Subgroups — #26247193

Order 12032.

778 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	C₄×#13094023	C₂	1	2	yes	maximal
2	#20244868	C₂	1	2	yes	maximal
3	#20244867	C₂	1	2	yes	maximal
4	#20244720	C₂	1	2	yes	maximal
5	#20244719	C₂	1	2	yes	maximal
6	C₄×#13093955	C₂	1	2	yes	maximal
7	C₄×#13093956	C₂	1	1	yes	maximal, characteristic
8	#20243133	C₂	1	1	yes	maximal, characteristic
9	C₄×#106×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094023	C₂²	1	2	yes	—
11	C₂×C₄×Dic₉₄	—	2	2	no	—
12	C₂×#13093956	C₂²	1	2	yes	—
13	C₂×#13093955	C₂²	1	2	yes	—
14	C₂×#13093955	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	#15689027	—	2	2	no	—
17	#15689022	C₂²	1	2	yes	—
18	#15689020	—	2	2	no	—
19	#15689019	C₂²	1	2	yes	—
20	#15688992	C₂²	1	2	yes	—
21	C₄×Dic₄₇⋊C₄	—	2	2	no	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
24	#15688886	C₂²	1	1	yes	characteristic
25	#15688886	C₂²	1	1	yes	characteristic
26	#15688885	C₂²	1	2	yes	—
27	#15688885	C₂²	1	2	yes	—
28	#15688877	C₂²	1	2	yes	—
29	#15688877	C₂²	1	2	yes	—
30	C₂×#106×C₄₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
33	C₄₇×#340	C₂²	1	1	yes	characteristic
34	C₄₇×#340	C₂²	1	1	yes	characteristic
35	C₂²×Dic₉₄	—	2	2	no	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	#13094023	—	2	4	no	—
38	#13094023	—	2	4	no	—
39	C₄×Dic₉₄	—	4	8	no	—
40	C₂×C₄⋊Dic₄₇	—	2	2	no	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
43	C₂×Dic₄₇⋊C₄	—	2	2	no	—
44	C₂×Dic₄₇⋊C₄	—	2	2	no	—
45	C₂×Dic₄₇⋊C₄	—	2	2	no	—
46	C₂×Dic₄₇⋊C₄	—	2	2	no	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
52	#13093956	—	2	4	no	—
53	#13093955	—	2	4	no	—
54	#13093955	—	2	4	no	—
55	C₄×C₄₇⋊C₈	Q₈	1	4	yes	complemented
56	C₄₇⋊#95	—	2	2	no	—
57	C₄₇⋊#95	—	2	2	no	—
58	C₄₇⋊#95	—	2	2	no	—
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₄₇	—	2	2	no	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	#106×C₄₇	—	2	4	no	—
66	C₄×C₈×C₄₇	Q₈	1	4	yes	complemented
67	#95×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	—	2	2	no	—
75	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
76	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
77	C₂×Dic₉₄	—	2	8	no	—
78	C₂×Dic₉₄	—	4	4	no	—
79	C₂×Dic₉₄	—	4	4	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	#41	1	4	yes	—
88	C₄²×C₄₇	—	4	4	no	—
89	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄⋊Dic₄₇	—	2	2	no	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	4	4	no	—
95	C₄⋊Dic₄₇	#41	1	4	yes	—
96	Dic₄₇⋊C₄	—	2	8	no	—
97	Dic₄₇⋊C₄	—	4	4	no	—
98	Dic₄₇⋊C₄	—	4	4	no	—
99	Dic₄₇⋊C₄	—	4	4	no	—
100	C₄×Dic₄₇	—	2	2	no	—
101	C₄×Dic₄₇	—	2	2	no	—
102	C₄×Dic₄₇	—	4	1	no	—
103	C₄×Dic₄₇	—	4	1	no	—
104	C₄×Dic₄₇	—	4	4	no	—
105	C₄×Dic₄₇	SD₁₆	1	8	yes	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	—	2	2	no	—
