Subgroups — #26422130

Order 12032.

7538 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689634	C₂	1	3	yes	maximal, complemented
2	#20251680	C₂	1	6	yes	maximal, complemented
3	#20251675	C₂	1	4	yes	maximal, complemented
4	#20251670	C₂	1	4	yes	maximal, complemented
5	#20251665	C₂	1	2	yes	maximal, complemented
6	C₄×#13093993	C₂	1	8	yes	maximal, complemented
7	#20245771	C₂	1	3	yes	maximal, complemented
8	C₂⁵×C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10625350	C₂²	1	1	yes	characteristic, complemented
10	#15689634	—	2	24	no	—
11	C₂×#13094007	C₂²	1	6	yes	complemented
12	C₂×#13094003	C₂²	1	6	yes	complemented
13	C₂×#13093998	C₂²	1	6	yes	complemented
14	C₂×#13093993	C₂²	1	8	yes	—
15	C₂×#13093993	C₂²	1	6	yes	—
16	C₂×C₄×C₂²⋊D₄₇	C₂²	1	12	yes	complemented
17	C₂×#13093983	C₂²	1	3	yes	complemented
18	#15689002	C₂²	1	24	yes	complemented
19	#15688996	C₂²	1	24	yes	complemented
20	#15688986	C₂²	1	6	yes	complemented
21	#15688983	C₂²	1	12	yes	complemented
22	#15688979	C₂²	1	6	yes	—
23	#15688979	C₂²	1	8	yes	—
24	C₄×C₂²⋊Dic₄₇	C₂²	1	12	yes	complemented
25	C₂⁶×C₄₇	C₂²	1	1	yes	characteristic
26	C₂⁴×C₄×C₄₇	—	2	24	no	—
27	C₂⁴×C₄×C₄₇	C₂²	1	4	yes	—
28	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
29	C₂⁴×C₄×C₄₇	C₂²	1	4	yes	—
30	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
31	C₂×#10625350	—	2	12	no	—
32	C₂²×Dic₉₄	C₂³	1	1	yes	characteristic, complemented
33	C₂²×D₁₈₈	C₂³	1	1	yes	characteristic, complemented
34	C₂²×C₂²⋊D₄₇	C₂³	1	12	yes	—
35	C₂²×C₂²⋊D₄₇	C₂³	1	2	yes	—
36	C₂²×C₄×D₄₇	C₂³	1	2	yes	complemented
37	#13094007	—	2	24	no	—
38	#13094003	—	2	24	no	—
39	#13093998	—	2	24	no	—
40	#13093993	—	2	24	no	—
41	#13093993	—	2	32	no	—
42	C₄×C₂²⋊D₄₇	—	2	48	no	—
43	#13093983	—	2	12	no	—
44	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	complemented
45	C₂×Dic₄₇⋊C₄	C₂³	1	12	yes	complemented
46	C₂×C₂²⋊Dic₄₇	C₂³	1	24	yes	—
47	C₂×C₂²⋊Dic₄₇	C₂³	1	6	yes	—
48	C₂×C₂²⋊Dic₄₇	C₂³	1	12	yes	—
49	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	12	yes	complemented
50	C₂×C₄×Dic₄₇	C₂³	1	6	yes	complemented
51	C₄₇⋊#95	C₂³	1	24	yes	complemented
52	C₂⁵×C₄₇	—	2	12	no	—
53	C₂⁵×C₄₇	—	2	16	no	—
54	C₂⁵×C₄₇	C₂³	1	3	yes	—
55	C₂⁵×C₄₇	C₂³	1	4	yes	—
56	C₂³×C₄×C₄₇	—	2	128	no	—
57	C₂³×C₄×C₄₇	—	2	12	no	—
58	C₂³×C₄×C₄₇	—	2	48	no	—
59	C₂³×C₄×C₄₇	—	2	48	no	—
60	C₂³×C₄×C₄₇	—	2	24	no	—
61	C₂³×C₄×C₄₇	—	2	24	no	—
