Subgroups — #26234708

Order 12032.

926 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689217	C₂	1	2	yes	maximal
3	#20246300	C₂	1	2	yes	maximal
4	#20244119	C₂	1	2	yes	maximal
5	#20243639	C₂	1	4	yes	maximal
6	#20243390	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
11	C₂×#13093989	C₂²	1	4	yes	—
12	C₄×Dic₁₈₈	C₄	1	8	yes	complemented
13	#15689217	—	2	8	no	—
14	C₂×#13093958	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×#13093949	C₂²	1	2	yes	—
17	C₂×#13093947	C₂²	1	4	yes	—
18	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
19	#15689019	C₂²	1	4	yes	—
20	#15689018	—	2	4	no	—
21	#15688992	—	2	2	no	—
22	C₄⋊C₄×Dic₄₇	—	2	2	no	—
23	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
24	#15688885	C₂²	1	4	yes	—
25	#15688877	C₂²	1	2	yes	—
26	C₂×#107×C₄₇	C₂²	1	2	yes	—
27	C₂×#107×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#384	C₂²	1	2	yes	—
30	C₂²×Dic₉₄	C₂³	1	2	yes	—
31	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
32	C₂×Dic₁₈₈	D₄	1	4	yes	—
33	C₂×Dic₁₈₈	Q₈	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₄×Dic₉₄	—	2	4	no	—
36	C₄×Dic₉₄	C₂×C₄	1	8	yes	—
37	#13093989	—	2	8	no	—
38	#13093989	—	4	4	no	—
39	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×Dic₄₇⋊C₄	—	2	2	no	—
43	C₂×Dic₄₇⋊C₄	—	2	4	no	—
44	C₂×Dic₄₇⋊C₄	—	2	4	no	—
45	C₂×C₄×Dic₄₇	—	2	2	no	—
46	C₂×C₄×Dic₄₇	—	2	4	no	—
47	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
48	#13093958	C₂×C₄	1	4	yes	—
49	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
50	#13093949	C₂×C₄	1	8	yes	—
51	#13093947	—	2	8	no	—
52	C₈×Dic₄₇	—	2	4	no	—
53	C₄₇⋊#95	—	2	4	no	—
54	C₄₇⋊#95	D₄	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	#107×C₄₇	—	2	4	no	—
62	#107×C₄₇	—	4	2	no	—
63	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
71	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
72	C₂×Dic₉₄	—	2	2	no	—
73	C₂×Dic₉₄	—	2	2	no	—
74	C₂×Dic₉₄	—	2	2	no	—
75	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
76	C₂×Dic₉₄	C₂×D₄	1	4	yes	—
77	C₂×Dic₉₄	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	4	4	no	—
87	C₄⋊C₄×C₄₇	—	8	2	no	—
88	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₇	#41	1	2	yes	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄⋊Dic₄₇	Q₁₆	1	4	yes	complemented
93	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
94	Dic₄₇⋊C₄	—	2	8	no	—
95	Dic₄₇⋊C₄	—	4	8	no	—
96	Dic₄₇⋊C₄	—	8	4	no	—
97	C₄×Dic₄₇	—	2	8	no	—
98	C₄×Dic₄₇	—	2	2	no	—
99	C₄×Dic₄₇	—	2	2	no	—
100	C₄×Dic₄₇	—	4	8	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	Dic₁₈₈	C₄⋊C₄	1	16	yes	complemented
105	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
114	C₂×C₄×C₄₇	#135	1	2	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	2	4	no	—
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	4	no	—
127	C₂×Dic₄₇	—	2	2	no	—
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	2	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	C₂×Q₁₆	1	4	yes	—
133	Dic₉₄	—	2	4	no	—
