Subgroups — #26122090

Order 12032.

1110 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245795	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688884	C₂	1	4	yes	maximal
4	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₄₇	—	2	2	no	—
7	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
8	#15688993	C₄	1	2	yes	complemented
9	#15688989	C₄	1	2	yes	complemented
10	#15688983	—	2	2	no	—
11	C₄×C₄⋊Dic₄₇	—	2	2	no	—
12	C₄²×Dic₄₇	—	2	2	no	—
13	C₄²×Dic₄₇	C₄	1	4	yes	complemented
14	#15688884	—	2	8	no	—
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#423	—	2	2	no	—
18	C₄₇×#423	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
20	C₂³×Dic₄₇	—	2	2	no	—
21	#13093983	—	2	4	no	—
22	#13093983	—	2	4	no	—
23	C₂×C₄⋊Dic₄₇	—	2	4	no	—
24	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
25	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	16	no	—
27	C₂×C₄×Dic₄₇	—	2	8	no	—
28	C₂×C₄×Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
34	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
35	C₄₇⋊#95	—	2	4	no	—
36	C₂³×C₄×C₄₇	D₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	2	4	no	—
40	C₂×M₁₆×C₄₇	—	2	4	no	—
41	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
50	C₄⋊C₈×C₄₇	—	2	8	no	—
51	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×Dic₄₇	—	2	16	no	—
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	2	8	no	—
73	M₁₆×C₄₇	—	4	4	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₄²×C₄₇	C₄²	1	2	yes	—
78	C₄²×C₄₇	C₄²	1	2	yes	—
79	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
82	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
83	C₂²⋊Dic₄₇	—	4	4	no	—
84	C₄⋊Dic₄₇	—	2	8	no	—
85	C₄⋊Dic₄₇	—	4	8	no	—
86	C₄×Dic₄₇	—	2	32	no	—
87	C₄×Dic₄₇	—	2	4	no	—
88	C₄×Dic₄₇	—	2	4	no	—
89	C₄×Dic₄₇	—	2	4	no	—
90	C₄×Dic₄₇	—	4	2	no	—
91	C₄×Dic₄₇	—	4	2	no	—
92	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	complemented
93	C₂³×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₇	#121	1	2	yes	—
98	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	#95	1	4	yes	—
103	C₂×C₄×C₄₇	#95	1	4	yes	—
104	C₂×C₄×C₄₇	#104	1	8	yes	—
105	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#117	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	#121	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#125	1	2	yes	—
120	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
123	C₂×Dic₄₇	—	2	16	no	—
124	C₂×Dic₄₇	—	2	16	no	—
125	C₂×Dic₄₇	—	2	16	no	—
126	C₂×Dic₄₇	—	2	8	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	8	no	—
129	C₈×C₄₇	—	4	4	no	—
130	#10839	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
136	C₂²×C₄₇	#380	1	2	yes	—
137	C₂²×C₄₇	#389	1	2	yes	—
138	C₂²×C₄₇	#390	1	1	yes	characteristic
139	C₂²×C₄₇	#391	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#104	1	4	yes	—
141	C₄×C₄₇	—	2	4	no	—
142	C₄×C₄₇	#339	1	8	yes	complemented
143	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
144	C₄×C₄₇	#382	1	1	yes	characteristic
145	C₄×C₄₇	#384	1	1	yes	characteristic
146	C₄×C₄₇	#389	1	2	yes	—
147	C₄×C₄₇	#390	1	1	yes	characteristic
148	C₄×C₄₇	#391	1	1	yes	characteristic
149	Dic₄₇	—	2	16	no	—
150	Dic₄₇	—	8	4	no	—
151	C₂×#423	D₄₇	1	1	yes	characteristic
152	#2285	—	47	1	no	—
153	C₂×C₄³	—	47	1	no	—
154	C₂×#339	—	47	4	no	—
155	C₂×C₄₇	—	2	2	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#339	1	2	yes	—
159	C₂×C₄₇	#1962	1	2	yes	—
160	C₂×C₄₇	#1994	1	2	yes	—
161	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
162	C₂×#117	—	47	2	no	—
163	C₂²×C₄²	—	47	2	no	—
164	C₂²×C₄²	—	94	2	no	—
165	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
166	#423	—	2	2	no	—
167	#423	Dic₄₇	1	4	yes	complemented
168	C₂×C₄⋊C₈	—	2	2	no	—
169	#384	—	47	2	no	—
170	#382	—	47	2	no	—
171	#380	—	94	2	no	—
172	C₄×C₄⋊C₄	—	94	2	no	—
173	C₄³	—	47	4	no	—
174	C₄³	—	94	2	no	—
175	#339	—	94	8	no	—
176	C₄₇	#10839	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	—	94	2	no	—
178	C₂³×C₄	D₁₈₈	1	2	yes	—
179	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
184	C₂²×C₈	—	2	2	no	—
185	#117	—	94	4	no	—
186	#117	—	94	4	no	—
187	C₂×C₄⋊C₄	—	47	4	no	—