62	C₂³×C₄×C₄₇	D₄	1	24	yes	complemented
63	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
64	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
65	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
66	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
67	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
68	C₂⁴×C₄₇	—	2	32	no	—
69	C₂⁴×C₄₇	—	2	192	no	—
70	C₂⁴×C₄₇	—	2	24	no	—
71	C₂⁴×C₄₇	—	2	12	no	—
72	C₂⁴×C₄₇	—	2	24	no	—
73	C₂⁴×C₄₇	—	2	24	no	—
74	C₂⁴×C₄₇	C₂×D₄	1	12	yes	—
75	C₂⁴×C₄₇	#41	1	16	yes	complemented
76	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₂⁴×C₄₇	C₂⁴	1	6	yes	—
78	C₂³×D₄₇	C₂⁴	1	2	yes	—
79	C₂²×C₄×C₄₇	—	2	96	no	—
80	C₂²×C₄×C₄₇	—	2	128	no	—
81	C₂²×C₄×C₄₇	—	2	128	no	—
82	C₂²×C₄×C₄₇	—	2	24	no	—
83	C₂²×C₄×C₄₇	—	2	12	no	—
84	C₂²×C₄×C₄₇	—	2	96	no	—
85	C₂²×C₄×C₄₇	—	2	48	no	—
86	C₂²×C₄×C₄₇	—	2	24	no	—
87	C₂²×C₄×C₄₇	—	2	24	no	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	24	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	24	yes	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
92	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
93	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₂×C₂²⋊D₄₇	—	2	24	no	—
96	C₂×C₂²⋊D₄₇	—	2	12	no	—
97	C₂×C₂²⋊D₄₇	—	2	48	no	—
98	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
99	C₂²×Dic₄₇	C₂⁴	1	2	yes	—
100	#10625350	—	4	16	no	—
101	C₂×D₁₈₈	—	2	6	no	—
102	C₂×C₄×D₄₇	—	2	12	no	—
103	C₂×Dic₉₄	—	2	6	no	—
104	C₂²⋊Dic₄₇	—	2	12	no	—
105	C₂²⋊Dic₄₇	—	2	24	no	—
106	C₂²⋊Dic₄₇	—	2	48	no	—
107	(C₂×D₄₇)⋊C₄	—	2	24	no	—
108	C₄⋊Dic₄₇	—	2	12	no	—
109	Dic₄₇⋊C₄	—	2	24	no	—
110	C₄×Dic₄₇	—	2	12	no	—
111	C₂³×C₄₇	—	2	32	no	—
112	C₂³×C₄₇	—	2	256	no	—
113	C₂³×C₄₇	—	2	96	no	—
114	C₂³×C₄₇	—	2	96	no	—
115	C₂³×C₄₇	—	2	24	no	—
116	C₂³×C₄₇	—	2	48	no	—
117	C₂³×C₄₇	—	2	96	no	—
118	C₂³×C₄₇	—	2	12	no	—
119	C₂³×C₄₇	—	2	4	no	—
120	C₂³×C₄₇	—	2	12	no	—
121	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
122	C₂³×C₄₇	C₂²×D₄	1	12	yes	—
123	C₂³×C₄₇	C₂×#41	1	24	yes	—
124	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
125	C₂²×D₄₇	—	2	6	no	—
126	C₂²×D₄₇	—	2	8	no	—
127	C₂×C₄×C₄₇	—	2	96	no	—
128	C₂×C₄×C₄₇	—	2	24	no	—
129	C₂×C₄×C₄₇	—	2	12	no	—
130	C₂×C₄×C₄₇	—	2	48	no	—
131	C₂×C₄×C₄₇	—	2	48	no	—
132	C₂×C₄×C₄₇	#120	1	16	yes	complemented
133	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
134	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
