Subgroups — #26200098

Order 12032.

694 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253483	C₂	1	2	yes	maximal
2	C₂×C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#20242808	C₂	1	4	yes	maximal, complemented
4	#20242774	C₂	1	2	yes	maximal, complemented
5	#20242750	C₂	1	2	yes	maximal, complemented
6	#20242151	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242088	C₂	1	2	yes	maximal, complemented
8	C₄₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	—	2	1	no	—
10	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#15689456	—	2	2	no	—
12	#15689400	C₂²	1	4	yes	—
13	#15689325	C₂²	1	4	yes	—
14	#15689273	—	2	2	no	—
15	#15689221	—	2	4	no	—
16	C₈×D₁₈₈	—	2	1	no	—
17	C₈×D₁₈₈	—	2	1	no	—
18	C₈×D₁₈₈	C₂²	1	2	yes	—
19	C₈×D₁₈₈	C₂²	1	2	yes	—
20	#15689180	C₂²	1	2	yes	—
21	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
23	#15688876	C₂²	1	2	yes	complemented
24	#15688874	C₂²	1	2	yes	complemented
25	#15688869	C₂²	1	4	yes	complemented
26	#15688869	C₂²	1	4	yes	complemented
27	C₄₇×#533	C₂²	1	2	yes	—
28	C₄₇×#500	C₂²	1	1	yes	characteristic, complemented
29	C₄₇×#494	C₂²	1	1	yes	characteristic, complemented
30	C₄₇×#487	C₂²	1	2	yes	complemented
31	C₄₇×#467	—	2	1	no	—
32	C₄₇×#467	—	2	1	no	—
33	C₄×Q₁₆×C₄₇	—	2	1	no	—
34	C₄×Q₁₆×C₄₇	—	2	1	no	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₄₇	—	2	1	no	—
38	C₂×#10625336	—	2	4	no	—
39	C₂²×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₈×D₄₇	—	2	2	no	—
41	C₂×C₈×D₄₇	—	2	2	no	—
42	C₂×C₈×D₄₇	—	4	4	no	—
43	C₄×D₁₈₈	—	2	1	no	—
44	C₄×D₁₈₈	—	2	1	no	—
45	C₄×D₁₈₈	D₄	1	2	yes	—
46	C₄×D₁₈₈	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
48	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
49	#13093957	—	2	4	no	—
50	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
51	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
52	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
53	#13093948	—	2	4	no	—
54	#13093944	—	2	4	no	—
55	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
56	(C₂×D₄₇)⋊C₈	—	2	2	no	—
57	(C₂×D₄₇)⋊C₈	—	2	2	no	—
58	#13093935	—	2	2	no	—
59	C₂×Q₁₆×C₄₇	—	2	1	no	—
60	C₂×Q₁₆×C₄₇	—	2	1	no	—
61	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
65	#125×C₄₇	—	2	2	no	—
66	C₂²⋊Q₈×C₄₇	—	2	2	no	—
67	C₂²⋊Q₈×C₄₇	—	2	2	no	—
68	C₄×Q₈×C₄₇	—	2	2	no	—
69	C₄×Q₈×C₄₇	—	2	2	no	—
70	#117×C₄₇	—	2	2	no	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	#107×C₄₇	—	2	1	no	—
73	#107×C₄₇	—	2	1	no	—
74	#103×C₄₇	—	2	2	no	—
75	#103×C₄₇	—	2	2	no	—
76	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₂³×D₄₇	—	2	1	no	—
81	C₂×Q₈×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
86	C₂²×Dic₄₇	—	2	1	no	—
87	C₂×D₁₈₈	—	2	1	no	—
88	C₂×D₁₈₈	—	2	4	no	—
89	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₄₇	—	2	2	no	—
91	C₂×C₄×D₄₇	—	2	2	no	—
92	C₂×C₄×D₄₇	—	4	2	no	—
93	C₂×C₄×D₄₇	—	4	2	no	—
94	Q₁₆×C₄₇	—	4	2	no	—
95	Q₁₆×C₄₇	—	4	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	4	1	no	—
98	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₄⋊C₄×C₄₇	—	2	2	no	—
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	4	2	no	—
107	C₄⋊C₄×C₄₇	—	4	2	no	—
108	C₂²⋊C₄×C₄₇	—	4	2	no	—
109	C₄²×C₄₇	—	4	2	no	—
110	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₇	#41	1	1	yes	characteristic
113	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
114	#10625336	—	4	8	no	—
115	(C₂×D₄₇)⋊C₄	—	2	2	no	—
116	(C₂×D₄₇)⋊C₄	—	2	2	no	—
