Subgroups — #26216699

Order 12032.

1086 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×#15689161	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689159	C₂	1	2	yes	maximal
3	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic
4	#20245791	C₂	1	2	yes	maximal
5	#20243555	C₂	1	4	yes	maximal, complemented
6	#20243553	C₂	1	4	yes	maximal, complemented
7	C₂×C₄₇×#427	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#13093983	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15689161	—	2	4	no	—
12	#15689159	—	2	8	no	—
13	C₂×#13093958	C₂²	1	2	yes	—
14	C₂×#13093957	C₂²	1	4	yes	—
15	C₂×#13093956	C₂²	1	2	yes	—
16	C₂×#13093955	C₂²	1	4	yes	—
17	C₂×#13093934	C₄	1	8	yes	complemented
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×#13093934	C₂²	1	2	yes	—
20	#15688992	C₂²	1	4	yes	—
21	#15688988	C₂²	1	4	yes	—
22	#15688981	—	2	4	no	—
23	C₄×Dic₄₇⋊C₄	—	2	4	no	—
24	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₄₇	C₂²	1	2	yes	—
26	C₄₇×#427	—	2	4	no	—
27	C₂×#107×C₄₇	C₂²	1	2	yes	—
28	C₂×#106×C₄₇	C₂²	1	2	yes	—
29	C₂³×Dic₄₇	D₄	1	2	yes	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
32	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	#13093983	—	2	4	no	—
35	#13093983	—	2	4	no	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	C₂×Dic₄₇⋊C₄	—	2	8	no	—
38	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
42	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
44	#13093958	—	2	4	no	—
45	#13093957	—	2	8	no	—
46	#13093956	—	2	4	no	—
47	#13093955	—	2	8	no	—
48	#13093934	C₂×C₄	1	8	yes	—
49	#13093934	C₂×C₄	1	8	yes	—
50	C₄₇⋊#95	—	2	8	no	—
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₄₇	D₄	1	4	yes	—
54	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	#117×C₄₇	—	2	4	no	—
58	#117×C₄₇	—	2	4	no	—
59	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
60	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
61	C₂×C₄²×C₄₇	—	2	2	no	—
62	#107×C₄₇	—	2	4	no	—
63	#106×C₄₇	—	2	4	no	—
64	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
77	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
79	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₂²⋊C₄×C₄₇	—	4	4	no	—
88	C₄²×C₄₇	—	4	4	no	—
89	C₂²⋊Dic₄₇	—	4	4	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	Dic₄₇⋊C₄	—	4	16	no	—
93	C₄×Dic₄₇	—	4	4	no	—
94	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
95	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
96	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
97	C₂×C₄₇⋊C₈	C₄⋊C₄	1	16	yes	complemented
98	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
100	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
101	C₂³×C₄₇	—	2	4	no	—
102	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₄₇	#127	1	2	yes	—
104	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
105	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	4	8	no	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₇	#127	1	2	yes	—
112	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
113	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
114	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×Dic₄₇	—	2	8	no	—
122	C₂×Dic₄₇	—	4	8	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	8	no	—
125	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
126	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
127	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
128	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
129	#20799	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	—	2	4	no	—
133	C₂²×C₄₇	#384	1	4	yes	—
134	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	C₂²×C₄₇	#575	1	4	yes	—
138	C₄×C₄₇	—	8	4	no	—
139	C₄×C₄₇	#384	1	4	yes	—
140	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	Dic₄₇	—	2	8	no	—
145	Dic₄₇	—	8	4	no	—
146	C₂×#427	—	47	1	no	—
147	C₂×#427	—	47	2	no	—
148	C₂×#427	D₄₇	1	1	yes	characteristic
149	C₂²×#97	—	47	1	no	—
150	#2285	—	47	2	no	—
151	#1847	—	47	4	no	—
152	#1847	—	47	4	no	—
153	C₂×C₄₇	—	2	4	no	—
154	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
155	C₂×C₄₇	#2911	1	2	yes	—
156	C₂×C₄₇	#3150	1	2	yes	—
157	C₂×C₄₇	#3162	1	2	yes	—
158	C₂³×C₈	—	47	1	no	—
159	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