134	Dic₉₄	—	2	4	no	—
135	Dic₉₄	—	2	4	no	—
136	Dic₉₄	C₂×C₄⋊C₄	1	8	yes	—
137	C₈×C₄₇	—	2	2	no	—
138	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
139	C₄₇⋊C₈	—	4	2	no	—
140	C₄⋊C₄×Q₁₆	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₄₇	D₄²	1	1	yes	characteristic
143	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
144	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
146	C₂²×C₄₇	#576	1	1	yes	characteristic
147	C₂²×C₄₇	#578	1	1	yes	characteristic
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	8	4	no	—
150	C₄×C₄₇	C₄×Q₁₆	1	4	yes	complemented
151	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₄₇	#520	1	1	yes	characteristic
153	C₄×C₄₇	#547	1	1	yes	characteristic
154	C₄×C₄₇	#554	1	1	yes	characteristic
155	Dic₄₇	—	2	8	no	—
156	Dic₄₇	—	2	4	no	—
157	Dic₄₇	—	4	4	no	—
158	Dic₄₇	—	4	2	no	—
159	C₂×C₄×Q₁₆	—	47	2	no	—
160	C₂×C₄×Q₁₆	—	47	1	no	—
161	Q₈×C₄⋊C₄	—	47	2	no	—
162	#1942	—	47	2	no	—
163	#1863	—	47	4	no	—
164	#1845	D₄₇	1	1	yes	characteristic
165	#1803	—	47	2	no	—
166	C₈×C₄⋊C₄	—	47	1	no	—
167	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
169	C₂×C₄₇	#2951	1	1	yes	characteristic
170	C₂×C₄₇	D₄×Q₁₆	1	1	yes	characteristic
171	C₂×C₄₇	#3287	1	1	yes	characteristic
172	C₂×C₄₇	Q₈×Q₁₆	1	1	yes	characteristic
173	C₂×C₄₇	#3383	1	1	yes	characteristic
174	C₂²×Q₁₆	—	47	1	no	—
175	C₂×C₄×Q₈	—	47	4	no	—
176	C₂×C₄×Q₈	—	47	2	no	—
177	C₄×Q₁₆	—	47	8	no	—
178	C₄×Q₁₆	—	94	8	no	—
179	C₂×#107	—	47	1	no	—
180	C₂×#107	C₂×D₄₇	1	2	yes	—
181	C₂×#107	C₂×D₄₇	1	2	yes	—
182	C₂×C₄⋊C₈	—	47	1	no	—
183	C₂×#103	—	47	2	no	—
184	C₂×#103	—	47	4	no	—
185	C₂×C₄×C₈	—	47	2	no	—
186	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
187	#389	—	47	4	no	—
188	#389	—	94	4	no	—
189	#384	—	94	2	no	—
190	#384	C₂×D₄₇	1	2	yes	—
191	C₄×C₄⋊C₄	—	47	1	no	—
192	C₄×C₄⋊C₄	—	94	2	no	—
193	#340	—	47	4	no	—
194	#336	—	47	2	no	—
195	C₄₇	C₄⋊C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	—	47	2	no	—
197	C₂×Q₁₆	—	47	4	no	—
198	C₂×Q₁₆	—	47	4	no	—
199	C₂×Q₁₆	—	47	4	no	—
200	C₂²×C₈	—	47	2	no	—
201	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
203	C₄×Q₈	—	47	8	no	—
204	C₄×Q₈	—	94	4	no	—
205	C₄×Q₈	—	94	8	no	—
206	C₄×Q₈	—	188	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	47	1	no	—
210	C₂×C₄⋊C₄	—	47	2	no	—
211	C₂×C₄⋊C₄	—	47	1	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄⋊C₄	—	94	4	no	—
214	C₂×C₄⋊C₄	—	94	4	no	—
215	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
216	C₂×C₄²	—	47	2	no	—
217	C₂×C₄²	—	94	2	no	—
218	C₂×C₄²	—	94	4	no	—
219	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
220	#107	—	2	4	no	—
221	#107	—	4	2	no	—
222	#107	—	47	4	no	—
223	C₄⋊C₈	—	94	2	no	—
224	#103	—	47	8	no	—
225	#103	—	94	8	no	—
226	C₄×C₈	—	94	4	no	—
227	C₄×C₈	C₄×D₄₇	1	4	yes	—
228	#95	—	47	2	no	—
229	#95	—	94	4	no	—
230	C₂×Q₈	—	47	4	no	—
231	C₂×Q₈	—	47	4	no	—
232	C₂×Q₈	—	47	4	no	—
233	C₂×Q₈	—	94	2	no	—
234	C₂×Q₈	—	94	2	no	—
235	C₂×Q₈	—	94	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	47	2	no	—
