Subgroups — #26122299

Order 12032.

890 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688882	C₂	1	2	yes	maximal
3	C₂×#15688882	C₂	1	2	yes	maximal
4	C₄×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2296	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
8	C₄×#10625329	—	2	4	no	—
9	C₂×#13093935	—	2	1	no	—
10	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
11	C₂×C₄₇⋊#95	—	2	2	no	—
12	#15688882	—	2	4	no	—
13	#15688882	C₄	1	8	yes	complemented
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#389	C₄	1	2	yes	complemented
18	C₄₇×#389	C₄	1	2	yes	complemented
19	C₄₇×#385	—	2	2	no	—
20	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
21	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
22	C₂³×Dic₄₇	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	C₂×C₄	1	4	yes	—
27	C₂×#10625329	D₄	1	4	yes	—
28	C₂×#10625329	Q₈	1	4	yes	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
36	#13093935	—	2	4	no	—
37	C₄×C₄₇⋊C₈	—	2	4	no	—
38	C₄₇⋊#95	—	2	8	no	—
39	C₂³×C₄×C₄₇	D₄	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	#117×C₄₇	—	2	2	no	—
42	#117×C₄₇	—	2	2	no	—
43	#117×C₄₇	—	2	2	no	—
44	#117×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
52	C₂×C₄²×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	—	2	1	no	—
54	C₂×C₄²×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	—	2	1	no	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
59	#95×C₄₇	—	2	2	no	—
60	#95×C₄₇	—	2	2	no	—
61	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₄₇	#41	1	2	yes	—
76	C₂²×C₄×C₄₇	#41	1	2	yes	—
77	C₂²×C₄×C₄₇	#41	1	2	yes	—
78	C₂²×Dic₄₇	—	2	8	no	—
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂²×Dic₄₇	—	4	4	no	—
81	C₂²×Dic₄₇	—	4	2	no	—
82	C₂²×Dic₄₇	—	4	2	no	—
83	C₄⋊C₄×C₄₇	—	2	1	no	—
84	C₄⋊C₄×C₄₇	—	2	1	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	4	4	no	—
87	C₄⋊C₄×C₄₇	—	4	4	no	—
88	C₄⋊C₄×C₄₇	C₄²	1	2	yes	—
89	C₄⋊C₄×C₄₇	C₄²	1	2	yes	—
90	C₂²⋊C₄×C₄₇	—	2	4	no	—
91	C₂²⋊C₄×C₄₇	—	4	2	no	—
92	C₄²×C₄₇	—	2	4	no	—
93	C₄²×C₄₇	—	2	4	no	—
94	C₄²×C₄₇	—	2	1	no	—
95	C₄²×C₄₇	—	2	1	no	—
96	C₄²×C₄₇	—	2	1	no	—
97	C₄²×C₄₇	—	2	1	no	—
98	C₄²×C₄₇	—	4	1	no	—
99	C₄²×C₄₇	—	4	1	no	—
100	C₄×Dic₄₇	—	2	4	no	—
101	C₄×Dic₄₇	—	2	4	no	—
102	C₄×Dic₄₇	—	4	2	no	—
103	C₄×Dic₄₇	—	4	2	no	—
104	#10625329	—	2	4	no	—
105	#10625329	—	2	4	no	—
106	#10625329	—	2	4	no	—
107	#10625329	C₄⋊C₄	1	8	yes	complemented
108	C₂×C₄₇⋊C₈	—	2	4	no	—
109	C₂×C₄₇⋊C₈	—	2	4	no	—
110	C₂×C₄₇⋊C₈	—	2	4	no	—
111	C₂³×C₄₇	—	2	2	no	—
112	C₂³×C₄₇	—	2	2	no	—
113	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂³×C₄₇	C₄×D₄	1	2	yes	—
115	C₂³×C₄₇	#121	1	2	yes	—
116	C₂³×C₄₇	#124	1	1	yes	characteristic
117	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	4	8	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	#104	1	4	yes	—
132	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₄₇	#117	1	2	yes	—
135	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#123	1	2	yes	—
146	C₂×C₄×C₄₇	#125	1	2	yes	—
147	C₂×C₄×C₄₇	#126	1	2	yes	—
148	C₂×Dic₄₇	—	2	8	no	—
149	C₂×Dic₄₇	—	4	8	no	—
150	C₂×Dic₄₇	—	4	8	no	—
151	C₂×Dic₄₇	—	4	4	no	—
