Subgroups — #26106344

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689178	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689075	C₂	1	1	yes	maximal, characteristic
3	#382×D₄₇	C₂	1	1	yes	maximal, characteristic
4	#20241270	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241270	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241269	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1295	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	—	2	1	no	—
9	C₂×C₄⋊C₄×D₄₇	C₄	1	2	yes	—
10	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
11	#15689178	—	2	2	no	—
12	#15689178	C₄	1	4	yes	complemented
13	C₂×Dic₄₇⋊C₈	—	2	1	no	—
14	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
15	#15689075	—	2	2	no	—
16	#15689075	C₄	1	4	yes	complemented
17	C₂×#13093935	C₂²	1	1	yes	characteristic
18	C₂×#13093935	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₄₇	—	2	1	no	—
20	#15688989	C₂²	1	1	yes	characteristic
21	#15688988	C₄	1	2	yes	—
22	#15688987	—	2	1	no	—
23	#15688987	—	2	1	no	—
24	#95×D₄₇	—	2	1	no	—
25	#95×D₄₇	—	2	1	no	—
26	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#382	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₄₇	—	2	2	no	—
30	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
33	C₂×#10625324	—	2	8	no	—
34	C₂×#10625324	—	2	4	no	—
35	C₂²×C₄₇⋊C₈	—	2	1	no	—
36	C₂²×C₄₇⋊C₈	—	2	1	no	—
37	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×Dic₄₇⋊C₄	—	2	2	no	—
39	C₄⋊C₄×D₄₇	—	4	4	no	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
43	C₄²×D₄₇	C₂×C₄	1	2	yes	—
44	C₄²×D₄₇	C₂×C₄	1	2	yes	—
45	C₄²×D₄₇	D₄	1	2	yes	—
46	C₄²×D₄₇	Q₈	1	2	yes	—
47	Dic₄₇⋊C₈	—	2	4	no	—
48	(C₂×D₄₇)⋊C₈	—	2	4	no	—
49	#13093935	—	2	1	no	—
50	#13093935	—	2	1	no	—
51	#13093935	C₂×C₄	1	2	yes	complemented
52	#13093935	C₂×C₄	1	2	yes	complemented
53	C₈×Dic₄₇	—	2	4	no	—
54	C₄₇⋊#95	—	2	1	no	—
55	C₄₇⋊#95	—	2	1	no	—
56	C₄₇⋊#95	—	2	2	no	—
57	C₄₇⋊#95	—	2	2	no	—
58	C₂²×C₈×C₄₇	—	2	1	no	—
59	C₂²×C₈×C₄₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	#97×C₄₇	—	2	1	no	—
63	#97×C₄₇	—	2	1	no	—
64	#97×C₄₇	C₂×C₄	1	2	yes	complemented
65	#97×C₄₇	C₂×C₄	1	2	yes	complemented
66	#95×C₄₇	—	2	1	no	—
67	#95×C₄₇	—	2	1	no	—
68	C₂³×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	—	2	2	no	—
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×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×D₄₇	—	2	4	no	—
78	C₂×C₄×D₄₇	—	2	2	no	—
79	C₂×C₄×D₄₇	—	2	2	no	—
80	C₂×C₄×D₄₇	—	4	2	no	—
81	C₂×C₄×D₄₇	—	4	4	no	—
82	C₂×C₄×D₄₇	—	4	4	no	—
83	C₂×C₄×D₄₇	—	4	4	no	—
84	C₂×C₄×D₄₇	C₄²	1	2	yes	—
85	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
87	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
88	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
89	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
90	C₂×C₈×C₄₇	—	2	4	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	4	2	no	—
94	C₄⋊C₄×C₄₇	—	4	2	no	—
95	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
99	C₄⋊Dic₄₇	—	4	2	no	—
100	Dic₄₇⋊C₄	—	4	4	no	—
101	C₄×Dic₄₇	C₄²	1	2	yes	—
102	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
103	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
104	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
105	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
106	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
107	C₂×C₄₇⋊C₈	—	2	4	no	—
108	C₂×C₄₇⋊C₈	—	2	2	no	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	—	4	2	no	—
