Subgroups — #26219660

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 304 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×#15689078	C₂	1	1	yes	maximal, characteristic
3	#20246006	C₂	1	2	yes	maximal
4	#20244127	C₂	1	4	yes	maximal, complemented
5	#20244122	C₂	1	4	yes	maximal, complemented
6	#20243107	C₂	1	2	yes	maximal
7	C₂²×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	1	yes	characteristic
9	C₂×#13093983	C₂²	1	2	yes	—
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	#15689161	—	2	4	no	—
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093956	C₂²	1	2	yes	—
14	#15689078	—	2	4	no	—
15	C₂×#13093934	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
17	#15689022	—	2	4	no	—
18	#15689022	C₂²	1	8	yes	—
19	#15688983	—	2	4	no	—
20	#15688885	C₂²	1	8	yes	—
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×#107×C₄₇	C₄	1	8	yes	complemented
24	C₂×#107×C₄₇	C₂²	1	2	yes	—
25	C₂×#107×C₄₇	C₂²	1	2	yes	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	Q₈	1	4	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	#13093983	—	2	2	no	—
31	#13093983	—	2	2	no	—
32	#13093983	—	2	4	no	—
33	#13093983	—	2	4	no	—
34	C₂×C₄⋊Dic₄₇	—	2	8	no	—
35	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
38	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	#13093958	—	2	4	no	—
42	#13093956	—	2	4	no	—
43	#13093934	—	2	4	no	—
44	C₈×Dic₄₇	—	2	4	no	—
45	C₄₇⋊#95	—	2	8	no	—
46	C₂³×C₄×C₄₇	—	2	2	no	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₄₇	D₄	1	4	yes	—
50	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
57	#107×C₄₇	C₂×C₄	1	8	yes	—
58	#107×C₄₇	C₂×C₄	1	8	yes	—
59	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	8	4	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
74	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
80	C₄⋊C₄×C₄₇	—	2	8	no	—
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
83	C₂²⋊Dic₄₇	—	2	4	no	—
84	C₂²⋊Dic₄₇	—	4	4	no	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	—	8	8	no	—
89	C₄×Dic₄₇	—	2	2	no	—
90	C₄×Dic₄₇	—	2	2	no	—
91	C₄×Dic₄₇	—	4	4	no	—
92	#10625329	—	2	8	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
96	C₂³×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	C₄×D₄	1	2	yes	—
99	C₂³×C₄₇	#121	1	2	yes	—
100	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
101	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	—	4	8	no	—
104	C₂×C₄×C₄₇	—	8	8	no	—
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
108	C₂×C₄×C₄₇	#121	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₄₇	#125	1	4	yes	—
111	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	2	4	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
125	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
126	C₄₇⋊C₈	—	2	4	no	—
127	#21158	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	#389	1	4	yes	—
133	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₂²×C₄₇	#576	1	2	yes	—
137	C₂²×C₄₇	#578	1	2	yes	—
138	C₄×C₄₇	—	4	8	no	—
139	C₄×C₄₇	#389	1	4	yes	—
140	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
144	Dic₄₇	—	4	4	no	—
145	Dic₄₇	—	8	4	no	—
146	C₂×#427	—	47	1	no	—
147	C₂²×#107	D₄₇	1	1	yes	characteristic
148	C₂×#405	—	47	1	no	—
149	#2316	—	47	2	no	—
150	#1943	—	47	4	no	—
151	#1942	—	47	4	no	—
152	#1736	—	47	2	no	—
153	C₂×C₄₇	—	2	4	no	—
154	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
155	C₂×C₄₇	#2948	1	1	yes	characteristic
156	C₂×C₄₇	#2950	1	1	yes	characteristic
157	C₂×C₄₇	#3061	1	2	yes	—
158	C₂×C₄₇	#3164	1	2	yes	—
159	C₂²×M₁₆	—	47	1	no	—
160	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
161	C₂×#117	—	47	1	no	—
162	C₂×#117	—	47	2	no	—
163	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
164	#427	—	94	4	no	—
165	C₂×#107	—	47	2	no	—
166	C₂×#107	Dic₄₇	1	8	yes	complemented
167	C₂×#107	C₂×D₄₇	1	2	yes	—
168	C₂×#107	C₂×D₄₇	1	2	yes	—
169	C₂×#106	—	47	2	no	—
170	#405	—	94	4	no	—
171	C₂×#97	—	47	2	no	—
172	C₂×C₄×C₈	—	47	2	no	—
173	#389	—	47	8	no	—
174	#389	—	94	4	no	—
175	#380	—	94	4	no	—
176	#340	—	47	8	no	—
177	C₄₇	#21158	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	2	2	no	—
179	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×M₁₆	—	47	4	no	—
181	C₂×M₁₆	—	94	2	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂²×C₈	—	47	2	no	—
184	C₂²×C₈	Dic₉₄	1	4	yes	—
185	C₂²×C₈	D₁₈₈	1	4	yes	—