152	C₂×Dic₄₇	—	4	4	no	—
153	C₄₇⋊C₈	—	2	8	no	—
154	#10886	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	—	2	4	no	—
156	C₂²×C₄₇	—	2	2	no	—
157	C₂²×C₄₇	—	2	2	no	—
158	C₂²×C₄₇	—	2	2	no	—
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄₇	#386	1	1	yes	characteristic
162	C₂²×C₄₇	#388	1	2	yes	—
163	C₂²×C₄₇	#389	1	2	yes	—
164	C₂²×C₄₇	#391	1	1	yes	characteristic
165	C₂²×C₄₇	C₂×#104	1	2	yes	—
166	C₂²×C₄₇	#421	1	2	yes	—
167	C₄×C₄₇	—	2	4	no	—
168	C₄×C₄₇	—	4	2	no	—
169	C₄×C₄₇	—	4	2	no	—
170	C₄×C₄₇	—	4	2	no	—
171	C₄×C₄₇	—	8	2	no	—
172	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
173	C₄×C₄₇	#385	1	2	yes	—
174	C₄×C₄₇	#387	1	2	yes	—
175	C₄×C₄₇	#387	1	2	yes	—
176	Dic₄₇	—	4	8	no	—
177	C₂×C₄×M₁₆	—	47	1	no	—
178	#2296	D₄₇	1	1	yes	characteristic
179	C₂×#339	—	47	2	no	—
180	C₂×#339	—	47	2	no	—
181	C₄×#95	—	47	1	no	—
182	C₂×C₄₇	—	2	2	no	—
183	C₂×C₄₇	—	2	2	no	—
184	C₂×C₄₇	#1436	1	1	yes	characteristic
185	C₂×C₄₇	C₄×#104	1	1	yes	characteristic
186	C₂×C₄₇	#1759	1	1	yes	characteristic
187	C₂×C₄₇	#1871	1	2	yes	—
188	C₂×C₄₇	#1986	1	2	yes	—
189	C₂²×M₁₆	—	47	2	no	—
190	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
191	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
192	C₂²×C₄²	—	47	2	no	—
193	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
194	C₄×M₁₆	—	94	4	no	—
195	C₂×#97	—	94	1	no	—
196	C₂×C₄×C₈	—	94	1	no	—
197	#389	Dic₄₇	1	2	yes	complemented
198	#389	Dic₄₇	1	2	yes	complemented
199	#385	—	2	2	no	—
200	C₄×C₄⋊C₄	—	2	1	no	—
201	C₄×C₄⋊C₄	—	2	1	no	—
202	C₂×#95	—	94	2	no	—
203	#339	—	47	8	no	—
204	#339	—	94	4	no	—
205	C₄₇	#10886	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₄	—	94	2	no	—
207	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
208	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
209	C₂×M₁₆	—	47	4	no	—
210	C₂×M₁₆	—	47	4	no	—
211	C₂×M₁₆	—	47	4	no	—
212	C₂×M₁₆	—	94	2	no	—
213	C₂×M₁₆	—	94	2	no	—
214	C₂×M₁₆	—	94	2	no	—
215	C₂²×C₈	—	94	2	no	—
216	#117	—	2	2	no	—
217	#117	—	2	2	no	—
218	#117	—	2	2	no	—
219	#117	C₄×D₄₇	1	4	yes	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
223	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
224	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
225	C₂×C₂²⋊C₄	—	2	1	no	—
226	C₂×C₂²⋊C₄	—	2	1	no	—
227	C₂×C₄²	—	2	2	no	—
228	C₂×C₄²	—	2	2	no	—
229	C₂×C₄²	—	2	1	no	—
230	C₂×C₄²	—	2	1	no	—
231	C₂×C₄²	—	47	4	no	—
232	C₂×C₄²	—	94	2	no	—
233	C₂×C₄²	—	94	2	no	—
234	C₂×C₄²	—	94	2	no	—
235	C₂×C₄²	—	94	2	no	—
236	C₂×C₄²	—	94	2	no	—
237	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
238	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
240	#97	—	94	4	no	—
241	C₄×C₈	—	94	4	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	#95	—	94	8	no	—
245	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	94	8	no	—
253	C₂²×C₄	—	94	4	no	—
254	C₂²×C₄	—	188	4	no	—
255	C₂²×C₄	—	188	2	no	—
256	C₂²×C₄	—	188	2	no	—
257	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
258	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
259	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
260	C₂²×C₄	#10625350	1	2	yes	—