188	C₂×C₄⋊C₄	—	94	4	no	—
189	C₂×C₂²⋊C₄	—	94	2	no	—
190	C₂×C₄²	—	47	4	no	—
191	C₂×C₄²	—	47	4	no	—
192	C₂×C₄²	—	47	4	no	—
193	C₂×C₄²	—	94	16	no	—
194	C₂×C₄²	—	94	8	no	—
195	C₂×C₄²	—	94	2	no	—
196	C₂×C₄²	—	94	2	no	—
197	C₂×C₄²	—	94	2	no	—
198	C₂×C₄²	—	94	2	no	—
199	C₂×C₄²	Dic₉₄	1	2	yes	—
200	C₂×C₄²	C₄×D₄₇	1	2	yes	—
201	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
202	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
204	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
205	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
206	C₄⋊C₈	—	2	8	no	—
207	#95	—	94	4	no	—
208	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	94	16	no	—
211	C₂²×C₄	—	94	4	no	—
212	C₂²×C₄	—	94	4	no	—
213	C₂²×C₄	—	94	4	no	—
214	C₂²×C₄	—	94	4	no	—
215	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
216	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
217	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
218	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
219	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
222	C₂²×C₄	D₄×D₄₇	1	2	yes	—
223	C₂²×C₄	#10625352	1	2	yes	—
224	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
225	C₂²×C₄	#10625354	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
229	M₁₆	—	2	8	no	—
230	M₁₆	—	4	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₄⋊C₄	—	94	8	no	—
235	C₄⋊C₄	—	188	8	no	—
236	C₂²⋊C₄	—	188	4	no	—
237	C₄²	—	47	8	no	—
238	C₄²	—	94	32	no	—
239	C₄²	—	94	4	no	—
240	C₄²	—	94	4	no	—
241	C₄²	—	94	4	no	—
242	C₄²	—	188	2	no	—
243	C₄²	—	188	2	no	—
244	C₄²	C₄×Dic₄₇	1	2	yes	—
245	C₄²	C₄×Dic₄₇	1	2	yes	—
246	C₄²	Dic₄₇⋊C₄	1	2	yes	—
247	C₄²	Dic₄₇⋊C₄	1	4	yes	—
248	C₄²	Dic₄₇⋊C₄	1	2	yes	—
249	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
252	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
253	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
254	C₂³	#13093999	1	2	yes	—
255	C₂³	#13094004	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	94	16	no	—
260	C₂×C₄	—	94	16	no	—
261	C₂×C₄	—	94	16	no	—
262	C₂×C₄	—	94	8	no	—
263	C₂×C₄	—	188	4	no	—
264	C₂×C₄	—	188	8	no	—
265	C₂×C₄	C₄₇⋊#95	1	4	yes	—
266	C₂×C₄	C₄₇⋊#95	1	4	yes	—
267	C₂×C₄	#13093950	1	8	yes	—
268	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
270	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
271	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
273	C₂×C₄	#13093982	1	2	yes	—
274	C₂×C₄	#13093986	1	2	yes	—
275	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
276	C₂×C₄	#13093989	1	2	yes	—
277	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
278	C₂×C₄	#13093995	1	2	yes	—
279	C₂×C₄	#13094000	1	2	yes	—
280	C₂×C₄	#13094011	1	1	yes	characteristic
281	C₂×C₄	#13094013	1	1	yes	characteristic
282	C₂×C₄	#13094015	1	2	yes	—
283	C₂×C₄	#13094020	1	1	yes	characteristic
284	C₂×C₄	#13094022	1	1	yes	characteristic
285	C₂×C₄	#13094023	1	2	yes	—
286	C₈	—	4	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
292	C₂²	#15688980	1	2	yes	—
293	C₂²	#15689022	1	2	yes	—
294	C₂²	#15689024	1	1	yes	characteristic
295	C₂²	#15689028	1	1	yes	characteristic
296	C₂²	C₂×#13093950	1	4	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	94	16	no	—
299	C₄	—	376	4	no	—
300	C₄	#15688884	1	8	yes	complemented
301	C₄	C₄⋊C₄×Dic₄₇	1	2	yes	—
302	C₄	#15688988	1	1	yes	characteristic
303	C₄	#15688992	1	1	yes	characteristic
304	C₄	#15689019	1	2	yes	—
305	C₄	#15689026	1	1	yes	characteristic
306	C₄	#15689030	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#20241617	1	1	yes	characteristic
310	C₂	C₂×#15688884	1	2	yes	—
311	C₂	#20244215	1	2	yes	—
312	C₂	#20244364	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#10839
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
8. #15688993	C₄	semidirect
9. #15688989	C₄	semidirect
13. C₄²×Dic₄₇	C₄	semidirect
18. C₄₇×#423	C₄	semidirect
92. C₄×Dic₄₇	C₄⋊C₄	semidirect
142. C₄×C₄₇	#339	semidirect
167. #423	Dic₄₇	semidirect
176. C₄₇	#10839	semidirect
300. C₄	#15688884	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
C₂×C₄²×Dic₄₇	2
C₂×#15688884	2
C₂×C₄₇×#423	2
#10839	47