238	C₂²×C₄	—	47	2	no	—
239	C₂²×C₄	—	94	2	no	—
240	C₂²×C₄	—	94	2	no	—
241	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
244	Q₁₆	—	47	16	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	94	2	no	—
248	C₂×C₈	—	94	2	no	—
249	C₂×C₈	—	94	2	no	—
250	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
252	C₂×C₈	D₄×D₄₇	1	2	yes	—
253	C₂×C₈	Q₈×D₄₇	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	—	47	4	no	—
259	C₄⋊C₄	—	47	4	no	—
260	C₄⋊C₄	—	94	8	no	—
261	C₄⋊C₄	—	94	2	no	—
262	C₄⋊C₄	—	94	2	no	—
263	C₄⋊C₄	—	376	4	no	—
264	C₄⋊C₄	—	188	8	no	—
265	C₄²	—	94	8	no	—
266	C₄²	—	94	2	no	—
267	C₄²	—	94	2	no	—
268	C₄²	—	188	8	no	—
269	C₄²	C₂×C₄×D₄₇	1	2	yes	—
270	C₄²	#10625354	1	2	yes	—
271	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
272	Q₈	—	47	8	no	—
273	Q₈	—	94	4	no	—
274	Q₈	—	94	4	no	—
275	Q₈	—	94	4	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	47	4	no	—
282	C₂×C₄	—	94	2	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	94	2	no	—
287	C₂×C₄	—	94	2	no	—
288	C₂×C₄	—	94	4	no	—
289	C₂×C₄	—	94	2	no	—
290	C₂×C₄	—	94	2	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	#13093986	1	2	yes	—
294	C₂×C₄	#13093986	1	2	yes	—
295	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
296	C₂×C₄	#13094043	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
301	C₂×C₄	#13094069	1	1	yes	characteristic
302	C₂×C₄	#13094072	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	188	2	no	—
305	C₈	C₄⋊C₄×D₄₇	1	4	yes	—
306	C₂²	C₂×#13093986	1	1	yes	characteristic
307	C₂²	#15689758	1	1	yes	characteristic
308	C₂²	#15689788	1	1	yes	characteristic
309	C₂²	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
310	C₂²	C₂×#13094043	1	1	yes	characteristic
311	C₂²	#15689924	1	1	yes	characteristic
312	C₂²	#15689936	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	8	4	no	—
315	C₄	—	94	8	no	—
316	C₄	—	94	4	no	—
317	C₄	—	188	4	no	—
318	C₄	—	188	2	no	—
319	C₄	#15689217	1	4	yes	complemented
320	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
321	C₄	#15689604	1	1	yes	characteristic
322	C₄	#15689776	1	1	yes	characteristic
323	C₄	#15689834	1	1	yes	characteristic
324	C₂	#20246288	1	1	yes	characteristic
325	C₂	C₂×#15689217	1	1	yes	characteristic
326	C₂	#20252262	1	1	yes	characteristic
327	C₂	#20255138	1	1	yes	characteristic
328	C₂	#20255146	1	1	yes	characteristic
329	C₂	#20256271	1	1	yes	characteristic
330	C₂	#20256281	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1845	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1845	D₄₇
hypocenter	C₄₇	C₄⋊C₄×Q₁₆
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
12. C₄×Dic₁₈₈	C₄	semidirect
92. C₄⋊Dic₄₇	Q₁₆	semidirect
104. Dic₁₈₈	C₄⋊C₄	semidirect
150. C₄×C₄₇	C₄×Q₁₆	semidirect
195. C₄₇	C₄⋊C₄×Q₁₆	semidirect
319. C₄	#15689217	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈₈	2
C₂×#15689217	2
#20246300	2
#20244119	2
#20243639	2
#20243390	2
C₈×C₄⋊Dic₄₇	2
C₄₇×#1845	2
C₄⋊C₄×Q₁₆	47