Subgroups — #26215166

Order 12032.

1222 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×C₈×D₁₈₈	C₂	1	2	yes	maximal
2	C₄²×D₁₈₈	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₄₇	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243158	C₂	1	4	yes	maximal, complemented
6	#20243145	C₂	1	2	yes	maximal, complemented
7	#336×D₄₇	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	C₂²	1	4	yes	—
10	C₂×C₄×D₁₈₈	C₄	1	2	yes	—
11	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₄₇	C₂²	1	2	yes	—
13	C₈×D₁₈₈	C₄	1	16	yes	complemented
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	4	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
17	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₄×(C₂×D₄₇)⋊C₄	C₄	1	4	yes	—
21	C₄×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
22	#15688878	C₂²	1	2	yes	—
23	#15688877	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
25	C₄³×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#336	C₂²	1	1	yes	characteristic
27	C₄₇×#336	C₂²	1	1	yes	characteristic
28	C₄₇×#336	C₂²	1	2	yes	—
29	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₄₇	C₂×C₄	1	4	yes	—
31	C₂²×C₄×D₄₇	C₂³	1	2	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₂×C₈×D₄₇	—	4	4	no	—
34	C₂×C₈×D₄₇	—	4	4	no	—
35	C₂×C₈×D₄₇	C₂×C₄	1	16	yes	complemented
36	C₄×D₁₈₈	C₈	1	16	yes	complemented
37	C₄×D₁₈₈	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
40	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
44	C₄²×D₄₇	—	4	4	no	—
45	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
46	(C₂×D₄₇)⋊C₈	C₂×C₄	1	16	yes	complemented
47	C₄×C₄₇⋊C₈	—	4	2	no	—
48	C₈×Dic₄₇	—	4	4	no	—
49	C₄₇⋊#95	C₂×C₄	1	2	yes	—
50	C₄₇⋊#95	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
59	C₂³×D₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
67	C₂×D₁₈₈	C₄²	1	4	yes	—
68	C₂×D₁₈₈	C₂×C₈	1	8	yes	—
69	C₂×C₄×D₄₇	—	2	4	no	—
70	C₂×C₄×D₄₇	—	2	2	no	—
71	C₂×C₄×D₄₇	—	2	2	no	—
72	C₂×C₄×D₄₇	—	4	4	no	—
73	C₂×C₄×D₄₇	—	4	4	no	—
74	C₂×C₄×D₄₇	C₂×C₈	1	16	yes	complemented
75	C₂×C₄×D₄₇	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
82	C₄²×C₄₇	—	2	1	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	1	no	—
85	C₄²×C₄₇	—	4	2	no	—
86	C₄²×C₄₇	C₂×C₈	1	8	yes	complemented
87	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
88	(C₂×D₄₇)⋊C₄	C₄²	1	8	yes	—
89	(C₂×D₄₇)⋊C₄	C₂×C₈	1	16	yes	complemented
90	C₄⋊Dic₄₇	C₄²	1	4	yes	—
91	C₄⋊Dic₄₇	C₂×C₈	1	8	yes	complemented
92	C₄×Dic₄₇	—	2	4	no	—
93	C₄×Dic₄₇	—	4	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
97	C₈×D₄₇	—	4	16	no	—
98	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂²×D₄₇	—	2	2	no	—
100	C₂²×D₄₇	—	2	2	no	—
101	C₂²×D₄₇	—	2	4	no	—
102	C₂²×D₄₇	C₂×C₄²	1	4	yes	—
103	C₂²×D₄₇	C₂²×C₈	1	8	yes	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
112	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
113	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
117	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
123	C₂×Dic₄₇	C₂²×C₈	1	8	yes	—
124	D₁₈₈	C₄×C₈	1	16	yes	complemented
125	C₄×D₄₇	—	2	8	no	—
126	C₄×D₄₇	—	2	8	no	—
127	C₄×D₄₇	—	4	16	no	—
128	C₈×C₄₇	—	2	4	no	—
129	C₈×C₄₇	—	2	4	no	—
130	C₄₇⋊C₈	—	2	8	no	—
131	#20555	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
133	C₂²×C₄₇	#518	1	1	yes	characteristic
134	C₂²×C₄₇	#520	1	1	yes	characteristic
135	C₂²×C₄₇	C₂³×C₈	1	2	yes	—
136	C₂²×C₄₇	#568	1	2	yes	—
137	C₂×D₄₇	—	2	4	no	—
138	C₂×D₄₇	—	2	8	no	—
139	C₂×D₄₇	—	2	8	no	—
140	C₂×D₄₇	—	2	4	no	—
141	C₂×D₄₇	C₂×C₄×C₈	1	8	yes	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	4	no	—
144	C₄×C₄₇	—	2	4	no	—
145	C₄×C₄₇	C₂×C₄×C₈	1	4	yes	—
146	C₄×C₄₇	#518	1	2	yes	—
147	C₄×C₄₇	#520	1	2	yes	—
148	Dic₄₇	—	2	8	no	—
149	Dic₄₇	—	2	4	no	—
150	C₂×C₈×D₄	—	47	2	no	—
151	C₄²×D₄	—	47	1	no	—
152	C₈×C₄⋊C₄	—	47	2	no	—
153	C₄×C₄⋊C₈	—	47	1	no	—
154	C₈×C₂²⋊C₄	—	47	4	no	—
155	C₄×C₂²⋊C₈	—	47	2	no	—
156	#1744	D₄₇	1	1	yes	characteristic, complemented
157	C₂×#336	—	47	2	no	—
158	C₂×C₄₇	#2275	1	1	yes	characteristic
159	C₂×C₄₇	C₂²×C₄×C₈	1	1	yes	characteristic
160	C₂×C₄₇	#2875	1	1	yes	characteristic
161	C₂×C₄₇	#2972	1	2	yes	—
162	C₂×C₄₇	#2988	1	2	yes	—
163	D₄₇	—	2	8	no	—
164	C₂³×C₈	—	47	4	no	—
