Subgroups — #26221119

Order 12032.

1002 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20244146	C₂	1	4	yes	maximal, complemented
4	#20243576	C₂	1	4	yes	maximal
5	C₄×#13093957	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093983	—	2	1	no	—
8	#15689176	C₄	1	8	yes	complemented
9	C₂×#13093957	C₂²	1	8	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
13	#15688993	C₂²	1	2	yes	—
14	#15688989	—	2	1	no	—
15	#15688983	—	2	1	no	—
16	#15688983	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₄₇	—	2	1	no	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	#15688885	C₂²	1	8	yes	—
20	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₇	—	2	1	no	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
24	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
25	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
27	C₂×#95×C₄₇	—	2	1	no	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	#13093983	—	4	4	no	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	#13093957	—	2	16	no	—
36	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
37	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
38	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
39	C₄₇⋊#95	—	2	2	no	—
40	C₄₇⋊#95	—	2	2	no	—
41	C₂³×C₄×C₄₇	—	2	2	no	—
42	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	—	4	4	no	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#95×C₄₇	—	2	2	no	—
58	#95×C₄₇	—	2	2	no	—
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	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	—	4	4	no	—
65	C₂²×C₄×C₄₇	—	4	4	no	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	D₈	1	4	yes	—
69	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	4	no	—
79	C₄⋊C₄×C₄₇	—	2	8	no	—
80	C₄⋊C₄×C₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	#41	1	8	yes	—
82	C₄²×C₄₇	—	4	4	no	—
83	C₄²×C₄₇	—	4	4	no	—
84	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
85	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
89	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
90	C₂²⋊Dic₄₇	—	4	4	no	—
91	C₄⋊Dic₄₇	—	2	8	no	—
92	C₄⋊Dic₄₇	—	4	4	no	—
93	C₄⋊Dic₄₇	—	4	4	no	—
94	C₄×Dic₄₇	—	4	4	no	—
95	C₂×C₄₇⋊C₈	—	2	8	no	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂³×C₄₇	C₂×D₈	1	2	yes	—
101	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
102	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	16	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	#107	1	16	yes	complemented
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
123	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₄₇⋊C₈	—	2	8	no	—
130	#21358	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	—	2	4	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	C₂×#107	1	4	yes	—
136	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₇	#525	1	2	yes	—
138	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
139	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂²×C₄₇	#575	1	2	yes	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	4	8	no	—
145	C₄×C₄₇	C₂×#107	1	4	yes	—
146	C₄×C₄₇	C₂×#107	1	4	yes	—
147	C₄×C₄₇	C₄×#41	1	2	yes	—
148	C₄×C₄₇	#533	1	2	yes	—
149	Dic₄₇	—	8	4	no	—
150	C₂×#431	—	47	1	no	—
151	#2285	—	47	1	no	—
152	C₂×C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic
153	#1947	—	47	4	no	—
154	#1852	—	47	4	no	—
155	C₄×#107	—	47	4	no	—
156	C₂×C₄₇	—	2	4	no	—
157	C₂×C₄₇	#2296	1	1	yes	characteristic
158	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
159	C₂×C₄₇	#2911	1	1	yes	characteristic
160	C₂×C₄₇	#3100	1	2	yes	—
161	C₂×C₄₇	#3104	1	2	yes	—
162	C₂×#117	—	94	1	no	—
163	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
164	C₂²×C₄²	—	2	1	no	—
165	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
166	#431	—	47	8	no	—
167	C₂×#107	—	47	8	no	—
168	C₂×C₄⋊C₈	—	47	2	no	—
169	C₂×C₂²⋊C₈	—	47	2	no	—
170	C₂×C₄×C₈	—	47	2	no	—
171	#384	—	47	2	no	—
172	#382	—	94	1	no	—
173	#380	—	47	2	no	—
174	#380	—	94	1	no	—
