Subgroups — #26122156

Order 12032.

918 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688892	C₂	1	4	yes	maximal
4	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	2	yes	—
6	C₂²×#10625329	C₂²	1	2	yes	—
7	#15689147	—	2	2	no	—
8	#15689147	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
10	C₄×#10625329	—	2	2	no	—
11	C₄×#10625329	C₄	1	4	yes	complemented
12	C₂×#13093935	—	2	1	no	—
13	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
14	#15688892	—	2	8	no	—
15	C₂²×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#384	—	2	1	no	—
19	C₄₇×#384	—	2	1	no	—
20	C₄₇×#384	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	C₂×C₄	1	4	yes	—
28	C₂×#10625329	C₂×C₄	1	4	yes	—
29	C₂×#10625329	D₄	1	4	yes	—
30	C₂×#10625329	Q₈	1	4	yes	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×C₄₇⋊C₈	—	2	2	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
34	#13093935	—	2	4	no	—
35	C₄×C₄₇⋊C₈	—	2	4	no	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₇	—	4	8	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	2	yes	—
52	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
53	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	8	no	—
58	C₂²×C₄×C₄₇	—	4	8	no	—
59	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₄⋊C₄×C₄₇	—	4	8	no	—
74	C₄⋊C₄×C₄₇	—	4	8	no	—
75	C₄⋊C₄×C₄₇	—	4	4	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₄⋊C₄×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
81	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
83	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
84	#10625329	—	2	4	no	—
85	#10625329	—	2	4	no	—
86	#10625329	—	2	4	no	—
87	#10625329	—	2	8	no	—
88	#10625329	—	4	4	no	—
89	#10625329	C₄⋊C₄	1	8	yes	complemented
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂³×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₇	C₄×D₄	1	2	yes	—
102	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	8	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	4	8	no	—
109	C₂×C₄×C₄₇	—	8	8	no	—
110	C₂×C₄×C₄₇	#95	1	4	yes	—
111	C₂×C₄×C₄₇	#95	1	4	yes	—
112	C₂×C₄×C₄₇	#100	1	4	yes	—
113	C₂×C₄×C₄₇	#101	1	4	yes	—
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#117	1	2	yes	—
120	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	#121	1	2	yes	—
123	C₂×C₄×C₄₇	#121	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#125	1	2	yes	—
129	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
132	C₄₇⋊C₈	—	2	8	no	—
133	C₄₇⋊C₈	—	4	4	no	—
134	#10857	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
141	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
142	C₂²×C₄₇	#389	1	2	yes	—
143	C₂²×C₄₇	#390	1	1	yes	characteristic
144	C₂²×C₄₇	#391	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#100	1	2	yes	—
146	C₂²×C₄₇	C₂×#101	1	2	yes	—
147	C₄×C₄₇	—	2	4	no	—
148	C₄×C₄₇	—	8	4	no	—
149	C₄×C₄₇	#343	1	8	yes	complemented
150	C₄×C₄₇	#380	1	2	yes	—
151	C₄×C₄₇	#382	1	1	yes	characteristic
152	C₄×C₄₇	#384	1	1	yes	characteristic
153	C₄×C₄₇	#389	1	2	yes	—
154	C₄×C₄₇	#390	1	1	yes	characteristic
155	C₄×C₄₇	#391	1	1	yes	characteristic
156	C₂×#423	—	47	1	no	—
157	C₂×C₄×M₁₆	—	47	1	no	—
158	C₂×#384	D₄₇	1	1	yes	characteristic
159	C₂×#343	—	47	4	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
163	C₂×C₄₇	C₂×#343	1	2	yes	—
164	C₂×C₄₇	#1965	1	1	yes	characteristic
165	C₂×C₄₇	#1966	1	1	yes	characteristic
166	C₂×C₄₇	#1997	1	2	yes	—
167	C₂²×M₁₆	—	47	2	no	—
168	C₂²×M₁₆	—	47	2	no	—
169	C₂²×C₄⋊C₄	Dic₄₇	1	4	yes	—
170	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	#423	—	47	4	no	—
173	#423	—	94	2	no	—
174	C₂×C₄⋊C₈	—	94	2	no	—
175	C₄×M₁₆	—	47	4	no	—
176	C₄×M₁₆	—	94	2	no	—
177	C₂×#97	—	94	1	no	—
178	C₂×C₄×C₈	—	94	1	no	—
179	#384	—	2	1	no	—
180	#384	—	2	1	no	—
181	#384	—	2	2	no	—
182	#343	—	94	8	no	—
183	C₄₇	#10857	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
185	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
186	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
187	C₂×M₁₆	—	47	4	no	—