165	C₂×C₄×D₄	—	47	2	no	—
166	C₂×C₄×D₄	—	47	1	no	—
167	C₂²×C₄²	—	47	2	no	—
168	C₈×D₄	—	47	16	no	—
169	C₂×C₄⋊C₈	—	47	2	no	—
170	C₂×C₂²⋊C₈	—	47	4	no	—
171	C₂×C₄×C₈	—	47	4	no	—
172	C₂×C₄×C₈	—	47	2	no	—
173	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
174	C₄×C₄⋊C₄	—	47	2	no	—
175	C₄×C₄⋊C₄	—	47	1	no	—
176	C₄×C₂²⋊C₄	—	47	4	no	—
177	C₄×C₂²⋊C₄	—	47	2	no	—
178	C₄³	C₂×D₄₇	1	1	yes	characteristic
179	#336	—	47	4	no	—
180	#336	—	47	2	no	—
181	#336	C₂×D₄₇	1	1	yes	characteristic
182	#336	C₂×D₄₇	1	1	yes	characteristic
183	#336	C₂×D₄₇	1	2	yes	—
184	C₄₇	#20555	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₄	—	47	1	no	—
186	C₂³×C₄	—	47	2	no	—
187	C₂³×C₄	—	47	4	no	—
188	C₂²×C₈	—	47	4	no	—
189	C₂²×C₈	—	47	16	no	—
190	C₂²×C₈	—	188	4	no	—
191	C₂²×C₈	—	188	4	no	—
192	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
193	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
194	C₄×D₄	—	47	16	no	—
195	C₄×D₄	—	47	8	no	—
196	C₂×C₄⋊C₄	—	47	1	no	—
197	C₂×C₄⋊C₄	—	47	2	no	—
198	C₂×C₂²⋊C₄	—	47	2	no	—
199	C₂×C₂²⋊C₄	—	47	4	no	—
200	C₂×C₄²	—	47	2	no	—
201	C₂×C₄²	—	47	4	no	—
202	C₂×C₄²	—	188	4	no	—
203	C₂×C₄²	C₄×D₄₇	1	2	yes	—
204	C₂×C₄²	C₄×D₄₇	1	2	yes	—
205	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
208	C₄⋊C₈	—	47	8	no	—
209	C₂²⋊C₈	—	47	16	no	—
210	C₄×C₈	—	188	2	no	—
211	C₄×C₈	—	188	4	no	—
212	C₄×C₈	C₄×D₄₇	1	8	yes	complemented
213	#95	—	47	2	no	—
214	#95	—	47	4	no	—
215	C₂⁴	—	47	2	no	—
216	C₂×D₄	—	47	4	no	—
217	C₂×D₄	—	47	8	no	—
218	C₂²×C₄	—	47	2	no	—
219	C₂²×C₄	—	47	4	no	—
220	C₂²×C₄	—	47	8	no	—
221	C₂²×C₄	—	47	16	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	—	94	2	no	—
224	C₂²×C₄	—	94	2	no	—
225	C₂²×C₄	—	188	4	no	—
226	C₂²×C₄	—	188	4	no	—
227	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
229	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
230	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	47	8	no	—
237	C₂×C₈	—	94	4	no	—
238	C₂×C₈	—	94	4	no	—
239	C₂×C₈	—	188	16	no	—
240	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
241	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
242	C₄⋊C₄	—	47	4	no	—
243	C₄⋊C₄	—	47	8	no	—
244	C₂²⋊C₄	—	47	8	no	—
245	C₂²⋊C₄	—	47	16	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	4	2	no	—
250	C₄²	—	94	4	no	—
251	C₄²	—	188	4	no	—
252	C₄²	C₈×D₄₇	1	8	yes	complemented
253	C₄²	C₂×C₄×D₄₇	1	4	yes	—
254	C₂³	—	47	4	no	—
255	C₂³	—	47	8	no	—
256	C₂³	—	94	2	no	—
257	C₂³	—	94	2	no	—
258	C₂³	—	94	4	no	—
259	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
260	D₄	—	47	16	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	47	4	no	—
267	C₂×C₄	—	47	8	no	—
268	C₂×C₄	—	94	4	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	—	94	2	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	94	8	no	—
273	C₂×C₄	—	94	8	no	—
274	C₂×C₄	—	188	16	no	—
275	C₂×C₄	C₄²×D₄₇	1	2	yes	—
276	C₂×C₄	C₄²×D₄₇	1	2	yes	—
277	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
278	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
279	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
280	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
281	C₂×C₄	#13094070	1	1	yes	characteristic
282	C₂×C₄	#13094070	1	2	yes	—
283	C₂×C₄	#13094071	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	2	4	no	—
286	C₈	—	94	8	no	—
287	C₂²	—	47	8	no	—
288	C₂²	—	94	8	no	—
289	C₂²	—	94	4	no	—
290	C₂²	—	94	4	no	—
291	C₂²	—	94	8	no	—
292	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
293	C₂²	#15689597	1	1	yes	characteristic
294	C₂²	#15689603	1	1	yes	characteristic
295	C₂²	C₂²×C₈×D₄₇	1	2	yes	—
296	C₂²	#15689889	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	4	no	—
300	C₄	—	94	4	no	—
301	C₄	—	94	8	no	—
302	C₄	C₄×C₈×D₄₇	1	4	yes	—
303	C₄	#15689597	1	2	yes	—
304	C₄	#15689603	1	2	yes	—
305	C₂	—	94	8	no	—
306	C₂	#20245750	1	1	yes	characteristic
307	C₂	C₂×C₄×C₈×D₄₇	1	1	yes	characteristic
308	C₂	#20251813	1	1	yes	characteristic
309	C₂	#20252411	1	2	yes	—
310	C₂	#20252496	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂²×C₄×C₈
Fitting	C₄₇×#1744	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#20555
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1744