135	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
136	C₂²⋊D₄₇	—	4	8	no	—
137	C₂²⋊D₄₇	—	4	48	no	—
138	C₂×Dic₄₇	—	2	12	no	—
139	C₂×Dic₄₇	—	2	6	no	—
140	C₂×Dic₄₇	—	2	24	no	—
141	D₁₈₈	—	4	4	no	—
142	C₄×D₄₇	—	4	8	no	—
143	Dic₉₄	—	4	4	no	—
144	#36614	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	2	96	no	—
146	C₂²×C₄₇	—	2	128	no	—
147	C₂²×C₄₇	—	2	12	no	—
148	C₂²×C₄₇	—	2	12	no	—
149	C₂²×C₄₇	—	2	12	no	—
150	C₂²×C₄₇	—	2	48	no	—
151	C₂²×C₄₇	C₂×#120	1	4	yes	—
152	C₂²×C₄₇	#525	1	24	yes	complemented
153	C₂²×C₄₇	C₂³×D₄	1	3	yes	—
154	C₂²×C₄₇	C₂²×#41	1	4	yes	—
155	C₂×D₄₇	—	4	2	no	—
156	C₂×D₄₇	—	4	12	no	—
157	C₄×C₄₇	—	2	12	no	—
158	C₄×C₄₇	C₂×#120	1	4	yes	—
159	C₄×C₄₇	C₂×#120	1	4	yes	—
160	Dic₄₇	—	4	2	no	—
161	Dic₄₇	—	4	12	no	—
162	C₂⁵×C₄	D₄₇	1	1	yes	characteristic, complemented
163	C₂×#525	—	47	3	no	—
164	#2849	—	47	6	no	—
165	#2848	—	47	4	no	—
166	#2847	—	47	4	no	—
167	#2846	—	47	2	no	—
168	C₄×#120	—	47	8	no	—
169	#2280	—	47	3	no	—
170	C₂×C₄₇	—	2	4	no	—
171	C₂×C₄₇	—	2	24	no	—
172	C₂×C₄₇	C₂²×#120	1	1	yes	characteristic
173	C₂×C₄₇	C₂×#525	1	6	yes	—
174	D₄₇	—	8	2	no	—
175	C₂⁶	C₂×D₄₇	1	1	yes	characteristic
176	C₂²×#41	—	47	1	no	—
177	C₂⁴×C₄	—	2	24	no	—
178	C₂⁴×C₄	C₂×D₄₇	1	4	yes	—
179	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
180	C₂⁴×C₄	C₂×D₄₇	1	4	yes	—
181	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
182	#525	—	94	24	no	—
183	C₂×#123	—	47	6	no	—
184	C₂×C₂²⋊Q₈	—	47	6	no	—
185	C₂×#121	—	47	6	no	—
186	C₂×#120	—	47	8	no	—
187	C₂×#120	—	47	6	no	—
188	C₂×C₄×D₄	—	47	12	no	—
189	C₂×#117	—	47	3	no	—
190	#386	—	47	24	no	—
191	#385	—	47	24	no	—
192	#381	—	47	6	no	—
193	#380	—	47	12	no	—
194	#379	—	47	8	no	—
195	#379	—	47	6	no	—
196	C₄×C₂²⋊C₄	—	47	12	no	—
197	C₄₇	#36614	1	1	yes	characteristic, complemented, Sylow
198	C₂⁵	—	2	16	no	—
199	C₂⁵	—	2	12	no	—
200	C₂⁵	C₂²×D₄₇	1	4	yes	—
201	C₂⁵	C₂²×D₄₇	1	3	yes	—
202	C₂×#41	—	94	12	no	—
203	C₂²×Q₈	—	47	1	no	—
204	C₂²×D₄	—	47	12	no	—
205	C₂²×D₄	—	47	2	no	—
206	C₂²×D₄	—	47	1	no	—
207	C₂³×C₄	—	2	128	no	—
208	C₂³×C₄	—	2	12	no	—
209	C₂³×C₄	—	2	48	no	—
210	C₂³×C₄	—	2	48	no	—
211	C₂³×C₄	—	2	24	no	—
212	C₂³×C₄	—	2	24	no	—
213	C₂³×C₄	—	47	2	no	—
214	C₂³×C₄	C₂²⋊D₄₇	1	24	yes	complemented
215	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
216	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
