Subgroups — #26216137

Order 12032.

1276 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689097	C₂	1	2	yes	maximal
2	#20243496	C₂	1	8	yes	maximal, complemented
3	#20243120	C₂	1	4	yes	maximal, complemented
4	C₄₇×#3428	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₂²	1	4	yes	—
6	#15689097	—	2	4	no	—
7	#15689097	—	2	4	no	—
8	C₂×#13093939	C₂²	1	8	yes	—
9	#15688889	C₂²	1	16	yes	—
10	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#520	—	2	4	no	—
12	C₄₇×#519	C₄	1	2	yes	complemented
13	C₄₇×#518	C₄	1	2	yes	complemented
14	C₄₇×#518	C₂²	1	4	yes	—
15	C₄×#41×C₄₇	—	2	4	no	—
16	C₂×#117×C₄₇	C₄	1	4	yes	complemented
17	C₂×#117×C₄₇	C₂²	1	2	yes	—
18	#13093983	—	2	4	no	—
19	#13093983	—	2	4	no	—
20	#13093983	—	2	4	no	—
21	#13093983	—	2	4	no	—
22	C₂×C₄⋊Dic₄₇	—	2	4	no	—
23	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
24	C₂×C₄×Dic₄₇	—	2	4	no	—
25	#13093939	—	2	16	no	—
26	C₂×#41×C₄₇	—	2	4	no	—
27	C₂×#41×C₄₇	—	2	4	no	—
28	C₂×#41×C₄₇	D₄	1	4	yes	—
29	C₂×#41×C₄₇	D₄	1	4	yes	—
30	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₄₇	—	2	8	no	—
36	C₄×D₄×C₄₇	—	2	8	no	—
37	C₄×D₄×C₄₇	—	2	16	no	—
38	#117×C₄₇	—	2	4	no	—
39	#117×C₄₇	—	2	4	no	—
40	#117×C₄₇	—	2	2	no	—
41	#117×C₄₇	—	2	2	no	—
42	#117×C₄₇	C₂×C₄	1	4	yes	—
43	#117×C₄₇	C₂×C₄	1	8	yes	—
44	#117×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
48	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₄₇	—	2	4	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
53	#41×C₄₇	—	2	8	no	—
54	#41×C₄₇	—	2	8	no	—
55	#41×C₄₇	—	2	8	no	—
56	#41×C₄₇	—	2	8	no	—
57	C₂×Q₈×C₄₇	—	2	4	no	—
58	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
59	C₂×D₄×C₄₇	—	2	8	no	—
60	C₂×D₄×C₄₇	—	2	4	no	—
61	C₂×D₄×C₄₇	C₂²⋊C₄	1	4	yes	—
62	C₂×D₄×C₄₇	C₂×D₄	1	8	yes	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	8	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	#41	1	4	yes	—
75	C₂²×C₄×C₄₇	#41	1	4	yes	—
76	C₂²×Dic₄₇	—	2	8	no	—
77	C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₄⋊C₄×C₄₇	—	2	8	no	—
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₂²⋊C₄×C₄₇	—	2	8	no	—
81	C₂²⋊C₄×C₄₇	—	2	4	no	—
82	C₂²⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	8	no	—
85	C₄²×C₄₇	—	2	4	no	—
86	C₂²⋊Dic₄₇	—	2	8	no	—
87	C₂²⋊Dic₄₇	—	2	8	no	—
88	C₄⋊Dic₄₇	—	4	4	no	—
89	C₄⋊Dic₄₇	—	4	4	no	—
90	C₄×Dic₄₇	—	4	4	no	—
91	C₄×Dic₄₇	—	4	4	no	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	—	8	1	no	—
94	C₂³×C₄₇	—	8	2	no	—
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
96	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
97	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
98	C₂³×C₄₇	C₂×#41	1	4	yes	—
99	Q₈×C₄₇	—	2	8	no	—
100	D₄×C₄₇	—	2	16	no	—
101	D₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	8	4	no	—
108	C₂×C₄×C₄₇	—	8	8	no	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#117	1	2	yes	—
114	C₂×C₄×C₄₇	#117	1	2	yes	—
115	C₂×C₄×C₄₇	#117	1	4	yes	—
116	C₂×C₄×C₄₇	#123	1	4	yes	—
117	C₂×C₄×C₄₇	#123	1	4	yes	—
118	C₂×C₄×C₄₇	#123	1	4	yes	—
119	C₂×C₄×C₄₇	#123	1	4	yes	—
120	C₂×Dic₄₇	—	4	8	no	—
121	C₂×Dic₄₇	—	8	8	no	—
122	#20710	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	4	no	—
124	C₂²×C₄₇	—	4	4	no	—
125	C₂²×C₄₇	—	4	2	no	—
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	—	4	4	no	—
128	C₂²×C₄₇	—	8	4	no	—
129	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#117	1	2	yes	—
131	C₂²×C₄₇	C₂×#123	1	4	yes	—
132	C₄×C₄₇	—	2	4	no	—
133	C₄×C₄₇	—	4	8	no	—
134	C₄×C₄₇	#381	1	1	yes	characteristic
135	C₄×C₄₇	#381	1	2	yes	—
136	C₄×C₄₇	#381	1	4	yes	—
137	C₄×C₄₇	#381	1	1	yes	characteristic
138	Dic₄₇	—	8	8	no	—
139	#3428	D₄₇	1	1	yes	characteristic
140	C₂×#410	—	47	2	no	—
141	#1828	—	47	8	no	—
142	#1740	—	47	4	no	—
143	C₂×C₄₇	—	2	4	no	—
144	C₂×C₄₇	—	2	1	no	—
145	C₂×C₄₇	—	2	2	no	—
146	C₂×C₄₇	—	4	4	no	—
147	C₂×C₄₇	C₂×#381	1	1	yes	characteristic
148	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
149	#520	—	2	4	no	—
150	#519	Dic₄₇	1	2	yes	complemented
151	#518	Dic₄₇	1	2	yes	complemented
152	#518	C₂×D₄₇	1	4	yes	—
153	C₄×#41	—	2	4	no	—
154	C₂×#117	—	47	4	no	—
155	C₂×#117	Dic₄₇	1	4	yes	complemented
156	C₂×#117	C₂×D₄₇	1	2	yes	—