188	C₂×M₁₆	—	47	4	no	—
189	C₂×M₁₆	—	47	4	no	—
190	C₂×M₁₆	—	47	4	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂×M₁₆	—	94	2	no	—
193	C₂×M₁₆	—	94	2	no	—
194	C₂×M₁₆	—	94	2	no	—
195	C₂×M₁₆	—	94	4	no	—
196	C₂×M₁₆	—	94	4	no	—
197	C₂²×C₈	—	94	2	no	—
198	C₂²×C₈	—	94	2	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	4	8	no	—
206	C₂×C₄²	Dic₉₄	1	2	yes	—
207	C₂×C₄²	C₄×D₄₇	1	2	yes	—
208	C₂×C₄²	C₄×D₄₇	1	2	yes	—
209	C₂×C₄²	D₁₈₈	1	2	yes	—
210	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
213	C₄⋊C₈	—	94	8	no	—
214	#97	—	94	4	no	—
215	C₄×C₈	—	94	4	no	—
216	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	4	8	no	—
222	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
223	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
224	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
225	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
226	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₄₇	1	2	yes	—
231	C₂²×C₄	#10625352	1	2	yes	—
232	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
233	C₂²×C₄	#10625354	1	2	yes	—
234	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	M₁₆	—	47	8	no	—
237	M₁₆	—	94	4	no	—
238	M₁₆	—	94	4	no	—
239	M₁₆	—	94	4	no	—
240	M₁₆	—	94	8	no	—
241	M₁₆	—	188	4	no	—
242	C₂×C₈	—	94	4	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	94	4	no	—
245	C₂×C₈	—	94	4	no	—
246	C₂×C₈	—	188	2	no	—
247	C₂×C₈	—	188	2	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₄⋊C₄	—	4	8	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄²	Dic₄₇⋊C₄	1	2	yes	—
254	C₄²	Dic₄₇⋊C₄	1	4	yes	—
255	C₄²	Dic₄₇⋊C₄	1	2	yes	—
256	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
257	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
258	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
262	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
263	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
264	C₂³	D₄×Dic₄₇	1	2	yes	—
265	C₂³	Q₈×Dic₄₇	1	2	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	—	8	8	no	—
273	C₂×C₄	C₄₇⋊#95	1	4	yes	—
274	C₂×C₄	C₄₇⋊#95	1	4	yes	—
275	C₂×C₄	#13093941	1	4	yes	—
276	C₂×C₄	#13093943	1	4	yes	—
277	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
280	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
282	C₂×C₄	#13093982	1	2	yes	—
283	C₂×C₄	#13093986	1	2	yes	—
284	C₂×C₄	#13093989	1	2	yes	—
285	C₂×C₄	#13093995	1	2	yes	—
286	C₂×C₄	#13093999	1	2	yes	—
287	C₂×C₄	#13094000	1	2	yes	—
288	C₂×C₄	#13094004	1	2	yes	—
289	C₂×C₄	#13094011	1	1	yes	characteristic
290	C₂×C₄	#13094013	1	1	yes	characteristic
291	C₂×C₄	#13094015	1	2	yes	—
292	C₂×C₄	#13094020	1	1	yes	characteristic
293	C₂×C₄	#13094022	1	1	yes	characteristic
294	C₂×C₄	#13094023	1	2	yes	—
295	C₈	—	94	8	no	—
296	C₈	—	188	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
303	C₂²	C₄⋊C₄×Dic₄₇	1	2	yes	—
304	C₂²	#15689022	1	2	yes	—
305	C₂²	#15689026	1	1	yes	characteristic
306	C₂²	#15689030	1	1	yes	characteristic
307	C₂²	C₂×#13093941	1	2	yes	—
308	C₂²	C₂×#13093943	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	8	4	no	—
311	C₄	#15688892	1	8	yes	complemented
312	C₄	#15688980	1	2	yes	—
313	C₄	#15688988	1	1	yes	characteristic
314	C₄	#15688992	1	1	yes	characteristic
315	C₄	#15689019	1	2	yes	—
316	C₄	#15689024	1	1	yes	characteristic
317	C₄	#15689028	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	#20241617	1	1	yes	characteristic
321	C₂	C₂×#15688892	1	2	yes	—
322	C₂	#20244229	1	1	yes	characteristic
323	C₂	#20244233	1	1	yes	characteristic
324	C₂	#20244378	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#384	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#384	D₄₇
hypocenter	C₄₇	#10857
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
8. #15689147	C₄	semidirect
11. C₄×#10625329	C₄	semidirect
89. #10625329	C₄⋊C₄	semidirect
149. C₄×C₄₇	#343	semidirect
183. C₄₇	#10857	semidirect
311. C₄	#15688892	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×C₄×#10625329	2
C₂×#15688892	2
C₂×C₄₇×#384	2
#10857	47