Complements

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

N	H ≅ G/N	type
3. C₈×C₄⋊Dic₄₇	C₂	semidirect
4. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
5. #20243158	C₂	semidirect
6. #20243145	C₂	semidirect
7. #336×D₄₇	C₂	semidirect
8. C₄₇×#1744	C₂	semidirect
13. C₈×D₁₈₈	C₄	semidirect
35. C₂×C₈×D₄₇	C₂×C₄	semidirect
36. C₄×D₁₈₈	C₈	semidirect
45. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
46. (C₂×D₄₇)⋊C₈	C₂×C₄	semidirect
58. C₄×C₈×C₄₇	C₂×C₄	semidirect
74. C₂×C₄×D₄₇	C₂×C₈	semidirect
86. C₄²×C₄₇	C₂×C₈	semidirect
89. (C₂×D₄₇)⋊C₄	C₂×C₈	semidirect
91. C₄⋊Dic₄₇	C₂×C₈	semidirect
124. D₁₈₈	C₄×C₈	semidirect
156. #1744	D₄₇	semidirect
184. C₄₇	#20555	semidirect
212. C₄×C₈	C₄×D₄₇	semidirect
252. C₄²	C₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₁₈₈	2
C₄²×D₁₈₈	2
C₈×C₄⋊Dic₄₇	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243158	2
#20243145	2
#336×D₄₇	2
C₄₇×#1744	2
#20555	47