186	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
187	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
189	#117	—	94	2	no	—
190	#117	—	94	2	no	—
191	#117	—	94	4	no	—
192	#117	—	94	4	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	47	2	no	—
196	C₂×C₄⋊C₄	—	47	4	no	—
197	C₂×C₄⋊C₄	—	94	8	no	—
198	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
199	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
200	C₂×C₂²⋊C₄	—	47	2	no	—
201	C₂×C₂²⋊C₄	—	94	2	no	—
202	C₂×C₄²	—	47	2	no	—
203	C₂×C₄²	—	94	2	no	—
204	#107	—	94	4	no	—
205	#107	C₂×Dic₄₇	1	8	yes	—
206	#107	C₂×Dic₄₇	1	8	yes	—
207	#106	—	94	4	no	—
208	#97	—	94	4	no	—
209	C₄×C₈	—	94	4	no	—
210	#95	—	94	8	no	—
211	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	8	4	no	—
215	C₂²×C₄	—	47	4	no	—
216	C₂²×C₄	—	94	4	no	—
217	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
218	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
219	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	#10625352	1	2	yes	—
221	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
222	C₂²×C₄	#10625354	1	2	yes	—
223	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
224	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
225	M₁₆	—	94	8	no	—
226	C₂×C₈	—	47	4	no	—
227	C₂×C₈	—	94	2	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	C₄⋊Dic₄₇	1	16	yes	complemented
230	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
231	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
232	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
233	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
234	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
235	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	94	4	no	—
239	C₄⋊C₄	—	94	4	no	—
240	C₄⋊C₄	—	94	4	no	—
241	C₄⋊C₄	—	376	8	no	—
242	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
243	C₂²⋊C₄	—	94	4	no	—
244	C₂²⋊C₄	—	188	4	no	—
245	C₄²	—	94	2	no	—
246	C₄²	—	94	2	no	—
247	C₄²	—	188	4	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	D₄×Dic₄₇	1	2	yes	—
251	C₂³	#13093999	1	2	yes	—
252	C₂³	#13094023	1	2	yes	—
253	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	—	94	4	no	—
258	C₂×C₄	—	94	4	no	—
259	C₂×C₄	—	94	4	no	—
260	C₂×C₄	—	188	4	no	—
261	C₂×C₄	—	188	4	no	—
262	C₂×C₄	—	188	4	no	—
263	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
264	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
265	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
266	C₂×C₄	#13093999	1	2	yes	—
267	C₂×C₄	#13094004	1	4	yes	—
268	C₂×C₄	#13094015	1	4	yes	—
269	C₂×C₄	#13094023	1	2	yes	—
270	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
272	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
273	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
274	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
275	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
276	C₈	—	94	4	no	—
277	C₈	C₂×C₄⋊Dic₄₇	1	4	yes	—
278	C₈	C₂×C₄⋊Dic₄₇	1	4	yes	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	#15689022	1	4	yes	—
284	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
285	C₂²	C₂×#13093999	1	1	yes	characteristic
286	C₂²	C₂×#13094023	1	1	yes	characteristic
287	C₂²	#15689924	1	2	yes	—
288	C₂²	#15689936	1	2	yes	—
289	C₄	—	4	8	no	—
290	C₄	—	376	4	no	—
291	C₄	—	188	4	no	—
292	C₄	#15689022	1	4	yes	—
293	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
294	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
295	C₄	C₂×#13094004	1	1	yes	characteristic
296	C₄	C₂×#13094015	1	1	yes	characteristic
297	C₂	—	2	4	no	—
298	C₂	C₂×#15689022	1	1	yes	characteristic
299	C₂	#20252240	1	1	yes	characteristic
300	C₂	#20252256	1	1	yes	characteristic
301	C₂	#20253060	1	2	yes	—
302	C₂	#20253924	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#107×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#107	D₄₇
hypocenter	C₄₇	#21158
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
4. #20244127	C₂	semidirect
5. #20244122	C₂	semidirect
23. C₂×#107×C₄₇	C₄	semidirect
73. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
166. C₂×#107	Dic₄₇	semidirect
177. C₄₇	#21158	semidirect
229. C₂×C₈	C₄⋊Dic₄₇	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×#15689078	2
#20246006	2
#20244127	2
#20244122	2
#20243107	2
C₂²×#107×C₄₇	2
#21158	47