217	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
218	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
219	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
220	#123	—	94	24	no	—
221	C₂²⋊Q₈	—	94	24	no	—
222	#121	—	94	24	no	—
223	#120	—	94	32	no	—
224	#120	—	94	24	no	—
225	C₄×D₄	—	94	48	no	—
226	#117	—	94	12	no	—
227	C₂×C₄⋊C₄	—	47	6	no	—
228	C₂×C₄⋊C₄	—	47	12	no	—
229	C₂×C₂²⋊C₄	—	47	6	no	—
230	C₂×C₂²⋊C₄	—	47	12	no	—
231	C₂×C₂²⋊C₄	—	47	24	no	—
232	C₂×C₂²⋊C₄	—	47	12	no	—
233	C₂×C₄²	—	47	6	no	—
234	#95	—	47	24	no	—
235	C₂⁴	—	2	32	no	—
236	C₂⁴	—	2	192	no	—
237	C₂⁴	—	2	24	no	—
238	C₂⁴	—	2	12	no	—
239	C₂⁴	—	2	24	no	—
240	C₂⁴	—	2	24	no	—
241	C₂⁴	—	47	2	no	—
242	C₂⁴	#10625350	1	16	yes	complemented
243	C₂⁴	C₂×C₂²⋊D₄₇	1	12	yes	—
244	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
245	C₂⁴	C₂³×D₄₇	1	6	yes	—
246	#41	—	188	16	no	—
247	C₂×Q₈	—	94	6	no	—
248	C₂×D₄	—	94	24	no	—
249	C₂×D₄	—	94	12	no	—
250	C₂×D₄	—	94	48	no	—
251	C₂×D₄	—	94	6	no	—
252	C₂²×C₄	—	2	96	no	—
253	C₂²×C₄	—	2	128	no	—
254	C₂²×C₄	—	2	128	no	—
255	C₂²×C₄	—	2	24	no	—
256	C₂²×C₄	—	2	12	no	—
257	C₂²×C₄	—	2	96	no	—
258	C₂²×C₄	—	2	48	no	—
259	C₂²×C₄	—	2	24	no	—
260	C₂²×C₄	—	2	24	no	—
261	C₂²×C₄	—	47	12	no	—
262	C₂²×C₄	—	47	2	no	—
263	C₂²×C₄	—	94	12	no	—
264	C₂²×C₄	C₂×C₂²⋊D₄₇	1	24	yes	—
265	C₂²×C₄	C₂×C₂²⋊D₄₇	1	24	yes	—
266	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
267	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
268	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
269	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
270	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
271	C₄⋊C₄	—	94	24	no	—
272	C₄⋊C₄	—	94	12	no	—
273	C₂²⋊C₄	—	94	12	no	—
274	C₂²⋊C₄	—	94	24	no	—
275	C₂²⋊C₄	—	94	48	no	—
276	C₂²⋊C₄	—	94	24	no	—
277	C₄²	—	94	12	no	—
278	C₂³	—	2	32	no	—
279	C₂³	—	2	256	no	—
280	C₂³	—	2	96	no	—
281	C₂³	—	2	96	no	—
282	C₂³	—	2	24	no	—
283	C₂³	—	2	96	no	—
284	C₂³	—	2	48	no	—
285	C₂³	—	2	12	no	—
286	C₂³	—	2	4	no	—
287	C₂³	—	2	12	no	—
288	C₂³	—	94	6	no	—
289	C₂³	—	94	8	no	—
290	C₂³	C₂²×C₂²⋊D₄₇	1	6	yes	—
291	C₂³	C₂²×C₂²⋊D₄₇	1	12	yes	—
292	C₂³	C₂×#10625350	1	24	yes	—
293	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
294	Q₈	—	188	4	no	—
295	D₄	—	188	8	no	—
296	D₄	—	188	48	no	—
297	D₄	—	188	4	no	—
298	C₂×C₄	—	2	96	no	—
299	C₂×C₄	—	2	24	no	—
300	C₂×C₄	—	2	12	no	—
301	C₂×C₄	—	2	48	no	—
