Subgroups — #26185412

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252505	C₂	1	1	yes	maximal, characteristic
2	#20252032	C₂	1	1	yes	maximal, characteristic
3	#20242176	C₂	1	2	yes	maximal, complemented
4	#20242171	C₂	1	2	yes	maximal, complemented
5	#20242166	C₂	1	2	yes	maximal, complemented
6	#20242162	C₂	1	2	yes	maximal, complemented
7	#20242141	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242138	C₂	1	1	yes	maximal, characteristic, complemented
9	#20242134	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15688875	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2988	C₂	1	1	yes	maximal, characteristic
12	#15689254	—	2	1	no	—
13	#15689201	—	2	1	no	—
14	#15689195	—	2	2	no	—
15	#15689184	C₄	1	4	yes	—
16	#15689180	C₂²	1	2	yes	—
17	#15689176	C₂²	1	1	yes	characteristic
18	#15689176	C₂²	1	2	yes	—
19	#15689176	C₂²	1	2	yes	—
20	#15689150	C₄	1	2	yes	—
21	#15689147	C₂²	1	2	yes	—
22	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
23	#15689090	C₄	1	2	yes	—
24	C₄×#10625329	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
26	#15688875	C₂²	1	2	yes	—
27	#15688875	C₂²	1	2	yes	—
28	#15688875	C₂²	1	2	yes	—
29	#15688875	C₂²	1	1	yes	characteristic
30	#15688875	C₂²	1	1	yes	characteristic
31	#15688875	C₂²	1	2	yes	—
32	#15688873	C₂²	1	1	yes	characteristic
33	#15688873	C₂²	1	2	yes	—
34	#15688873	C₂²	1	1	yes	characteristic
35	#15688873	C₂²	1	2	yes	—
36	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
37	C₈×Q₈×C₄₇	—	2	1	no	—
38	C₈×D₄×C₄₇	—	2	2	no	—
39	C₈×D₄×C₄₇	—	2	1	no	—
40	C₄₇×#431	C₂²	1	2	yes	—
41	C₄₇×#431	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
44	C₂×#10625329	—	2	2	no	—
45	C₂×#10625329	C₂×C₄	1	2	yes	—
46	C₂²×C₄₇⋊C₈	—	2	1	no	—
47	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
48	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
51	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
52	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
53	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
54	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
55	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
56	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
57	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
58	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
59	#13093935	—	2	2	no	—
60	#13093935	C₂×C₄	1	2	yes	—
61	C₄×C₄₇⋊C₈	—	2	1	no	—
62	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
63	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
64	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
65	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
66	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	—	2	1	no	—
68	C₂²×C₈×C₄₇	—	2	2	no	—
69	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
70	C₄×Q₈×C₄₇	Q₈	1	1	yes	characteristic
71	C₄×D₄×C₄₇	D₄	1	2	yes	—
72	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
73	C₄×D₄×C₄₇	Q₈	1	2	yes	—
74	C₄×D₄×C₄₇	Q₈	1	1	yes	characteristic
75	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
76	#117×C₄₇	C₂×C₄	1	2	yes	—
77	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₄₇	—	2	1	no	—
80	C₄⋊C₈×C₄₇	—	2	2	no	—
81	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
84	C₂²⋊C₈×C₄₇	—	2	1	no	—
85	C₂²⋊C₈×C₄₇	—	2	2	no	—
86	C₄×C₈×C₄₇	—	2	1	no	—
87	C₄×C₈×C₄₇	—	2	2	no	—
88	#41×C₄₇	—	2	4	no	—
89	C₂×Q₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂×D₄×C₄₇	C₄⋊C₄	1	2	yes	—
91	C₂×D₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	1	no	—
100	C₂×C₈×C₄₇	—	2	1	no	—
101	C₂×C₈×C₄₇	—	8	1	no	—
102	C₂×C₈×C₄₇	—	8	2	no	—
103	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
104	C₄⋊C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
105	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