111	#10625324	—	2	16	no	—
112	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×D₄₇	—	2	1	no	—
114	C₂²×D₄₇	—	2	1	no	—
115	C₂²×D₄₇	—	2	2	no	—
116	C₂²×D₄₇	#95	1	2	yes	—
117	C₂²×D₄₇	#104	1	4	yes	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×Dic₄₇	—	2	2	no	—
132	C₂×Dic₄₇	—	2	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	#95	1	2	yes	—
137	C₂×Dic₄₇	#95	1	2	yes	—
138	C₂×Dic₄₇	#95	1	2	yes	—
139	C₂×Dic₄₇	#100	1	2	yes	—
140	C₂×Dic₄₇	#101	1	2	yes	—
141	C₄×D₄₇	—	2	4	no	—
142	C₄×D₄₇	—	2	4	no	—
143	C₄×D₄₇	—	2	4	no	—
144	C₄×D₄₇	—	2	4	no	—
145	C₄×D₄₇	—	4	2	no	—
146	C₄×D₄₇	—	4	2	no	—
147	C₄×D₄₇	—	8	8	no	—
148	C₈×C₄₇	—	2	4	no	—
149	C₈×C₄₇	—	4	2	no	—
150	C₄₇⋊C₈	—	2	4	no	—
151	C₄₇⋊C₈	—	4	2	no	—
152	#8033	—	47	1	no	maximal, Sylow
153	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#104	1	2	yes	—
157	C₂²×C₄₇	#421	1	2	yes	—
158	C₂×D₄₇	—	2	2	no	—
159	C₂×D₄₇	—	2	2	no	—
160	C₂×D₄₇	—	2	4	no	—
161	C₂×D₄₇	—	2	4	no	—
162	C₂×D₄₇	#339	1	4	yes	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	2	1	no	—
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	8	2	no	—
170	Dic₄₇	—	2	2	no	—
171	Dic₄₇	—	2	2	no	—
172	Dic₄₇	—	8	2	no	—
173	Dic₄₇	#343	1	4	yes	—
174	C₂×#432	—	47	1	no	—
175	C₂×C₄×M₁₆	—	47	1	no	—
176	C₂×#382	—	47	1	no	—
177	#1295	—	47	1	no	—
178	#1295	—	47	1	no	—
179	#1295	—	47	1	no	—
180	#1295	D₄₇	1	1	yes	characteristic, complemented
181	C₂×C₄₇	#1527	1	2	yes	—
182	C₂×C₄₇	#1528	1	2	yes	—
183	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
184	C₂×C₄₇	#1733	1	1	yes	characteristic
185	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
186	D₄₇	—	2	4	no	—
187	C₂²×M₁₆	—	94	1	no	—
188	C₂²×C₄⋊C₄	—	47	2	no	—
189	C₂²×C₄²	—	47	1	no	—
190	#432	—	47	4	no	—
191	#432	—	94	2	no	—
192	C₂×C₄⋊C₈	—	94	1	no	—
193	C₂×C₂²⋊C₈	—	94	1	no	—
194	C₄×M₁₆	—	47	4	no	—
195	C₄×M₁₆	—	94	2	no	—
196	C₂×#97	—	47	1	no	—
197	C₂×#97	—	47	1	no	—
198	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
199	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
200	C₂×C₄×C₈	—	94	1	no	—
201	#382	—	47	2	no	—
202	#382	—	47	1	no	—
203	#382	—	94	1	no	—
204	#382	—	94	1	no	—
205	#382	C₂×D₄₇	1	1	yes	characteristic
206	C₂×#95	—	94	1	no	—
207	C₂×#95	—	94	1	no	—
208	C₄₇	#8033	1	1	yes	characteristic, complemented, Sylow
209	C₂³×C₄	—	47	1	no	—
210	C₂³×C₄	—	47	1	no	—
211	C₂³×C₄	—	47	1	no	—
212	C₂³×C₄	—	94	2	no	—
213	C₂×M₁₆	—	94	4	no	—
214	C₂×M₁₆	—	94	8	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	94	1	no	—
218	C₂²×C₈	—	94	1	no	—
219	C₂×C₄⋊C₄	—	47	2	no	—
220	C₂×C₄⋊C₄	—	94	2	no	—
221	C₂×C₄⋊C₄	—	188	4	no	—
222	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
223	C₂×C₄²	—	47	1	no	—
224	C₂×C₄²	—	47	1	no	—
225	C₂×C₄²	—	47	1	no	—
226	C₂×C₄²	—	47	2	no	—
227	C₂×C₄²	—	47	2	no	—
228	C₂×C₄²	—	47	2	no	—
229	C₂×C₄²	—	47	2	no	—
230	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
231	C₄⋊C₈	—	94	4	no	—
232	C₂²⋊C₈	—	94	4	no	—
233	#97	—	2	1	no	—
234	#97	—	2	1	no	—
235	#97	—	47	2	no	—
236	#97	—	47	2	no	—
237	#97	—	94	1	no	—
238	#97	—	94	1	no	—
239	#97	C₄×D₄₇	1	2	yes	complemented
240	#97	C₄×D₄₇	1	2	yes	complemented
241	C₄×C₈	—	94	4	no	—
242	#95	—	2	1	no	—
243	#95	—	2	1	no	—
244	#95	—	94	2	no	—
245	#95	—	94	2	no	—
246	#95	—	94	1	no	—
247	#95	—	94	1	no	—
248	C₂⁴	—	47	1	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	47	1	no	—
251	C₂²×C₄	—	47	1	no	—
252	C₂²×C₄	—	47	1	no	—
253	C₂²×C₄	—	47	2	no	—
254	C₂²×C₄	—	47	2	no	—
