Subgroups — #26105806

Order 12032.

830 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×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20241228	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₄	1	2	yes	—
6	#15689176	—	2	2	no	—
7	#15689176	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
11	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
14	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
15	#15688969	—	2	1	no	—
16	C₄²×Dic₄₇	—	2	1	no	—
17	C₂³×C₈×C₄₇	—	2	1	no	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₇	—	2	1	no	—
20	C₂×C₄×C₈×C₄₇	—	2	2	no	—
21	C₂×C₄×C₈×C₄₇	C₄	1	4	yes	complemented
22	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	#13093983	—	2	4	no	—
25	#13093983	C₈	1	8	yes	complemented
26	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄×Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
33	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
34	C₄×C₄₇⋊C₈	—	2	4	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	8	no	—
41	C₂²×C₈×C₄₇	—	2	4	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₄₇	—	2	4	no	—
52	C₄×C₈×C₄₇	—	2	2	no	—
53	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
54	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
67	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	8	no	—
73	C₂×C₈×C₄₇	—	2	16	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	C₄²	1	2	yes	—
76	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
85	C₂²⋊Dic₄₇	—	2	8	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	C₂×C₈	1	8	yes	complemented
88	C₄×Dic₄₇	—	2	8	no	—
89	C₄×Dic₄₇	—	4	4	no	—
90	C₄×Dic₄₇	—	4	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	4	4	no	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	#95	1	2	yes	—
96	C₂³×C₄₇	C₂²⋊C₈	1	4	yes	—
97	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	#95	1	2	yes	—
105	C₂×C₄×C₄₇	#95	1	2	yes	—
106	C₂×C₄×C₄₇	#95	1	2	yes	—
107	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
108	C₂×C₄×C₄₇	#97	1	4	yes	—
109	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
110	C₂×C₄×C₄₇	#104	1	4	yes	—
111	C₂×C₄×C₄₇	C₄⋊C₈	1	8	yes	—
112	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₄₇	#135	1	4	yes	—
121	C₂×Dic₄₇	—	2	8	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₈×C₄₇	—	2	8	no	—
125	C₄₇⋊C₈	—	4	4	no	—
126	#7890	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	8	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	#336	1	4	yes	—
131	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
133	C₂²×C₄₇	#416	1	2	yes	—
134	C₂²×C₄₇	C₂×#104	1	2	yes	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	#336	1	4	yes	—
139	C₄×C₄₇	#339	1	4	yes	—
140	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
142	C₄×C₄₇	#416	1	2	yes	—
143	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
144	C₄×C₄₇	#427	1	2	yes	—
145	Dic₄₇	—	4	8	no	—
146	C₂×#431	—	47	1	no	—
147	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
148	C₄×#117	—	47	1	no	—
149	#1279	—	47	4	no	—
150	C₂×C₄₇	—	2	4	no	—
151	C₂×C₄₇	#1476	1	4	yes	—
152	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
153	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
154	C₂×C₄₇	#1735	1	1	yes	characteristic
155	C₂³×C₈	—	2	1	no	—
156	C₂×#117	—	47	2	no	—
157	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
158	#431	—	47	4	no	—
159	#431	—	94	2	no	—
160	C₂×C₄⋊C₈	—	47	2	no	—
161	C₂×C₂²⋊C₈	—	94	1	no	—
162	C₂×C₄×C₈	—	2	1	no	—
163	C₂×C₄×C₈	—	2	2	no	—
164	C₂×C₄×C₈	—	94	1	no	—
165	C₂×C₄×C₈	Dic₄₇	1	4	yes	complemented
166	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
167	C₄×C₄⋊C₄	—	47	2	no	—
168	C₄×C₄⋊C₄	—	47	2	no	—
169	C₄×C₂²⋊C₄	—	94	1	no	—
170	C₄×C₂²⋊C₄	—	94	1	no	—
171	#376	—	94	1	no	—
172	C₄³	—	94	1	no	—
173	C₄₇	#7890	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