108	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
109	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
121	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	2	no	—
127	C₂×Dic₄₇	—	2	2	no	—
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	C₂×SD₁₆	1	4	yes	—
136	C₂×Dic₄₇	C₂×SD₁₆	1	4	yes	—
137	Dic₉₄	—	4	16	no	—
138	C₈×C₄₇	—	2	2	no	—
139	C₈×C₄₇	—	2	2	no	—
140	C₄₇⋊C₈	—	2	4	no	—
141	#23837	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	#525	1	2	yes	—
143	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
145	C₂²×C₄₇	#575	1	1	yes	characteristic
146	C₂²×C₄₇	#575	1	1	yes	characteristic
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	4	4	no	—
149	C₄×C₄₇	—	8	2	no	—
150	C₄×C₄₇	#498	1	4	yes	—
151	C₄×C₄₇	#533	1	1	yes	characteristic
152	C₄×C₄₇	#533	1	2	yes	—
153	C₄×C₄₇	#533	1	1	yes	characteristic
154	Dic₄₇	—	2	4	no	—
155	Dic₄₇	—	2	4	no	—
156	Dic₄₇	—	4	4	no	—
157	Dic₄₇	—	8	2	no	—
158	C₄×C₄⋊Q₈	—	47	2	no	—
159	#2084	—	47	2	no	—
160	#2084	—	47	2	no	—
161	#2058	—	47	2	no	—
162	#2058	—	47	2	no	—
163	C₄×#106	—	47	1	no	—
164	C₄×#106	—	47	2	no	—
165	C₄×#106	D₄₇	1	1	yes	characteristic
166	#1744	—	47	1	no	—
167	C₂×C₄₇	#2399	1	1	yes	characteristic
168	C₂×C₄₇	#3101	1	2	yes	—
169	C₂×C₄₇	#3103	1	2	yes	—
170	C₂×C₄₇	C₂×#498	1	1	yes	characteristic
171	C₂×C₄₇	#3162	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	—	47	2	no	—
173	C₂×C₄×Q₈	—	94	2	no	—
174	C₂×#106	—	47	2	no	—
175	C₂×#106	—	47	2	no	—
176	C₂×#106	—	47	2	no	—
177	C₂×#106	C₂×D₄₇	1	2	yes	—
178	C₂×C₄×C₈	—	47	1	no	—
179	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
180	#391	—	94	2	no	—
181	#389	—	47	2	no	—
182	#389	—	47	2	no	—
183	#389	—	94	2	no	—
184	#384	—	47	2	no	—
185	C₄×C₄⋊C₄	—	47	2	no	—
186	C₄×C₄⋊C₄	—	94	2	no	—
187	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
188	C₄³	—	47	1	no	—
189	#340	—	47	1	no	—
190	#340	—	47	1	no	—
191	#340	—	47	2	no	—
192	#340	—	47	2	no	—
193	#340	C₂×D₄₇	1	1	yes	characteristic
194	#340	C₂×D₄₇	1	1	yes	characteristic
195	#336	—	47	2	no	—
196	#336	—	47	2	no	—
197	C₄₇	#23837	1	1	yes	characteristic, complemented, Sylow
198	C₂²×Q₈	—	94	2	no	—
199	C₂²×C₈	—	47	2	no	—
200	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊Q₈	—	94	4	no	—
203	C₄⋊Q₈	—	94	4	no	—
204	C₄×Q₈	—	188	8	no	—
205	C₂×C₄⋊C₄	—	47	2	no	—
206	C₂×C₄⋊C₄	—	47	2	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	—	94	2	no	—
209	C₂×C₄⋊C₄	—	94	2	no	—
210	C₂×C₄⋊C₄	—	94	2	no	—
211	C₂×C₄⋊C₄	—	94	2	no	—
212	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
213	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	47	2	no	—
216	C₂×C₄²	—	47	2	no	—
217	C₂×C₄²	—	47	1	no	—
218	C₂×C₄²	—	47	1	no	—
219	C₂×C₄²	—	94	2	no	—
220	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