106	C₄⋊C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
108	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
109	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
110	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₇	C₂×D₄	1	2	yes	—
114	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
115	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
116	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
117	#10625329	—	2	4	no	—
118	#10625329	—	8	2	no	—
119	C₂×C₄₇⋊C₈	—	2	4	no	—
120	C₂×C₄₇⋊C₈	—	2	1	no	—
121	C₂×C₄₇⋊C₈	—	2	1	no	—
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	8	1	no	—
124	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
125	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
126	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
127	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Q₈×C₄₇	—	2	2	no	—
130	D₄×C₄₇	—	2	4	no	—
131	D₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	4	no	—
133	C₂×C₄×C₄₇	—	2	1	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	1	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	4	1	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#131	1	4	yes	—
146	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
149	C₈×C₄₇	—	4	2	no	—
150	C₈×C₄₇	—	4	1	no	—
151	C₈×C₄₇	—	4	1	no	—
152	C₄₇⋊C₈	—	2	2	no	—
153	C₄₇⋊C₈	—	2	2	no	—
154	C₄₇⋊C₈	—	2	4	no	—
155	C₄₇⋊C₈	—	4	2	no	—
156	C₄₇⋊C₈	—	4	1	no	—
157	C₄₇⋊C₈	—	4	1	no	—
158	#17812	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	2	no	—
162	C₂²×C₄₇	—	2	2	no	—
163	C₂²×C₄₇	—	2	2	no	—
164	C₂²×C₄₇	#424	1	2	yes	—
165	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	2	2	no	—
172	C₄×C₄₇	#424	1	2	yes	—
173	C₄×C₄₇	C₂×#131	1	2	yes	—
174	C₄×C₄₇	#576	1	1	yes	characteristic
175	C₄×C₄₇	#578	1	1	yes	characteristic
176	#2989	—	47	1	no	—
177	#2988	D₄₇	1	1	yes	characteristic
178	#2922	—	47	1	no	—
179	#1574	—	47	2	no	—
180	#1573	—	47	2	no	—
181	#1572	—	47	2	no	—
182	#1571	—	47	2	no	—
183	#1565	—	47	1	no	—
184	#1565	—	47	1	no	—
185	#1564	—	47	1	no	—
186	C₂×#335	—	47	1	no	—
187	C₂×C₄₇	—	2	2	no	—
188	C₂×C₄₇	—	4	2	no	—
189	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
190	C₂×C₄₇	#2913	1	1	yes	characteristic
191	C₂×C₄₇	#2916	1	1	yes	characteristic
192	C₄×#41	C₂×D₄₇	1	1	yes	characteristic
193	#446	—	94	1	no	—
194	C₈×Q₈	—	2	1	no	—
195	#436	—	94	1	no	—
196	#435	—	94	2	no	—
197	C₈×D₄	—	2	2	no	—
198	C₈×D₄	—	2	1	no	—
199	#433	—	47	4	no	—
200	#432	—	47	2	no	—
201	#431	—	47	2	no	—
202	#431	—	47	2	no	—
203	#431	—	47	1	no	—
204	#431	C₂×D₄₇	1	1	yes	characteristic
205	#431	C₂×D₄₇	1	2	yes	—
206	#424	—	47	2	no	—
207	#423	—	47	2	no	—
208	C₂×C₄⋊C₈	—	47	1	no	—
209	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
210	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
211	#408	—	47	2	no	—
212	C₄×M₁₆	—	47	1	no	—
213	C₂×C₄×C₈	—	47	1	no	—
214	#335	—	47	2	no	—
215	#335	—	47	2	no	—
216	#335	—	47	1	no	—
217	#335	—	47	1	no	—
218	#335	—	47	2	no	—
219	#335	—	47	2	no	—
220	#334	—	47	1	no	—
221	#334	—	47	1	no	—
222	#334	—	47	2	no	—
223	#334	—	47	2	no	—
224	C₄₇	#17812	1	1	yes	characteristic, complemented, Sylow
225	C₂×#41	C₄×D₄₇	1	1	yes	characteristic
226	C₂×M₁₆	—	47	2	no	—
227	C₂×M₁₆	—	94	2	no	—
228	C₂²×C₈	—	2	1	no	—
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	—	47	2	no	—
231	C₂²×C₈	—	94	1	no	—
232	C₄×Q₈	Dic₉₄	1	1	yes	characteristic
233	C₄×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
234	C₄×D₄	Dic₉₄	1	2	yes	—
235	C₄×D₄	Dic₉₄	1	1	yes	characteristic
236	C₄×D₄	C₂²⋊D₄₇	1	2	yes	—
237	C₄×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
238	#117	C₄×D₄₇	1	1	yes	characteristic
239	#117	C₄×D₄₇	1	2	yes	—
240	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	—	47	4	no	—
245	C₄⋊C₈	—	47	4	no	—
246	C₄⋊C₈	—	47	1	no	—