117	C₄⋊Dic₄₇	—	2	1	no	—
118	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
119	C₂×C₄₇⋊C₈	—	2	4	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	—	4	2	no	—
122	C₈×D₄₇	—	4	4	no	—
123	C₈×D₄₇	—	4	4	no	—
124	C₂³×C₄₇	#121	1	1	yes	characteristic
125	C₂²×D₄₇	—	2	4	no	—
126	C₂²×D₄₇	—	2	2	no	—
127	C₂²×D₄₇	—	4	2	no	—
128	C₂²×D₄₇	—	4	2	no	—
129	Q₈×C₄₇	—	4	4	no	—
130	Q₈×C₄₇	—	4	2	no	—
131	Q₈×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	1	no	—
135	C₂×C₄×C₄₇	—	2	1	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	D₁₈₈	—	2	4	no	—
147	D₁₈₈	—	2	2	no	—
148	D₁₈₈	#135	1	4	yes	—
149	C₄×D₄₇	—	4	4	no	—
150	C₄×D₄₇	—	4	2	no	—
151	C₄×D₄₇	—	4	2	no	—
152	C₈×C₄₇	—	2	1	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₈×C₄₇	—	2	1	no	—
155	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
156	C₄₇⋊C₈	—	4	4	no	—
157	C₄₇⋊C₈	—	4	2	no	—
158	#19259	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	#473	1	2	yes	—
162	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
163	C₂×D₄₇	—	2	4	no	—
164	C₂×D₄₇	—	4	8	no	—
165	C₂×D₄₇	—	4	2	no	—
166	C₂×D₄₇	—	4	2	no	—
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	8	2	no	—
172	C₄×C₄₇	—	8	2	no	—
173	C₄×C₄₇	C₂×#135	1	2	yes	—
174	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₄×C₄₇	#575	1	1	yes	characteristic
176	Dic₄₇	—	4	2	no	—
177	#3170	D₄₇	1	1	yes	characteristic, complemented
178	#3110	—	47	2	no	—
179	C₂×C₈×D₄	—	47	1	no	—
180	#1679	—	47	4	no	—
181	#1673	—	47	2	no	—
182	#1669	—	47	2	no	—
183	#1568	—	47	1	no	—
184	#1552	—	47	2	no	—
185	C₂×C₄₇	—	2	2	no	—
186	C₂×C₄₇	#3036	1	1	yes	characteristic
187	C₂×C₄₇	#3050	1	1	yes	characteristic
188	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
189	D₄₇	—	4	4	no	—
190	C₂³×C₈	—	94	1	no	—
191	#533	C₂×D₄₇	1	2	yes	—
192	C₂×C₄×D₄	—	47	1	no	—
193	#500	C₂×D₄₇	1	1	yes	characteristic, complemented
194	#494	C₂×D₄₇	1	1	yes	characteristic, complemented
195	#487	C₂×D₄₇	1	2	yes	complemented
196	#484	—	94	2	no	—
197	#474	—	47	4	no	—
198	#467	—	2	1	no	—
199	#467	—	2	1	no	—
200	#461	—	47	4	no	—
201	#450	—	94	2	no	—
202	#440	—	94	4	no	—
203	C₄×Q₁₆	—	2	1	no	—
204	C₄×Q₁₆	—	2	1	no	—
205	C₈×D₄	—	47	2	no	—
206	C₈×D₄	—	47	2	no	—
207	C₈×D₄	—	94	1	no	—
208	C₈×D₄	—	94	1	no	—
209	#432	—	47	2	no	—
210	C₂×C₄⋊C₈	—	47	1	no	—
211	C₂×C₂²⋊C₈	—	94	1	no	—
212	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
213	#335	—	47	2	no	—
214	#334	—	47	2	no	—
215	#332	—	47	4	no	—
216	#332	—	47	4	no	—
217	C₄₇	#19259	1	1	yes	characteristic, complemented, Sylow
218	C₂²×D₄	—	47	1	no	—
219	C₂³×C₄	—	94	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	—	2	1	no	—
222	C₂×SD₁₆	—	94	4	no	—
223	C₂²×C₈	—	94	1	no	—
224	C₂²×C₈	—	94	2	no	—
225	C₂²×C₈	—	94	2	no	—
226	C₂²×C₈	—	188	4	no	—
227	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
228	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
229	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
230	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
231	#125	—	2	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×D₄	—	47	2	no	—
237	C₄×D₄	—	47	2	no	—
238	C₄×D₄	—	94	1	no	—
239	C₄×D₄	—	94	1	no	—
240	#117	—	2	2	no	—
241	C₂×C₄⋊C₄	—	47	1	no	—
242	C₂×C₂²⋊C₄	—	94	1	no	—
243	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
244	#107	—	2	1	no	—
245	#107	—	2	1	no	—
246	#107	—	94	4	no	—
247	C₄⋊C₈	—	47	2	no	—
248	C₄⋊C₈	—	47	4	no	—
249	C₄⋊C₈	—	94	1	no	—
