Subgroups — #26121546

Order 12032.

1678 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689087	C₂	1	2	yes	maximal
2	C₂×#15688892	C₂	1	4	yes	maximal, complemented
3	C₂³×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	C₂²	1	4	yes	—
5	#15689087	—	2	4	no	—
6	#15689087	C₄	1	8	yes	complemented
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
8	#15688892	—	2	8	no	—
9	C₂⁴×C₄×C₄₇	C₄	1	2	yes	—
10	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₄₇	—	2	4	no	—
12	C₂²×C₄⋊C₄×C₄₇	—	2	8	no	—
13	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂×#10625329	—	2	4	no	—
16	C₂×#10625329	—	2	8	no	—
17	C₂×#10625329	—	2	8	no	—
18	C₂×#10625329	C₂×C₄	1	8	yes	complemented
19	C₂²×C₄₇⋊C₈	—	2	4	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	16	no	—
21	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	—	2	4	no	—
23	C₂³×C₄×C₄₇	—	2	4	no	—
24	C₂³×C₄×C₄₇	—	2	4	no	—
25	C₂³×C₄×C₄₇	—	2	8	no	—
26	C₂³×C₄×C₄₇	—	4	8	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
30	C₂³×C₄×C₄₇	D₄	1	2	yes	—
31	C₂³×C₄×C₄₇	D₄	1	4	yes	—
32	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₄₇	—	2	32	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₇	D₄	1	8	yes	—
40	C₂⁴×C₄₇	—	2	4	no	—
41	C₂⁴×C₄₇	—	2	4	no	—
42	C₂⁴×C₄₇	—	2	2	no	—
43	C₂⁴×C₄₇	—	2	2	no	—
44	C₂⁴×C₄₇	C₄²	1	2	yes	—
45	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	16	no	—
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	—	2	8	no	—
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	—	2	16	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	4	32	no	—
59	C₂²×C₄×C₄₇	—	4	8	no	—
60	C₂²×C₄×C₄₇	—	4	8	no	—
61	C₂²×C₄×C₄₇	—	4	8	no	—
62	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	#41	1	4	yes	—
74	C₄⋊C₄×C₄₇	—	2	16	no	—
75	C₄⋊C₄×C₄₇	—	2	16	no	—
76	#10625329	—	2	16	no	—
77	#10625329	—	4	8	no	—
78	C₂×C₄₇⋊C₈	—	2	8	no	—
79	C₂×C₄₇⋊C₈	—	4	4	no	—
80	C₂×C₄₇⋊C₈	—	4	4	no	—
81	C₂³×C₄₇	—	2	8	no	—
82	C₂³×C₄₇	—	2	16	no	—
83	C₂³×C₄₇	—	2	8	no	—
84	C₂³×C₄₇	—	2	8	no	—
85	C₂³×C₄₇	—	2	8	no	—
86	C₂³×C₄₇	—	2	4	no	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	—	2	4	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	#95	1	4	yes	—
95	C₂³×C₄₇	#100	1	4	yes	—
96	C₂³×C₄₇	#101	1	4	yes	—
97	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
98	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₇	#117	1	2	yes	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	16	no	—
107	C₂×C₄×C₄₇	—	2	16	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	4	32	no	—
110	C₂×C₄×C₄₇	—	4	8	no	—
111	C₂×C₄×C₄₇	—	4	8	no	—
112	C₂×C₄×C₄₇	—	4	8	no	—
113	C₂×C₄×C₄₇	#95	1	4	yes	—
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
117	C₂×C₄×C₄₇	#120	1	4	yes	—
118	C₂×C₄×C₄₇	#121	1	4	yes	—
119	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
120	C₂×C₄×C₄₇	#123	1	4	yes	—
121	C₄₇⋊C₈	—	4	8	no	—
122	#10727	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	16	no	—
124	C₂²×C₄₇	—	2	16	no	—
125	C₂²×C₄₇	—	2	4	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	4	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	#343	1	8	yes	complemented
137	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
138	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
139	C₂²×C₄₇	#379	1	1	yes	characteristic
140	C₂²×C₄₇	#380	1	2	yes	—
141	C₂²×C₄₇	#381	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#100	1	2	yes	—
143	C₂²×C₄₇	C₂×#101	1	2	yes	—
144	C₄×C₄₇	—	2	4	no	—
145	C₄×C₄₇	—	4	8	no	—
146	C₄×C₄₇	#385	1	4	yes	—
147	C₄×C₄₇	#386	1	4	yes	—
148	C₂³×C₄⋊C₄	D₄₇	1	1	yes	characteristic
149	C₂×#407	—	47	2	no	—
150	C₂×#343	—	47	4	no	—
151	C₂×C₄₇	—	2	8	no	—
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	C₂²⋊#95	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#343	1	2	yes	—
156	C₂×C₄₇	#1783	1	1	yes	characteristic
157	C₂×C₄₇	#1784	1	1	yes	characteristic
158	C₂×C₄₇	#1821	1	2	yes	—
159	C₂⁴×C₄	Dic₄₇	1	2	yes	—
160	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
161	C₂²×M₁₆	—	47	4	no	—
162	C₂²×C₄⋊C₄	—	2	4	no	—
163	C₂²×C₄⋊C₄	—	2	8	no	—
164	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
165	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
166	#407	—	47	8	no	—
167	#407	—	94	4	no	—
