94	2	no	—
156	C₄×C₄⋊C₄	—	2	2	no	—
157	C₄×C₄⋊C₄	—	2	2	no	—
158	C₂×#95	Dic₄₇	1	2	yes	—
159	#343	—	94	8	no	—
160	C₄₇	#10847	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
162	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
163	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
164	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
165	C₂×M₁₆	—	47	8	no	—
166	C₂×M₁₆	—	94	4	no	—
167	C₂×M₁₆	—	94	8	no	—
168	C₂×M₁₆	—	94	8	no	—
169	C₂²×C₈	—	94	4	no	—
170	C₂×C₄⋊C₄	—	2	4	no	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₄⋊C₄	Dic₉₄	1	4	yes	—
173	C₂×C₄⋊C₄	D₁₈₈	1	4	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	—	4	4	no	—
177	C₂×C₄²	Dic₉₄	1	2	yes	—
178	C₂×C₄²	C₄×D₄₇	1	4	yes	—
179	C₂×C₄²	D₁₈₈	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
181	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
182	C₄⋊C₈	—	94	8	no	—
183	C₄⋊C₈	—	94	8	no	—
184	#95	—	2	4	no	—
185	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
194	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
195	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
196	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
197	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
198	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
199	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
202	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
203	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
204	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
206	C₂²×C₄	#10625350	1	4	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
208	M₁₆	—	94	16	no	—
209	M₁₆	—	188	4	no	—
210	M₁₆	—	188	4	no	—
211	C₂×C₈	—	94	8	no	—
212	C₂×C₈	—	188	8	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄²	—	4	4	no	—
216	C₄²	—	4	4	no	—
217	C₄²	Dic₄₇⋊C₄	1	4	yes	—
218	C₄²	Dic₄₇⋊C₄	1	4	yes	—
219	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
220	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	2	no	—
224	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
225	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
226	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
227	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
228	C₂³	#13093983	1	2	yes	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	C₄₇⋊#95	1	2	yes	—
242	C₂×C₄	C₄₇⋊#95	1	4	yes	—
243	C₂×C₄	C₄₇⋊#95	1	2	yes	—
244	C₂×C₄	#13093941	1	2	yes	—
245	C₂×C₄	#13093943	1	2	yes	—
246	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
247	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
248	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
249	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
250	C₂×C₄	#13093998	1	4	yes	—
251	C₂×C₄	#13094003	1	4	yes	—
252	C₂×C₄	#13094012	1	2	yes	—
253	C₂×C₄	#13094016	1	2	yes	—
254	C₂×C₄	#13094021	1	1	yes	characteristic
255	C₂×C₄	#13094024	1	2	yes	—
256	C₂×C₄	#13094024	1	1	yes	characteristic
257	C₈	—	188	8	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
267	C₂²	C₄×C₄⋊Dic₄₇	1	2	yes	—
268	C₂²	#15688983	1	2	yes	—
269	C₂²	#15688989	1	1	yes	characteristic
270	C₂²	#15688993	1	1	yes	characteristic
271	C₂²	C₂×#13093941	1	1	yes	characteristic
272	C₂²	C₂×#13093943	1	1	yes	characteristic
273	C₂²	#15689107	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	4	8	no	—
277	C₄	#15688892	1	4	yes	complemented
278	C₄	#15689021	1	4	yes	—
279	C₄	#15689025	1	2	yes	—
280	C₄	#15689029	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	#20241618	1	1	yes	characteristic
285	C₂	C₂×#15688892	1	1	yes	characteristic
286	C₂	#20243127	1	1	yes	characteristic
287	C₂	#20243380	1	2	yes	—
288	C₂	#20243384	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄×C₄⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#10847
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. #15689147	C₄	semidirect
133. C₄×C₄₇	#343	semidirect
160. C₄₇	#10847	semidirect
277. C₄	#15688892	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×#15688892	2
C₂×C₄×C₄⋊C₄×C₄₇	2
#10847	47