250	#103	—	2	2	no	—
251	#103	—	2	2	no	—
252	#103	—	94	4	no	—
253	#102	—	94	4	no	—
254	C₂²⋊C₈	—	94	2	no	—
255	C₂²⋊C₈	—	94	2	no	—
256	C₂²⋊C₈	—	94	2	no	—
257	#97	—	94	2	no	—
258	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
259	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
262	C₂⁴	—	94	1	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×D₄	—	47	2	no	—
266	C₂×D₄	—	94	4	no	—
267	C₂×D₄	—	94	1	no	—
268	C₂²×C₄	—	94	1	no	—
269	C₂²×C₄	—	94	2	no	—
270	C₂²×C₄	—	94	2	no	—
271	C₂²×C₄	—	188	2	no	—
272	C₂²×C₄	—	188	2	no	—
273	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
274	C₂²×C₄	#10625352	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
276	Q₁₆	—	4	2	no	—
277	Q₁₆	—	4	2	no	—
278	SD₁₆	—	188	8	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	94	4	no	—
282	C₂×C₈	—	94	2	no	—
283	C₂×C₈	—	188	4	no	—
284	C₂×C₈	—	188	2	no	—
285	C₂×C₈	—	188	4	no	—
286	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	4	2	no	—
296	C₄⋊C₄	—	47	2	no	—
297	C₄⋊C₄	—	94	1	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	94	2	no	—
300	C₂²⋊C₄	—	94	2	no	—
301	C₄²	—	4	2	no	—
302	C₄²	D₄×D₄₇	1	1	yes	characteristic
303	C₄²	#10625352	1	1	yes	characteristic
304	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
305	C₄²	C₂³×D₄₇	1	1	yes	characteristic
306	C₂³	—	94	4	no	—
307	C₂³	—	94	2	no	—
308	C₂³	—	188	2	no	—
309	C₂³	—	188	2	no	—
310	C₂³	#13093995	1	1	yes	characteristic
311	Q₈	—	4	4	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	D₄	—	47	4	no	—
315	D₄	—	94	4	no	—
316	D₄	—	94	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	188	4	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	—	188	2	no	—
328	C₂×C₄	#13093995	1	1	yes	characteristic
329	C₂×C₄	#13093995	1	1	yes	characteristic
330	C₂×C₄	#13093995	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
333	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	188	4	no	—
338	C₈	—	188	2	no	—
339	C₈	#13094049	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	94	4	no	—
343	C₂²	—	188	8	no	—
344	C₂²	—	188	2	no	—
345	C₂²	—	188	2	no	—
346	C₂²	#15689395	1	2	yes	—
347	C₂²	C₂×#13093995	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	8	2	no	—
353	C₄	—	8	2	no	—
354	C₄	—	188	2	no	—
355	C₄	#135×D₄₇	1	2	yes	—
356	C₄	C₂×#13094049	1	1	yes	characteristic
357	C₄	#15689921	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	188	4	no	—
360	C₂	#20252848	1	1	yes	characteristic
361	C₂	#20252965	1	1	yes	characteristic
362	C₂	C₂×#15689395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3170	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	#3170	D₄₇
hypocenter	C₄₇	#19259
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #20242808	C₂	semidirect
4. #20242774	C₂	semidirect
5. #20242750	C₂	semidirect
6. #20242151	C₂	semidirect
7. #20242088	C₂	semidirect
8. C₄₇×#3170	C₂	semidirect
23. #15688876	C₂²	semidirect
24. #15688874	C₂²	semidirect
25. #15688869	C₂²	semidirect
26. #15688869	C₂²	semidirect
28. C₄₇×#500	C₂²	semidirect
29. C₄₇×#494	C₂²	semidirect
30. C₄₇×#487	C₂²	semidirect
177. #3170	D₄₇	semidirect
193. #500	C₂×D₄₇	semidirect
194. #494	C₂×D₄₇	semidirect
195. #487	C₂×D₄₇	semidirect
217. C₄₇	#19259	semidirect

Maximal subgroups

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

subgroup	index
#20253483	2
C₂×C₈×D₁₈₈	2
#20242808	2
#20242774	2
#20242750	2
#20242151	2
#20242088	2
C₄₇×#3170	2
#19259	47