Subgroups — #26234801

Order 12032.

812 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252306	C₂	1	2	yes	maximal
2	C₂×#15689241	C₂	1	1	yes	maximal, characteristic
3	#20251996	C₂	1	2	yes	maximal, complemented
4	#20244178	C₂	1	2	yes	maximal, complemented
5	#20243629	C₂	1	4	yes	maximal, complemented
6	#20243560	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243439	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	1	yes	characteristic
10	#15689889	C₂²	1	4	yes	—
11	C₂×#13094030	C₂²	1	2	yes	—
12	#15689246	—	2	4	no	—
13	#15689241	—	2	4	no	—
14	#15689241	C₄	1	2	yes	complemented
15	#15689241	C₄	1	2	yes	complemented
16	#15689241	C₄	1	2	yes	complemented
17	#15689241	C₄	1	2	yes	complemented
18	#15689171	C₂²	1	4	yes	—
19	#15689170	—	2	4	no	—
20	C₂×#13093960	C₂²	1	2	yes	—
21	C₂×#13093960	C₂²	1	2	yes	—
22	C₂×#13093960	C₂²	1	1	yes	characteristic
23	#108×D₄₇	—	2	4	no	—
24	#15689147	C₂²	1	2	yes	—
25	#15689141	C₂²	1	4	yes	—
26	C₂×#13093952	C₂²	1	2	yes	—
27	#15688893	C₂²	1	4	yes	—
28	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#427	C₂²	1	1	yes	characteristic
30	C₄₇×#405	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#339	C₂²	1	2	yes	—
33	C₂×#10625350	C₂³	1	2	yes	—
34	#13094046	C₂×C₄	1	2	yes	—
35	#13094046	C₂×C₄	1	2	yes	—
36	#13094046	D₄	1	2	yes	—
37	#13094046	D₄	1	2	yes	—
38	C₂×Dic₁₈₈	Q₈	1	1	yes	characteristic
39	C₂×#10625325	Q₈	1	2	yes	—
40	C₂×D₃₇₆	Q₈	1	1	yes	characteristic
41	#13094032	—	4	4	no	—
42	#13094030	—	2	2	no	—
43	#13094030	—	2	2	no	—
44	#13094030	—	2	8	no	—
45	#13094030	C₂×C₄	1	4	yes	—
46	#13094030	C₂×C₄	1	4	yes	—
47	C₂×#10625329	C₂³	1	4	yes	—
48	C₂×#10625329	C₂³	1	2	yes	—
49	C₂×#10625324	—	2	2	no	—
50	M₁₆×D₄₇	—	2	8	no	—
51	C₂×C₈×D₄₇	—	2	2	no	—
52	#13093960	—	2	2	no	—
53	#13093960	—	2	2	no	—
54	#13093960	—	2	2	no	—
55	#13093960	—	2	2	no	—
56	#13093960	C₂×C₄	1	2	yes	—
57	#13093960	C₂×C₄	1	2	yes	—
58	#13093959	—	2	8	no	—
59	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
60	#13093952	—	2	8	no	—
61	#13093952	C₂×C₄	1	4	yes	—
62	#13093952	C₂×C₄	1	4	yes	—
63	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	#117×C₄₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	#108×C₄₇	D₄	1	2	yes	—
69	#108×C₄₇	D₄	1	2	yes	—
70	#107×C₄₇	—	2	1	no	—
71	#106×C₄₇	—	2	1	no	—
72	#97×C₄₇	—	2	2	no	—
73	C₄×C₈×C₄₇	—	2	2	no	—
74	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	—	2	1	no	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂×C₂²⋊D₄₇	—	2	2	no	—
81	#10625350	—	2	2	no	—
82	#10625350	—	2	2	no	—
83	#10625350	C₂²×C₄	1	4	yes	—
84	#10625350	C₂×D₄	1	4	yes	—
85	C₂×D₁₈₈	C₂×Q₈	1	2	yes	—
86	C₂×C₄×D₄₇	—	2	2	no	—
87	C₂×Dic₉₄	C₂×Q₈	1	2	yes	—
88	M₁₆×C₄₇	—	2	2	no	—
89	M₁₆×C₄₇	C₂×D₄	1	4	yes	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
102	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₂²⋊C₄×C₄₇	—	2	2	no	—
105	C₄²×C₄₇	—	2	1	no	—
106	C₄²×C₄₇	—	2	2	no	—
107	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	#10625329	—	2	2	no	—
110	#10625329	—	2	4	no	—
111	#10625329	—	2	4	no	—
112	#10625329	C₂²×C₄	1	4	yes	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	4	no	—
115	Dic₁₈₈	C₄⋊C₄	1	2	yes	—
116	Dic₁₈₈	C₄⋊C₄	1	2	yes	—
117	D₃₇₆	C₄⋊C₄	1	2	yes	—
118	D₃₇₆	C₄⋊C₄	1	2	yes	—
119	#10625325	C₄⋊C₄	1	4	yes	—
120	#10625325	C₄⋊C₄	1	4	yes	—
121	#10625324	—	2	4	no	—
122	#10625324	—	2	4	no	—
123	#10625324	—	4	8	no	—
124	C₈×D₄₇	—	2	4	no	—
125	C₈×D₄₇	—	2	4	no	—
126	C₈×D₄₇	—	4	8	no	—
127	C₂³×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂²×D₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	1	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂²⋊D₄₇	—	2	4	no	—
140	C₂²⋊D₄₇	—	2	4	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	D₁₈₈	—	2	2	no	—
143	D₁₈₈	C₂×C₄⋊C₄	1	4	yes	—
144	C₄×D₄₇	—	2	4	no	—