157	#410	—	94	4	no	—
158	#410	—	94	4	no	—
159	C₂×#99	—	47	8	no	—
160	#342	—	47	16	no	—
161	C₄₇	#20710	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	2	4	no	—
163	C₂×#41	—	2	4	no	—
164	C₂×#41	C₂²⋊D₄₇	1	4	yes	—
165	C₂×#41	C₂²⋊D₄₇	1	4	yes	—
166	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
167	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
169	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
170	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
171	C₄×Q₈	—	2	8	no	—
172	C₄×D₄	—	2	8	no	—
173	C₄×D₄	—	2	16	no	—
174	#117	—	2	4	no	—
175	#117	—	2	4	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	94	4	no	—
179	#117	—	94	4	no	—
180	#117	—	94	4	no	—
181	#117	—	94	4	no	—
182	#117	C₂×Dic₄₇	1	4	yes	—
183	#117	C₂×Dic₄₇	1	8	yes	—
184	#117	C₂×Dic₄₇	1	4	yes	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	94	4	no	—
188	C₂×C₂²⋊C₄	—	47	8	no	—
189	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
190	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
191	C₂×C₄²	—	2	4	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	94	4	no	—
194	#99	—	94	16	no	—
195	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
196	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
197	#41	—	2	8	no	—
198	#41	—	2	8	no	—
199	#41	—	2	8	no	—
200	#41	—	2	8	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	C₂²⋊Dic₄₇	1	4	yes	—
203	C₂×D₄	—	2	8	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	C₂²⋊Dic₄₇	1	4	yes	—
206	C₂×D₄	C₂×C₂²⋊D₄₇	1	8	yes	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	94	8	no	—
212	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
213	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
214	C₂²×C₄	#10625350	1	4	yes	—
215	C₂²×C₄	#10625350	1	4	yes	—
216	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	8	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	188	4	no	—
225	C₄⋊C₄	—	188	4	no	—
226	C₂²⋊C₄	—	2	8	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	94	8	no	—
229	C₂²⋊C₄	—	94	8	no	—
230	C₂²⋊C₄	C₂²×Dic₄₇	1	8	yes	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	8	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	188	4	no	—
235	C₄²	—	188	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	2	no	—
239	C₂³	C₂×C₂²⋊Dic₄₇	1	4	yes	—
240	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
241	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
242	C₂³	C₂×#10625350	1	4	yes	—
243	Q₈	—	2	8	no	—
244	D₄	—	2	16	no	—
245	D₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	—	8	8	no	—
253	C₂×C₄	—	376	8	no	—
254	C₂×C₄	—	188	8	no	—
255	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
256	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
257	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
259	C₂×C₄	#13093983	1	2	yes	—
260	C₂×C₄	#13093983	1	4	yes	—
261	C₂×C₄	#13093983	1	2	yes	—
262	C₂×C₄	#13094007	1	4	yes	—
263	C₂×C₄	#13094007	1	4	yes	—
264	C₂×C₄	#13094007	1	4	yes	—
265	C₂×C₄	#13094007	1	4	yes	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
273	C₂²	C₂×#13093983	1	2	yes	—
274	C₂²	C₂×#13094007	1	4	yes	—
275	C₄	—	2	4	no	—
276	C₄	—	4	8	no	—
277	C₄	—	376	8	no	—
278	C₄	#15688986	1	2	yes	—
279	C₄	#15688986	1	1	yes	characteristic
280	C₄	#15688986	1	4	yes	—
281	C₄	#15688986	1	1	yes	characteristic
282	C₂	—	2	1	no	—
283	C₂	—	2	4	no	—
284	C₂	—	2	2	no	—
285	C₂	—	4	4	no	—
286	C₂	C₂×#15688986	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15688986
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂⁴	C₂³×D₄₇
Fitting	C₄₇×#3428	C₂
socle	C₂×C₄₇	C₂×#381
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	#3428	D₄₇
hypocenter	C₄₇	#20710
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #3428

Complements

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

N	H ≅ G/N	type
2. #20243496	C₂	semidirect
3. #20243120	C₂	semidirect
12. C₄₇×#519	C₄	semidirect
13. C₄₇×#518	C₄	semidirect
16. C₂×#117×C₄₇	C₄	semidirect
150. #519	Dic₄₇	semidirect
151. #518	Dic₄₇	semidirect
155. C₂×#117	Dic₄₇	semidirect
161. C₄₇	#20710	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689097	2
#20243496	2
#20243120	2
C₄₇×#3428	2
#20710	47