Subgroups — #26221163

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 350 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	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20244152	C₂	1	2	yes	maximal, complemented
4	#20244147	C₂	1	2	yes	maximal, complemented
5	#20243589	C₂	1	4	yes	maximal
6	#20243285	C₂	1	4	yes	maximal, complemented
7	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	—	2	1	no	—
9	C₂×#13093983	C₂²	1	2	yes	—
10	C₂×#13093958	C₂²	1	4	yes	—
11	C₂×#13093956	C₂²	1	4	yes	—
12	#15688993	C₂²	1	2	yes	—
13	#15688989	—	2	1	no	—
14	#15688983	—	2	1	no	—
15	#15688983	C₂²	1	2	yes	—
16	C₄×C₄⋊Dic₄₇	—	2	1	no	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
22	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
23	#15688969	—	2	1	no	—
24	C₄²×Dic₄₇	—	2	1	no	—
25	#15688886	C₂²	1	4	yes	—
26	#15688886	C₂²	1	4	yes	—
27	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#432	C₄	1	8	yes	complemented
29	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
31	C₂×#97×C₄₇	C₂²	1	2	yes	—
32	#13093983	—	2	4	no	—
33	#13093983	—	2	4	no	—
34	#13093983	—	4	4	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
41	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	C₂×C₄×Dic₄₇	—	2	2	no	—
44	C₂×C₄×Dic₄₇	—	2	1	no	—
45	C₂×C₄×Dic₄₇	—	2	1	no	—
46	C₂×C₄×Dic₄₇	—	2	2	no	—
47	#13093958	—	2	8	no	—
48	#13093956	—	2	8	no	—
49	C₄₇⋊#95	—	2	2	no	—
50	C₄₇⋊#95	—	2	1	no	—
51	C₄₇⋊#95	—	2	1	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₄₇⋊#95	—	2	2	no	—
54	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
65	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
66	#97×C₄₇	C₂×C₄	1	8	yes	—
67	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	4	2	no	—
73	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₇	—	2	2	no	—
84	C₂²×Dic₄₇	—	2	2	no	—
85	C₂²×Dic₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
89	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
90	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
94	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
95	C₂²⋊Dic₄₇	—	2	8	no	—
96	C₂²⋊Dic₄₇	—	4	4	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄⋊Dic₄₇	—	4	4	no	—
102	C₄⋊Dic₄₇	—	4	4	no	—
103	C₄⋊Dic₄₇	#41	1	4	yes	—
104	C₄⋊Dic₄₇	#41	1	4	yes	—
105	C₄×Dic₄₇	—	2	8	no	—
106	C₄×Dic₄₇	—	4	4	no	—
107	C₄×Dic₄₇	—	4	4	no	—
108	C₄×Dic₄₇	—	4	2	no	—
109	C₄×Dic₄₇	—	4	2	no	—
110	C₂³×C₄₇	—	2	2	no	—
111	C₂³×C₄₇	—	2	1	no	—
112	C₂³×C₄₇	—	2	1	no	—
113	C₂³×C₄₇	—	2	2	no	—
114	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₄₇	#137	1	2	yes	—
133	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
137	C₂×Dic₄₇	—	2	4	no	—
138	C₂×Dic₄₇	—	2	4	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	4	no	—
144	C₂×Dic₄₇	—	4	4	no	—
145	C₂×Dic₄₇	—	4	4	no	—
146	C₈×C₄₇	—	2	8	no	—
147	#21364	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	—	4	2	no	—
152	C₂²×C₄₇	—	4	2	no	—
153	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₄₇	#525	1	2	yes	—
155	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
157	C₂²×C₄₇	#575	1	2	yes	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	#428	1	4	yes	—
163	C₄×C₄₇	C₄×#41	1	2	yes	—
164	C₄×C₄₇	#533	1	1	yes	characteristic
165	C₄×C₄₇	#533	1	1	yes	characteristic
166	Dic₄₇	—	4	4	no	—
167	Dic₄₇	—	4	4	no	—
168	Dic₄₇	—	8	4	no	—
169	C₂×#432	D₄₇	1	1	yes	characteristic
170	#2285	—	47	1	no	—
171	C₄×#117	—	47	1	no	—
172	#1948	—	47	2	no	—
173	#1947	—	47	2	no	—
174	#1854	—	47	4	no	—
175	#1776	—	47	4	no	—
176	C₂×C₄₇	—	4	2	no	—
177	C₂×C₄₇	#2296	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#428	1	1	yes	characteristic
179	C₂×C₄₇	#2911	1	1	yes	characteristic
180	C₂×C₄₇	#3115	1	2	yes	—
181	C₂×C₄₇	#3116	1	2	yes	—
182	C₂×#117	—	47	2	no	—
183	C₂×#117	—	94	1	no	—
184	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
185	#432	Dic₄₇	1	8	yes	complemented
186	C₂×#107	—	47	4	no	—
187	C₂×#106	—	47	4	no	—
188	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
189	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
190	C₂×#97	C₂×D₄₇	1	2	yes	—
191	#384	—	47	2	no	—