302	C₂×C₄	—	2	48	no	—
303	C₂×C₄	—	94	12	no	—
304	C₂×C₄	—	94	6	no	—
305	C₂×C₄	—	94	24	no	—
306	C₂×C₄	—	188	8	no	—
307	C₂×C₄	#13093993	1	16	yes	complemented
308	C₂×C₄	C₂²×C₂²⋊D₄₇	1	12	yes	—
309	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
310	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
311	C₂²	—	2	96	no	—
312	C₂²	—	2	128	no	—
313	C₂²	—	2	12	no	—
314	C₂²	—	2	12	no	—
315	C₂²	—	2	12	no	—
316	C₂²	—	2	48	no	—
317	C₂²	—	188	2	no	—
318	C₂²	—	188	12	no	—
319	C₂²	C₂×#13093993	1	4	yes	—
320	C₂²	#15689634	1	24	yes	complemented
321	C₂²	C₂³×C₂²⋊D₄₇	1	3	yes	—
322	C₂²	C₂²×#10625350	1	4	yes	—
323	C₄	—	2	12	no	—
324	C₄	—	188	2	no	—
325	C₄	—	188	12	no	—
326	C₄	C₂×#13093993	1	4	yes	—
327	C₄	C₂×#13093993	1	4	yes	—
328	C₂	—	2	4	no	—
329	C₂	—	2	24	no	—
330	C₂	—	376	2	no	—
331	C₂	C₂²×#13093993	1	1	yes	characteristic
332	C₂	C₂×#15689634	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator, socle, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂⁵×C₄×C₄₇	C₂
hypercenter	C₂⁵×C₄	D₄₇
hypocenter	C₄₇	#36614
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#15689634	C₂	semidirect
2. #20251680	C₂	semidirect
3. #20251675	C₂	semidirect
4. #20251670	C₂	semidirect
5. #20251665	C₂	semidirect
6. C₄×#13093993	C₂	semidirect
7. #20245771	C₂	semidirect
8. C₂⁵×C₄×C₄₇	C₂	semidirect
9. C₂²×#10625350	C₂²	semidirect
11. C₂×#13094007	C₂²	semidirect
12. C₂×#13094003	C₂²	semidirect
13. C₂×#13093998	C₂²	semidirect
16. C₂×C₄×C₂²⋊D₄₇	C₂²	semidirect
17. C₂×#13093983	C₂²	semidirect
18. #15689002	C₂²	semidirect
19. #15688996	C₂²	semidirect
20. #15688986	C₂²	semidirect
21. #15688983	C₂²	semidirect
24. C₄×C₂²⋊Dic₄₇	C₂²	semidirect
32. C₂²×Dic₉₄	C₂³	semidirect
33. C₂²×D₁₈₈	C₂³	semidirect
36. C₂²×C₄×D₄₇	C₂³	semidirect
44. C₂×C₄⋊Dic₄₇	C₂³	semidirect
45. C₂×Dic₄₇⋊C₄	C₂³	semidirect
49. C₂×(C₂×D₄₇)⋊C₄	C₂³	semidirect
50. C₂×C₄×Dic₄₇	C₂³	semidirect
51. C₄₇⋊#95	C₂³	semidirect
62. C₂³×C₄×C₄₇	D₄	semidirect
75. C₂⁴×C₄₇	#41	semidirect
132. C₂×C₄×C₄₇	#120	semidirect
152. C₂²×C₄₇	#525	semidirect
162. C₂⁵×C₄	D₄₇	semidirect
197. C₄₇	#36614	semidirect
214. C₂³×C₄	C₂²⋊D₄₇	semidirect
242. C₂⁴	#10625350	semidirect
307. C₂×C₄	#13093993	semidirect
320. C₂²	#15689634	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689634	2
#20251680	2
#20251675	2
#20251670	2
#20251665	2
C₄×#13093993	2
#20245771	2
C₂⁵×C₄×C₄₇	2
#36614	47