168	C₂×C₂²⋊C₈	—	94	4	no	—
169	#343	—	94	8	no	—
170	C₄₇	#10727	1	1	yes	characteristic, complemented, Sylow
171	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂³×C₄	—	2	4	no	—
173	C₂³×C₄	—	2	4	no	—
174	C₂³×C₄	—	2	4	no	—
175	C₂³×C₄	—	2	8	no	—
176	C₂³×C₄	—	4	8	no	—
177	C₂³×C₄	Dic₉₄	1	2	yes	—
178	C₂³×C₄	C₄×D₄₇	1	4	yes	—
179	C₂³×C₄	D₁₈₈	1	2	yes	—
180	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
182	C₂³×C₄	C₂²⋊D₄₇	1	4	yes	—
183	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂×M₁₆	—	47	8	no	—
185	C₂×M₁₆	—	94	4	no	—
186	C₂×M₁₆	—	94	8	no	—
187	C₂×M₁₆	—	94	8	no	—
188	C₂²×C₈	—	94	4	no	—
189	C₂×C₄⋊C₄	—	2	32	no	—
190	C₂×C₄⋊C₄	—	2	8	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	8	yes	—
195	C₂²⋊C₈	—	94	16	no	—
196	C₂⁴	—	2	4	no	—
197	C₂⁴	—	2	4	no	—
198	C₂⁴	—	2	2	no	—
199	C₂⁴	—	2	2	no	—
200	C₂⁴	C₄×Dic₄₇	1	2	yes	—
201	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
202	C₂⁴	C₂²⋊Dic₄₇	1	2	yes	—
203	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	16	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	16	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	32	no	—
215	C₂²×C₄	—	4	8	no	—
216	C₂²×C₄	—	4	8	no	—
217	C₂²×C₄	—	4	8	no	—
218	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
219	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
220	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
221	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	8	yes	—
222	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
223	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
224	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
226	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
227	C₂²×C₄	#10625350	1	4	yes	—
228	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
230	M₁₆	—	94	16	no	—
231	M₁₆	—	188	8	no	—
232	C₂×C₈	—	94	8	no	—
233	C₂×C₈	—	188	4	no	—
234	C₂×C₈	—	188	4	no	—
235	C₄⋊C₄	—	2	16	no	—
236	C₄⋊C₄	—	2	16	no	—
237	C₂³	—	2	8	no	—
238	C₂³	—	2	16	no	—
239	C₂³	—	2	8	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	8	no	—
249	C₂³	—	2	8	no	—
250	C₂³	C₄₇⋊#95	1	4	yes	—
251	C₂³	#13093941	1	4	yes	—
252	C₂³	#13093943	1	4	yes	—
253	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
254	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
255	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
256	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
257	C₂³	#13093983	1	2	yes	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	16	no	—
263	C₂×C₄	—	2	16	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	32	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	C₄₇⋊#95	1	4	yes	—
270	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
271	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
272	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
273	C₂×C₄	#13093993	1	4	yes	—
274	C₂×C₄	#13093998	1	4	yes	—
275	C₂×C₄	#13094003	1	4	yes	—
276	C₂×C₄	#13094007	1	4	yes	—
277	C₈	—	188	8	no	—
278	C₂²	—	2	16	no	—
279	C₂²	—	2	16	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	#15688892	1	8	yes	complemented
292	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
293	C₂²	C₄×C₂²⋊Dic₄₇	1	2	yes	—
294	C₂²	#15688979	1	1	yes	characteristic
295	C₂²	#15688983	1	2	yes	—
296	C₂²	#15688986	1	1	yes	characteristic
297	C₂²	C₂×#13093941	1	2	yes	—
298	C₂²	C₂×#13093943	1	2	yes	—
299	C₄	—	2	4	no	—
300	C₄	—	4	8	no	—
301	C₄	#15688996	1	4	yes	—
302	C₄	#15689002	1	4	yes	—
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	—	2	8	no	—
306	C₂	#20241611	1	1	yes	characteristic
307	C₂	C₂×#15688892	1	2	yes	—
308	C₂	#20243309	1	1	yes	characteristic
309	C₂	#20243312	1	1	yes	characteristic
310	C₂	#20243468	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂³×C₄⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂³×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#10727
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#15688892	C₂	semidirect
6. #15689087	C₄	semidirect
18. C₂×#10625329	C₂×C₄	semidirect
136. C₂²×C₄₇	#343	semidirect
170. C₄₇	#10727	semidirect
291. C₂²	#15688892	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689087	2
C₂×#15688892	2
C₂³×C₄⋊C₄×C₄₇	2
#10727	47