175	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
177	C₂²×C₈	—	2	8	no	—
178	C₂²×C₈	—	2	4	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	94	2	no	—
181	#117	—	47	8	no	—
182	#117	—	94	4	no	—
183	C₂×C₄⋊C₄	—	47	4	no	—
184	C₂×C₂²⋊C₄	—	94	2	no	—
185	C₂×C₄²	—	94	2	no	—
186	C₂×C₄²	—	94	2	no	—
187	C₂×C₄²	—	94	2	no	—
188	C₂×C₄²	Dic₉₄	1	2	yes	—
189	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₄×D₄₇	1	2	yes	—
191	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
192	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
193	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
195	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
196	C₄⋊C₈	—	47	4	no	—
197	C₄⋊C₈	—	94	2	no	—
198	C₂²⋊C₈	—	94	4	no	—
199	C₄×C₈	—	2	4	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	94	4	no	—
202	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
203	#95	—	94	2	no	—
204	#95	—	94	2	no	—
205	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	94	4	no	—
209	C₂²×C₄	C₈×D₄₇	1	4	yes	—
210	C₂²×C₄	#10625324	1	4	yes	—
211	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
212	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
213	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
214	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
215	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
217	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
220	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	—	2	16	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	188	4	no	—
228	C₄⋊C₄	—	47	8	no	—
229	C₄⋊C₄	—	94	4	no	—
230	C₂²⋊C₄	—	94	8	no	—
231	C₄²	—	94	8	no	—
232	C₄²	—	188	4	no	—
233	C₄²	—	188	4	no	—
234	C₄²	C₄×Dic₄₇	1	2	yes	—
235	C₄²	Dic₄₇⋊C₄	1	2	yes	—
236	C₄²	Dic₄₇⋊C₄	1	2	yes	—
237	C₄²	C₄⋊Dic₄₇	1	2	yes	—
238	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
239	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
240	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
241	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
243	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₄₇⋊#95	1	2	yes	—
247	C₂³	(C₂×D₄₇)⋊C₈	1	4	yes	—
248	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	94	8	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	C₄₇⋊#95	1	2	yes	—
259	C₂×C₄	C₄₇⋊#95	1	2	yes	—
260	C₂×C₄	C₄₇⋊#95	1	2	yes	—
261	C₂×C₄	C₈×Dic₄₇	1	4	yes	—
262	C₂×C₄	#13093934	1	4	yes	—
263	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
264	C₂×C₄	#13093952	1	4	yes	—
265	C₂×C₄	Dic₄₇⋊C₈	1	8	yes	—
266	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
273	C₂×C₄	C₂×#10625324	1	2	yes	—
274	C₂×C₄	#13094046	1	4	yes	—
275	C₈	—	2	8	no	—
276	C₈	—	188	4	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	#15688877	1	4	yes	—
281	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
282	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	2	yes	—
283	C₂²	#15689120	1	2	yes	—
284	C₂²	C₂×#13093952	1	2	yes	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	188	8	no	—
289	C₄	#15688877	1	4	yes	—
290	C₄	#15688883	1	4	yes	—
291	C₄	C₂×C₈×Dic₄₇	1	1	yes	characteristic
292	C₄	C₂×#13093934	1	1	yes	characteristic
293	C₄	#15689120	1	2	yes	—
294	C₄	C₂×Dic₄₇⋊C₈	1	2	yes	—
295	C₄	#15689161	1	2	yes	—
296	C₂	—	2	4	no	—
297	C₂	#20241771	1	4	yes	—
298	C₂	C₂×#15688877	1	1	yes	characteristic
299	C₂	C₂×#15688883	1	1	yes	characteristic
300	C₂	#20243105	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₁₈₈
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#7890
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #20241228	C₂	semidirect
7. #15689176	C₄	semidirect
21. C₂×C₄×C₈×C₄₇	C₄	semidirect
25. #13093983	C₈	semidirect
87. C₄⋊Dic₄₇	C₂×C₈	semidirect
165. C₂×C₄×C₈	Dic₄₇	semidirect
173. C₄₇	#7890	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
C₄×#13093983	2
#20241228	2
C₂²×C₄×C₈×C₄₇	2
#7890	47