175	C₄×C₄⋊C₄	—	2	1	no	—
176	C₄×C₄⋊C₄	—	2	1	no	—
177	C₄×C₄⋊C₄	—	47	2	no	—
178	C₄×C₄⋊C₄	—	94	1	no	—
179	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
180	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
181	C₂×#95	—	2	1	no	—
182	#340	—	47	8	no	—
183	C₄₇	#21358	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	2	2	no	—
185	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
188	C₂²×C₈	—	47	4	no	—
189	#117	—	188	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	8	no	—
192	C₂×C₄⋊C₄	—	47	4	no	—
193	C₂×C₄⋊C₄	—	94	2	no	—
194	C₂×C₄⋊C₄	—	94	2	no	—
195	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
196	C₂×C₂²⋊C₄	—	94	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	4	4	no	—
200	C₂×C₄²	—	94	2	no	—
201	C₂×C₄²	Dic₉₄	1	4	yes	—
202	C₂×C₄²	C₄×D₄₇	1	4	yes	—
203	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	#107	—	94	16	no	—
208	C₄⋊C₈	—	47	8	no	—
209	C₂²⋊C₈	—	47	8	no	—
210	C₄×C₈	—	47	8	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	94	2	no	—
214	#95	—	94	2	no	—
215	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
224	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
225	C₂²×C₄	#10625334	1	4	yes	—
226	C₂²×C₄	#10625337	1	4	yes	—
227	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
228	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
235	C₂×C₈	—	47	8	no	—
236	C₂×C₈	—	94	8	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	94	8	no	—
240	C₄⋊C₄	—	188	4	no	—
241	C₄⋊C₄	—	188	4	no	—
242	C₄⋊C₄	#10625350	1	8	yes	—
243	C₂²⋊C₄	—	188	4	no	—
244	C₄²	—	4	4	no	—
245	C₄²	—	4	4	no	—
246	C₄²	—	188	4	no	—
247	C₄²	Dic₄₇⋊C₄	1	4	yes	—
248	C₄²	Dic₄₇⋊C₄	1	4	yes	—
249	C₄²	C₂×Dic₉₄	1	2	yes	—
250	C₄²	C₂×Dic₉₄	1	2	yes	—
251	C₄²	C₂×C₄×D₄₇	1	2	yes	—
252	C₄²	C₂×C₄×D₄₇	1	2	yes	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
256	C₂³	C₂×#10625334	1	2	yes	—
257	C₂³	C₂×#10625337	1	2	yes	—
258	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	16	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	188	4	no	—
270	C₂×C₄	—	188	4	no	—
271	C₂×C₄	—	188	4	no	—
272	C₂×C₄	#13093957	1	16	yes	complemented
273	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
274	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
275	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
276	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
277	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
278	C₂×C₄	C₂×#10625334	1	2	yes	—
279	C₂×C₄	C₂×#10625334	1	2	yes	—
280	C₂×C₄	C₂×#10625337	1	2	yes	—
281	C₂×C₄	C₂×#10625337	1	2	yes	—
282	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
284	C₂×C₄	C₂×#10625350	1	4	yes	—
285	C₈	—	94	8	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	C₂×#13093957	1	4	yes	—
291	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
292	C₂²	#15689634	1	2	yes	—
293	C₂²	C₂²×#10625334	1	1	yes	characteristic
294	C₂²	C₂²×#10625337	1	1	yes	characteristic
295	C₂²	#15689921	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	8	no	—
300	C₄	—	376	4	no	—
301	C₄	C₂×#13093957	1	4	yes	—
302	C₄	C₂×#13093957	1	4	yes	—
303	C₄	C₄×#10625350	1	2	yes	—
304	C₄	#15689680	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	#20245871	1	1	yes	characteristic
307	C₂	C₂²×#13093957	1	1	yes	characteristic
308	C₂	#20251977	1	1	yes	characteristic
309	C₂	#20253388	1	2	yes	—
310	C₂	#20253427	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×C₄⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#21358
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #20244146	C₂	semidirect
5. C₄×#13093957	C₂	semidirect
8. #15689176	C₄	semidirect
36. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
38. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
113. C₂×C₄×C₄₇	#107	semidirect
183. C₄₇	#21358	semidirect
272. C₂×C₄	#13093957	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20245795	2
#20244146	2
#20243576	2
C₄×#13093957	2
C₂×C₄×C₄⋊C₄×C₄₇	2
#21358	47