261	C₂²×C₄	D₄×D₄₇	1	2	yes	—
262	C₂²×C₄	#10625352	1	2	yes	—
263	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
264	C₂²×C₄	#10625354	1	2	yes	—
265	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
267	M₁₆	—	47	8	no	—
268	M₁₆	—	94	4	no	—
269	M₁₆	—	94	4	no	—
270	M₁₆	—	94	4	no	—
271	C₂×C₈	—	94	4	no	—
272	C₂×C₈	—	94	4	no	—
273	C₂×C₈	—	94	4	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	C₄×Dic₄₇	1	2	yes	—
280	C₄⋊C₄	C₄×Dic₄₇	1	2	yes	—
281	C₂²⋊C₄	—	2	4	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	94	4	no	—
292	C₄²	—	94	4	no	—
293	C₄²	—	188	2	no	—
294	C₄²	—	188	2	no	—
295	C₂³	—	2	2	no	—
296	C₂³	—	2	2	no	—
297	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
298	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
299	C₂³	#13093996	1	2	yes	—
300	C₂³	#13094013	1	1	yes	characteristic
301	C₂³	#13094022	1	1	yes	characteristic
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	8	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	94	8	no	—
316	C₂×C₄	—	188	8	no	—
317	C₂×C₄	—	188	8	no	—
318	C₂×C₄	—	188	4	no	—
319	C₂×C₄	—	188	4	no	—
320	C₂×C₄	#13093951	1	4	yes	—
321	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
322	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
323	C₂×C₄	#13093982	1	2	yes	—
324	C₂×C₄	#13093983	1	1	yes	characteristic
325	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
326	C₂×C₄	#13093986	1	2	yes	—
327	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
328	C₂×C₄	#13093989	1	2	yes	—
329	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
330	C₂×C₄	#13093992	1	1	yes	characteristic
331	C₂×C₄	#13094000	1	2	yes	—
332	C₂×C₄	#13094002	1	2	yes	—
333	C₂×C₄	#13094005	1	1	yes	characteristic
334	C₂×C₄	#13094009	1	2	yes	—
335	C₂×C₄	#13094014	1	2	yes	—
336	C₂×C₄	#13094019	1	2	yes	—
337	C₈	—	94	8	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	4	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
344	C₂²	#15689003	1	1	yes	characteristic
345	C₂²	#15689013	1	2	yes	—
346	C₂²	#15689018	1	2	yes	—
347	C₂²	#15689030	1	1	yes	characteristic
348	C₂²	C₂×#13093951	1	2	yes	—
349	C₂²	#15689140	1	2	yes	—
350	C₄	—	2	4	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	188	8	no	—
356	C₄	C₄⋊C₄×Dic₄₇	1	2	yes	—
357	C₄	#15688994	1	2	yes	—
358	C₄	#15689004	1	2	yes	—
359	C₄	#15689009	1	2	yes	—
360	C₂	—	2	2	no	—
361	C₂	—	2	2	no	—
362	C₂	#20241620	1	1	yes	characteristic
363	C₂	C₄×#13093951	1	1	yes	characteristic
364	C₂	#20243198	1	1	yes	characteristic
365	C₂	#20243691	1	2	yes	—
366	C₂	#20244333	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2296	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2296	D₄₇
hypocenter	C₄₇	#10886
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2296

Complements

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

N	H ≅ G/N	type
13. #15688882	C₄	semidirect
17. C₄₇×#389	C₄	semidirect
18. C₄₇×#389	C₄	semidirect
107. #10625329	C₄⋊C₄	semidirect
197. #389	Dic₄₇	semidirect
198. #389	Dic₄₇	semidirect
205. C₄₇	#10886	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
C₂×#15688882	2
C₄×C₄₇⋊#95	2
C₄₇×#2296	2
#10886	47