192	#382	—	94	1	no	—
193	#380	—	47	2	no	—
194	#380	—	94	1	no	—
195	C₄×C₄⋊C₄	—	47	1	no	—
196	C₄×C₄⋊C₄	—	47	1	no	—
197	C₄×C₄⋊C₄	—	47	2	no	—
198	C₄×C₄⋊C₄	—	47	2	no	—
199	C₄×C₄⋊C₄	—	94	1	no	—
200	C₄×C₂²⋊C₄	—	94	1	no	—
201	C₄×C₂²⋊C₄	—	94	1	no	—
202	#376	—	94	1	no	—
203	C₄³	—	94	1	no	—
204	#340	—	47	4	no	—
205	#340	—	47	4	no	—
206	C₄₇	#21364	1	1	yes	characteristic, complemented, Sylow
207	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
208	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
209	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
211	#117	—	94	4	no	—
212	#117	—	94	4	no	—
213	#117	—	188	4	no	—
214	C₂×C₄⋊C₄	—	47	2	no	—
215	C₂×C₄⋊C₄	—	47	2	no	—
216	C₂×C₄⋊C₄	—	47	4	no	—
217	C₂×C₄⋊C₄	—	94	2	no	—
218	C₂×C₄⋊C₄	—	94	2	no	—
219	C₂×C₂²⋊C₄	—	94	2	no	—
220	C₂×C₂²⋊C₄	—	94	2	no	—
221	C₂×C₄²	—	94	2	no	—
222	C₂×C₄²	—	94	2	no	—
223	C₂×C₄²	—	94	2	no	—
224	C₂×C₄²	—	94	1	no	—
225	C₂×C₄²	—	94	1	no	—
226	C₂×C₄²	Dic₉₄	1	4	yes	—
227	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
228	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
230	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	#107	—	94	8	no	—
233	#106	—	94	8	no	—
234	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
235	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
236	#97	C₂×Dic₄₇	1	8	yes	—
237	#95	—	94	2	no	—
238	#95	—	94	1	no	—
239	#95	—	94	1	no	—
240	#95	—	94	2	no	—
241	#95	—	94	2	no	—
242	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	94	2	no	—
249	C₂²×C₄	—	94	2	no	—
250	C₂²×C₄	—	94	4	no	—
251	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
252	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
253	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
254	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
260	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
264	C₄⋊C₄	—	47	4	no	—
265	C₄⋊C₄	—	47	4	no	—
266	C₄⋊C₄	—	94	2	no	—
267	C₄⋊C₄	—	94	2	no	—
268	C₄⋊C₄	—	94	4	no	—
269	C₄⋊C₄	—	94	4	no	—
270	C₄⋊C₄	—	188	4	no	—
271	C₄⋊C₄	—	188	4	no	—
272	C₂²⋊C₄	—	94	8	no	—
273	C₂²⋊C₄	—	188	4	no	—
274	C₄²	—	94	8	no	—
275	C₄²	—	188	4	no	—
276	C₄²	—	188	4	no	—
277	C₄²	—	188	2	no	—
278	C₄²	—	188	2	no	—
279	C₄²	C₄⋊Dic₄₇	1	4	yes	—
280	C₄²	C₄⋊Dic₄₇	1	4	yes	—
281	C₄²	C₂×Dic₉₄	1	2	yes	—
282	C₄²	C₂×Dic₉₄	1	2	yes	—
283	C₄²	C₂²×Dic₄₇	1	2	yes	—
284	C₄²	C₂²×Dic₄₇	1	2	yes	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	2	no	—
289	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
290	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	94	4	no	—
302	C₂×C₄	—	94	4	no	—
303	C₂×C₄	—	188	4	no	—
304	C₂×C₄	—	188	4	no	—
305	C₂×C₄	—	188	4	no	—
306	C₂×C₄	—	188	2	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
314	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
315	C₂×C₄	#13094052	1	2	yes	—
316	C₂×C₄	#13094057	1	2	yes	—
317	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
318	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
319	C₂×C₄	#41×D₄₇	1	2	yes	—
320	C₂×C₄	#41×D₄₇	1	2	yes	—
321	C₈	—	2	8	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
328	C₂²	#15689635	1	2	yes	—
329	C₂²	C₂×#13094052	1	1	yes	characteristic
330	C₂²	C₂×#13094057	1	1	yes	characteristic
331	C₂²	#15689923	1	2	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	376	4	no	—
337	C₄	—	188	4	no	—
338	C₄	—	188	4	no	—
339	C₄	#15689165	1	4	yes	—
340	C₄	#41×Dic₄₇	1	2	yes	—
341	C₄	#15689676	1	1	yes	characteristic
342	C₄	#15689676	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	#20245872	1	1	yes	characteristic
345	C₂	C₂×#15689165	1	1	yes	characteristic
346	C₂	#20251978	1	1	yes	characteristic
347	C₂	#20253540	1	2	yes	—
348	C₂	#20253551	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#21364
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20244152	C₂	semidirect
4. #20244147	C₂	semidirect
6. #20243285	C₂	semidirect
28. C₄₇×#432	C₄	semidirect
185. #432	Dic₄₇	semidirect
206. C₄₇	#21364	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
C₄×#13093983	2
#20244152	2
#20244147	2
#20243589	2
#20243285	2
C₂×C₄₇×#432	2
#21364	47