221	#106	—	2	4	no	—
222	#106	—	94	4	no	—
223	#106	—	94	4	no	—
224	#106	—	94	4	no	—
225	C₄×C₈	—	47	4	no	—
226	C₄×C₈	Dic₉₄	1	4	yes	complemented
227	#95	—	2	2	no	—
228	#95	—	94	2	no	—
229	#95	—	94	2	no	—
230	#95	—	94	2	no	—
231	C₂×Q₈	—	94	8	no	—
232	C₂×Q₈	—	188	4	no	—
233	C₂×Q₈	—	188	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	47	2	no	—
237	C₂²×C₄	—	47	2	no	—
238	C₂²×C₄	—	94	2	no	—
239	C₂²×C₄	—	94	2	no	—
240	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	47	4	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	94	2	no	—
248	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
249	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	47	4	no	—
254	C₄⋊C₄	—	94	8	no	—
255	C₄⋊C₄	—	94	2	no	—
256	C₄⋊C₄	—	94	2	no	—
257	C₄⋊C₄	—	94	2	no	—
258	C₄⋊C₄	—	188	4	no	—
259	C₄⋊C₄	—	188	4	no	—
260	C₄⋊C₄	—	188	4	no	—
261	C₄⋊C₄	—	188	4	no	—
262	C₄⋊C₄	#10625350	1	4	yes	—
263	C₄²	—	4	4	no	—
264	C₄²	—	47	8	no	—
265	C₄²	—	94	2	no	—
266	C₄²	—	94	2	no	—
267	C₄²	—	188	1	no	—
268	C₄²	—	188	1	no	—
269	C₄²	—	188	4	no	—
270	C₄²	C₂×Dic₉₄	1	2	yes	—
271	C₄²	Q₈×D₄₇	1	2	yes	—
272	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
273	Q₈	—	188	16	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	47	4	no	—
283	C₂×C₄	—	47	4	no	—
284	C₂×C₄	—	94	2	no	—
285	C₂×C₄	—	94	2	no	—
286	C₂×C₄	—	94	2	no	—
287	C₂×C₄	—	94	2	no	—
288	C₂×C₄	—	94	4	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	#13094023	1	2	yes	—
295	C₂×C₄	#13094023	1	2	yes	—
296	C₂×C₄	SD₁₆×D₄₇	1	4	yes	—
297	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
299	C₂×C₄	#41×D₄₇	1	2	yes	—
300	C₂×C₄	C₂×#10625350	1	2	yes	—
301	C₈	—	2	2	no	—
302	C₈	—	2	2	no	—
303	C₈	—	94	4	no	—
304	C₂²	#15689632	1	2	yes	—
305	C₂²	C₂×#13094023	1	1	yes	characteristic
306	C₂²	C₂×SD₁₆×D₄₇	1	2	yes	—
307	C₂²	#15689920	1	1	yes	characteristic
308	C₂²	#15689920	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	8	2	no	—
312	C₄	—	94	4	no	—
313	C₄	—	94	4	no	—
314	C₄	—	376	2	no	—
315	C₄	—	188	4	no	—
316	C₄	#15689520	1	4	yes	—
317	C₄	#15689676	1	1	yes	characteristic
318	C₄	#15689677	1	2	yes	—
319	C₄	#15689680	1	1	yes	characteristic
320	C₂	#20246682	1	1	yes	characteristic
321	C₂	#20253394	1	2	yes	—
322	C₂	#20253417	1	2	yes	—
323	C₂	C₂×#15689520	1	1	yes	characteristic
324	C₂	#20253911	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#106×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₄×#106	D₄₇
hypocenter	C₄₇	#23837
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
55. C₄×C₄₇⋊C₈	Q₈	semidirect
66. C₄×C₈×C₄₇	Q₈	semidirect
197. C₄₇	#23837	semidirect
226. C₄×C₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094023	2
#20244868	2
#20244867	2
#20244720	2
#20244719	2
C₄×#13093955	2
C₄×#13093956	2
#20243133	2
C₄×#106×C₄₇	2
#23837	47