255	C₂²×C₄	—	47	2	no	—
256	C₂²×C₄	—	47	2	no	—
257	C₂²×C₄	—	47	2	no	—
258	C₂²×C₄	—	47	2	no	—
259	C₂²×C₄	—	94	2	no	—
260	C₂²×C₄	—	94	2	no	—
261	C₂²×C₄	—	94	4	no	—
262	C₂²×C₄	—	94	2	no	—
263	C₂²×C₄	—	188	2	no	—
264	C₂²×C₄	—	188	4	no	—
265	C₂²×C₄	—	188	4	no	—
266	C₂²×C₄	—	188	4	no	—
267	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
268	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
269	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
270	M₁₆	—	94	16	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	94	4	no	—
276	C₂×C₈	—	94	2	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	—	188	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	188	2	no	—
281	C₄⋊C₄	—	188	4	no	—
282	C₄²	—	47	2	no	—
283	C₄²	—	47	2	no	—
284	C₄²	—	47	2	no	—
285	C₄²	—	47	2	no	—
286	C₄²	—	47	2	no	—
287	C₄²	—	47	2	no	—
288	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
289	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
290	C₄²	D₄×D₄₇	1	1	yes	characteristic
291	C₄²	Q₈×D₄₇	1	1	yes	characteristic
292	C₂³	—	47	2	no	—
293	C₂³	—	47	4	no	—
294	C₂³	—	94	1	no	—
295	C₂³	—	94	1	no	—
296	C₂³	—	94	2	no	—
297	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	47	2	no	—
306	C₂×C₄	—	47	2	no	—
307	C₂×C₄	—	47	2	no	—
308	C₂×C₄	—	47	2	no	—
309	C₂×C₄	—	47	2	no	—
310	C₂×C₄	—	94	4	no	—
311	C₂×C₄	—	94	4	no	—
312	C₂×C₄	—	94	4	no	—
313	C₂×C₄	—	94	4	no	—
314	C₂×C₄	—	94	2	no	—
315	C₂×C₄	—	94	2	no	—
316	C₂×C₄	—	376	8	no	—
317	C₂×C₄	—	188	2	no	—
318	C₂×C₄	—	188	2	no	—
319	C₂×C₄	—	188	2	no	—
320	C₂×C₄	—	188	2	no	—
321	C₂×C₄	—	188	2	no	—
322	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
323	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
324	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
325	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
326	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
327	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
328	C₈	—	2	4	no	—
329	C₈	—	4	2	no	—
330	C₈	—	94	4	no	—
331	C₈	—	188	2	no	—
332	C₂²	—	47	4	no	—
333	C₂²	—	94	4	no	—
334	C₂²	—	94	4	no	—
335	C₂²	—	94	2	no	—
336	C₂²	—	94	2	no	—
337	C₂²	#95×D₄₇	1	1	yes	characteristic
338	C₂²	#100×D₄₇	1	1	yes	characteristic
339	C₂²	#101×D₄₇	1	1	yes	characteristic
340	C₂²	#104×D₄₇	1	2	yes	—
341	C₂²	#15689138	1	2	yes	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	47	4	no	—
350	C₄	—	94	2	no	—
351	C₄	—	94	2	no	—
352	C₄	—	376	2	no	—
353	C₂	—	94	4	no	—
354	C₂	#20241984	1	2	yes	—
355	C₂	#20241988	1	2	yes	—
356	C₂	#339×D₄₇	1	1	yes	characteristic
357	C₂	#20243096	1	1	yes	characteristic
358	C₂	#343×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1295	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂³×D₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1295	D₄₇
hypocenter	C₄₇	#8033
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1295

Complements

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

N	H ≅ G/N	type
4. #20241270	C₂	semidirect
5. #20241270	C₂	semidirect
6. #20241269	C₂	semidirect
7. C₄₇×#1295	C₂	semidirect
12. #15689178	C₄	semidirect
16. #15689075	C₄	semidirect
51. #13093935	C₂×C₄	semidirect
52. #13093935	C₂×C₄	semidirect
64. #97×C₄₇	C₂×C₄	semidirect
65. #97×C₄₇	C₂×C₄	semidirect
180. #1295	D₄₇	semidirect
208. C₄₇	#8033	semidirect
239. #97	C₄×D₄₇	semidirect
240. #97	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689178	2
C₂×#15689075	2
#382×D₄₇	2
#20241270	2
#20241269	2
C₄₇×#1295	2
#8033	47