247	C₄⋊C₈	—	47	2	no	—
248	C₄⋊C₈	—	47	1	no	—
249	C₄⋊C₈	—	94	1	no	—
250	C₄⋊C₈	—	94	2	no	—
251	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
252	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
253	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
254	C₂²⋊C₈	—	2	1	no	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	47	4	no	—
257	C₂²⋊C₈	—	47	4	no	—
258	C₂²⋊C₈	—	94	1	no	—
259	C₂²⋊C₈	—	94	2	no	—
260	#97	—	47	2	no	—
261	#97	—	94	2	no	—
262	C₄×C₈	—	2	1	no	—
263	C₄×C₈	—	2	2	no	—
264	C₄×C₈	—	47	2	no	—
265	C₄×C₈	—	47	1	no	—
266	C₄×C₈	—	47	2	no	—
267	C₄×C₈	—	47	1	no	—
268	C₄×C₈	—	94	1	no	—
269	#41	—	2	4	no	—
270	C₂×Q₈	Dic₄₇⋊C₄	1	1	yes	characteristic
271	C₂×D₄	Dic₄₇⋊C₄	1	2	yes	—
272	C₂×D₄	Dic₄₇⋊C₄	1	1	yes	characteristic
273	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
274	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
276	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
279	M₁₆	—	94	4	no	—
280	M₁₆	—	376	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	8	1	no	—
285	C₂×C₈	—	8	2	no	—
286	C₂×C₈	—	47	2	no	—
287	C₂×C₈	—	47	2	no	—
288	C₂×C₈	—	47	4	no	—
289	C₂×C₈	—	94	4	no	—
290	C₂×C₈	—	94	1	no	—
291	C₂×C₈	—	94	1	no	—
292	C₂×C₈	—	94	2	no	—
293	C₂×C₈	—	376	1	no	—
294	C₄⋊C₄	Dic₄₇⋊C₄	1	2	yes	—
295	C₄⋊C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
296	C₄⋊C₄	Dic₄₇⋊C₄	1	2	yes	—
297	C₄⋊C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
298	C₂²⋊C₄	Dic₄₇⋊C₄	1	2	yes	—
299	C₂²⋊C₄	Dic₄₇⋊C₄	1	2	yes	—
300	C₂²⋊C₄	Dic₄₇⋊C₄	1	2	yes	—
301	C₄²	C₂×Dic₉₄	1	2	yes	—
302	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
304	C₄²	C₂×C₄×D₄₇	1	2	yes	—
305	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
306	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
307	C₄²	C₂³×D₄₇	1	1	yes	characteristic
308	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
309	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
310	Q₈	—	2	2	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
321	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
322	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
323	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
324	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
325	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
326	C₂×C₄	#13094030	1	4	yes	—
327	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
329	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
330	C₈	—	4	2	no	—
331	C₈	—	4	1	no	—
332	C₈	—	4	1	no	—
333	C₈	—	94	2	no	—
334	C₈	—	94	2	no	—
335	C₈	—	94	4	no	—
336	C₈	—	188	2	no	—
337	C₈	—	188	1	no	—
338	C₈	—	188	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	#15689149	1	2	yes	—
345	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	#15689149	1	2	yes	—
353	C₄	C₂×#13094030	1	2	yes	—
354	C₄	#15689926	1	1	yes	characteristic
355	C₄	#15689938	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×#15689149	1	1	yes	characteristic
359	C₂	#20251986	1	1	yes	characteristic
360	C₂	#20251997	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#2988	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#2988	D₄₇
hypocenter	C₄₇	#17812
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2988

Complements

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

N	H ≅ G/N	type
3. #20242176	C₂	semidirect
4. #20242171	C₂	semidirect
5. #20242166	C₂	semidirect
6. #20242162	C₂	semidirect
7. #20242141	C₂	semidirect
8. #20242138	C₂	semidirect
9. #20242134	C₂	semidirect
10. C₂×#15688875	C₂	semidirect
224. C₄₇	#17812	semidirect

Maximal subgroups

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

subgroup	index
#20252505	2
#20252032	2
#20242176	2
#20242171	2
#20242166	2
#20242162	2
#20242141	2
#20242138	2
#20242134	2
C₂×#15688875	2
C₄₇×#2988	2
#17812	47