Subgroups — #26105894

Order 12032.

666 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×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
3	#20241231	C₂	1	4	yes	maximal
4	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
5	#15689180	—	2	2	no	—
6	#15689180	C₄	1	4	yes	complemented
7	#15689176	—	2	2	no	—
8	#15689176	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×#13093935	—	2	1	no	—
14	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#382	—	2	1	no	—
18	C₄₇×#382	—	2	1	no	—
19	C₄₇×#382	—	2	1	no	—
20	C₄₇×#382	—	2	1	no	—
21	C₂×#95×C₄₇	C₄	1	2	yes	—
22	C₂×#95×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
27	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
29	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
30	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
31	#13093935	—	2	4	no	—
32	C₄×C₄₇⋊C₈	—	2	4	no	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
48	#95×C₄₇	—	2	2	no	—
49	#95×C₄₇	—	2	2	no	—
50	#95×C₄₇	—	2	2	no	—
51	#95×C₄₇	—	2	2	no	—
52	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	8	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
62	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
75	C₄⋊C₄×C₄₇	—	4	4	no	—
76	C₄⋊C₄×C₄₇	—	4	4	no	—
77	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
83	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
84	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	4	4	no	—
88	C₂×C₄₇⋊C₈	—	4	4	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₄₇	#131	1	4	yes	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	4	1	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	—	4	1	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	8	8	no	—
114	C₂×C₄×C₄₇	#95	1	2	yes	—
115	C₂×C₄×C₄₇	#95	1	2	yes	—
116	C₂×C₄×C₄₇	#95	1	2	yes	—
117	C₂×C₄×C₄₇	#95	1	2	yes	—
118	C₂×C₄×C₄₇	#100	1	2	yes	—
119	C₂×C₄×C₄₇	#101	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₄₇⋊C₈	—	4	4	no	—
125	C₄₇⋊C₈	—	4	4	no	—
126	#7914	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
134	C₂²×C₄₇	#405	1	2	yes	—
135	C₂²×C₄₇	#408	1	2	yes	—
136	C₂²×C₄₇	#410	1	2	yes	—
137	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
139	C₂²×C₄₇	#424	1	2	yes	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	8	4	no	—
148	C₄×C₄₇	#343	1	4	yes	—
149	C₂×#432	—	47	1	no	—
150	C₂×#431	—	47	1	no	—
151	C₂×#382	D₄₇	1	1	yes	characteristic
152	#1280	—	47	4	no	—
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	#1469	1	2	yes	—
156	C₂×C₄₇	#1470	1	2	yes	—
157	C₂×C₄₇	#1729	1	1	yes	characteristic
158	C₂×C₄₇	#1740	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
160	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
161	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
162	#432	—	47	4	no	—
163	#432	—	94	2	no	—
164	#431	—	47	4	no	—
165	#431	—	94	2	no	—
166	C₂×C₄⋊C₈	—	94	1	no	—
167	C₂×C₄⋊C₈	—	94	1	no	—
168	C₂×C₂²⋊C₈	—	47	2	no	—
169	C₂×C₂²⋊C₈	—	47	2	no	—
170	C₂×#97	—	94	1	no	—
171	C₂×C₄×C₈	—	94	1	no	—
172	#382	—	2	1	no	—
173	#382	—	2	1	no	—
174	#382	—	2	1	no	—
175	#382	—	2	1	no	—
176	C₂×#95	Dic₄₇	1	2	yes	—
177	C₂×#95	C₂×D₄₇	1	2	yes	—
178	C₄₇	#7914	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
180	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
182	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
183	C₂²×C₈	—	94	2	no	—
184	C₂²×C₈	—	94	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	4	4	no	—
188	C₂×C₄²	Dic₉₄	1	2	yes	—
189	C₂×C₄²	C₄×D₄₇	1	2	yes	—
190	C₂×C₄²	C₄×D₄₇	1	2	yes	—
191	C₂×C₄²	D₁₈₈	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₄⋊C₈	—	94	4	no	—
197	C₄⋊C₈	—	94	4	no	—
198	C₂²⋊C₈	—	47	4	no	—
199	C₂²⋊C₈	—	47	4	no	—
200	C₂²⋊C₈	—	94	2	no	—
201	C₂²⋊C₈	—	94	2	no	—
202	#97	—	94	4	no	—
203	C₄×C₈	—	94	4	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	#95	—	2	2	no	—
207	#95	—	2	2	no	—
208	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
218	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
219	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
220	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
221	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
231	C₂×C₈	—	94	4	no	—
232	C₂×C₈	—	94	4	no	—
233	C₂×C₈	—	188	4	no	—
234	C₂×C₈	—	188	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄²	Dic₄₇⋊C₄	1	2	yes	—
238	C₄²	Dic₄₇⋊C₄	1	2	yes	—
239	C₄²	Dic₄₇⋊C₄	1	2	yes	—
240	C₄²	Dic₄₇⋊C₄	1	2	yes	—
241	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
242	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
243	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
244	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
249	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
250	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
251	C₂³	#13094031	1	4	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	8	no	—
270	C₂×C₄	C₄₇⋊#95	1	2	yes	—
271	C₂×C₄	C₄₇⋊#95	1	2	yes	—
272	C₂×C₄	C₄₇⋊#95	1	2	yes	—
273	C₂×C₄	C₄₇⋊#95	1	2	yes	—
274	C₂×C₄	#13093941	1	2	yes	—
275	C₂×C₄	#13093943	1	2	yes	—
276	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
280	C₈	—	188	4	no	—
281	C₈	—	188	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
289	C₂²	#15689079	1	2	yes	—
290	C₂²	#15689090	1	2	yes	—
291	C₂²	#15689097	1	2	yes	—
292	C₂²	C₂×#13093941	1	1	yes	characteristic
293	C₂²	C₂×#13093943	1	1	yes	characteristic
294	C₂²	#15689150	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	8	4	no	—
303	C₄	#15688892	1	4	yes	—
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	#20241740	1	2	yes	—
307	C₂	#20241744	1	2	yes	—
308	C₂	#20243085	1	1	yes	characteristic
309	C₂	#20243121	1	1	yes	characteristic
310	C₂	C₂×#15688892	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#382	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#382	D₄₇
hypocenter	C₄₇	#7914
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
6. #15689180	C₄	semidirect
8. #15689176	C₄	semidirect
178. C₄₇	#7914	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₂×#15689176	2
#20241231	2
C₂×C₄₇×#382	2
#7914	47