160	C₂×#117	—	47	2	no	—
161	C₂×#117	C₂×D₄₇	1	2	yes	—
162	C₂²×C₄²	—	47	1	no	—
163	#427	—	2	4	no	—
164	#427	—	94	8	no	—
165	#427	—	94	4	no	—
166	C₂×#107	—	47	2	no	—
167	C₂×#107	—	47	4	no	—
168	C₂×#107	C₂×D₄₇	1	2	yes	—
169	C₂×#106	—	47	2	no	—
170	C₂×#106	—	47	4	no	—
171	C₂×#106	C₂×D₄₇	1	2	yes	—
172	C₂×#97	—	47	8	no	—
173	C₂×#97	—	47	2	no	—
174	C₂×#97	—	47	2	no	—
175	#384	—	47	4	no	—
176	#382	—	47	4	no	—
177	#380	—	94	4	no	—
178	C₄×C₄⋊C₄	—	94	4	no	—
179	C₄₇	#20799	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	—	47	2	no	—
181	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
182	C₂²×C₈	—	47	4	no	—
183	C₂²×C₈	—	47	4	no	—
184	C₂²×C₈	—	47	4	no	—
185	C₂²×C₈	—	47	2	no	—
186	C₂²×C₈	Dic₉₄	1	4	yes	—
187	C₂²×C₈	C₄×D₄₇	1	4	yes	—
188	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
189	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
191	#117	—	2	4	no	—
192	#117	—	2	4	no	—
193	#117	—	94	4	no	—
194	#117	—	94	4	no	—
195	C₂×C₄⋊C₄	—	47	4	no	—
196	C₂×C₄⋊C₄	—	94	8	no	—
197	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	94	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	47	4	no	—
202	C₂×C₄²	—	47	2	no	—
203	C₂×C₄²	—	47	4	no	—
204	C₂×C₄²	—	47	2	no	—
205	C₂×C₄²	—	94	2	no	—
206	#107	—	2	4	no	—
207	#107	—	94	8	no	—
208	#107	—	94	4	no	—
209	#106	—	2	4	no	—
210	#106	—	94	8	no	—
211	#106	—	94	4	no	—
212	#97	—	47	8	no	—
213	#97	—	47	8	no	—
214	#95	—	94	8	no	—
215	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	47	4	no	—
218	C₂²×C₄	—	47	8	no	—
219	C₂²×C₄	—	94	4	no	—
220	C₂²×C₄	—	188	4	no	—
221	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
223	C₂²×C₄	D₄×D₄₇	1	2	yes	—
224	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
225	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
226	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	C₂×C₈	—	47	16	no	—
230	C₂×C₈	—	47	4	no	—
231	C₂×C₈	—	47	4	no	—
232	C₂×C₈	—	47	4	no	—
233	C₂×C₈	Dic₄₇⋊C₄	1	16	yes	complemented
234	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
235	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
238	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	94	4	no	—
243	C₄⋊C₄	—	94	4	no	—
244	C₄⋊C₄	—	188	16	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₂²⋊C₄	—	188	4	no	—
247	C₄²	—	4	4	no	—
248	C₄²	—	47	8	no	—
249	C₄²	—	47	4	no	—
250	C₄²	—	47	4	no	—
251	C₄²	—	188	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
254	C₂³	#13094020	1	2	yes	—
255	C₂³	#13094023	1	2	yes	—
256	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	47	8	no	—
260	C₂×C₄	—	94	8	no	—
261	C₂×C₄	—	188	8	no	—
262	C₂×C₄	—	188	4	no	—
263	C₂×C₄	—	188	8	no	—
264	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
265	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
266	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
267	C₂×C₄	#13094020	1	2	yes	—
268	C₂×C₄	#13094022	1	4	yes	—
269	C₂×C₄	#13094023	1	2	yes	—
270	C₂×C₄	#13094023	1	4	yes	—
271	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
277	C₈	—	47	8	no	—
278	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
279	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	#15688992	1	4	yes	—
284	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
285	C₂²	C₂×#13094020	1	1	yes	characteristic
286	C₂²	C₂×#13094023	1	1	yes	characteristic
287	C₂²	#15689920	1	4	yes	—
288	C₄	—	8	4	no	—
289	C₄	—	94	8	no	—
290	C₄	—	376	4	no	—
291	C₄	#15688992	1	4	yes	—
292	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
293	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
294	C₄	C₂×#13094022	1	1	yes	characteristic
295	C₄	C₂×#13094023	1	1	yes	characteristic
296	C₂	—	2	4	no	—
297	C₂	C₂×#15688992	1	1	yes	characteristic
298	C₂	#20251976	1	2	yes	—
299	C₂	#20253831	1	2	yes	—
300	C₂	#20253911	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	#20799
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
5. #20243555	C₂	semidirect
6. #20243553	C₂	semidirect
17. C₂×#13093934	C₄	semidirect
78. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
97. C₂×C₄₇⋊C₈	C₄⋊C₄	semidirect
179. C₄₇	#20799	semidirect
233. C₂×C₈	Dic₄₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
C₂×#15689159	2
C₂²×#13093934	2
#20245791	2
#20243555	2
#20243553	2
C₂×C₄₇×#427	2
#20799	47