145	C₄×D₄₇	—	2	4	no	—
146	Dic₉₄	—	2	2	no	—
147	Dic₉₄	C₂×C₄⋊C₄	1	4	yes	—
148	C₈×C₄₇	—	2	4	no	—
149	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
154	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
155	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
156	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
157	C₄₇⋊C₈	—	2	4	no	—
158	C₄₇⋊C₈	—	4	4	no	—
159	#22694	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	4	1	no	—
161	C₂²×C₄₇	#520	1	1	yes	characteristic
162	C₂²×C₄₇	#547	1	1	yes	characteristic
163	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
164	C₂×D₄₇	—	2	4	no	—
165	C₂×D₄₇	—	8	2	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	4	1	no	—
168	C₄×C₄₇	—	8	2	no	—
169	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₄₇	D₄²	1	1	yes	characteristic
172	C₄×C₄₇	#554	1	1	yes	characteristic
173	Dic₄₇	—	2	4	no	—
174	#2957	—	47	2	no	—
175	C₂×#443	—	47	1	no	—
176	#2916	—	47	2	no	—
177	#1953	—	47	2	no	—
178	#1861	—	47	4	no	—
179	#1848	—	47	1	no	—
180	#1814	—	47	2	no	—
181	#1772	D₄₇	1	1	yes	characteristic, complemented
182	C₂×C₄₇	—	2	1	no	—
183	C₂×C₄₇	—	2	1	no	—
184	C₂×C₄₇	—	2	1	no	—
185	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
186	D₄₇	—	4	4	no	—
187	C₂×#135	—	47	1	no	—
188	#568	—	47	4	no	—
189	C₂×#131	—	47	2	no	—
190	#444	—	94	4	no	—
191	#443	—	47	2	no	—
192	#443	—	47	2	no	—
193	#443	—	47	2	no	—
194	#443	—	47	2	no	—
195	#443	—	94	4	no	—
196	#430	—	47	4	no	—
197	#430	—	94	4	no	—
198	C₂×#108	—	47	1	no	—
199	C₂×#108	—	47	2	no	—
200	C₂×#108	—	47	2	no	—
201	C₂×#108	—	94	4	no	—
202	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
203	#427	C₂×D₄₇	1	1	yes	characteristic
204	#423	—	47	2	no	—
205	#421	—	47	4	no	—
206	C₂×#104	—	47	2	no	—
207	#405	C₂×D₄₇	1	2	yes	—
208	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
209	#344	—	47	4	no	—
210	#339	C₂×D₄₇	1	2	yes	—
211	C₄₇	#22694	1	1	yes	characteristic, complemented, Sylow
212	C₂×#41	—	47	2	no	—
213	#135	—	47	2	no	—
214	#135	—	47	2	no	—
215	#135	—	47	2	no	—
216	#135	—	47	2	no	—
217	C₂×Q₁₆	—	47	1	no	—
218	C₂×SD₁₆	—	47	2	no	—
219	C₂×D₈	—	47	1	no	—
220	#131	—	47	4	no	—
221	#131	—	47	4	no	—
222	#131	—	94	8	no	—
223	#131	—	94	2	no	—
224	#131	—	94	2	no	—
225	#131	—	188	4	no	—
226	C₂×M₁₆	—	47	2	no	—
227	C₂×M₁₆	—	47	4	no	—
228	C₂×M₁₆	—	94	2	no	—
229	C₂×M₁₆	—	94	8	no	—
230	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
231	C₂²×C₈	—	94	2	no	—
232	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
234	#117	C₂²×D₄₇	1	2	yes	—
235	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
236	#108	—	47	2	no	—
237	#108	—	47	2	no	—
238	#108	—	94	8	no	—
239	#108	—	94	2	no	—
240	#108	—	94	2	no	—
241	#108	—	94	2	no	—
242	#108	—	94	2	no	—
243	#108	D₁₈₈	1	2	yes	—
244	#108	D₁₈₈	1	2	yes	—
245	#107	—	2	1	no	—
246	#106	—	2	1	no	—
247	C₄⋊C₈	—	188	2	no	—
248	#104	—	47	4	no	—
249	#104	—	47	4	no	—
250	#104	—	94	8	no	—
251	#97	—	2	2	no	—
252	C₄×C₈	—	2	2	no	—
253	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
254	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
255	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
256	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
257	#41	—	47	4	no	—
258	#41	—	47	4	no	—
259	#41	—	94	2	no	—
260	#41	—	94	2	no	—
261	C₂×Q₈	—	47	2	no	—
262	C₂×D₄	—	47	2	no	—
263	C₂×D₄	—	94	2	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	94	2	no	—
266	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
267	Q₁₆	—	47	2	no	—
268	Q₁₆	—	47	2	no	—
269	SD₁₆	—	47	4	no	—
270	SD₁₆	—	47	4	no	—
271	D₈	—	47	2	no	—
272	D₈	—	47	2	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	47	4	no	—
275	M₁₆	—	94	2	no	—
276	M₁₆	—	94	4	no	—
277	M₁₆	—	94	4	no	—
278	M₁₆	—	94	4	no	—
279	M₁₆	—	94	4	no	—
280	M₁₆	—	188	8	no	—
281	M₁₆	C₂×D₁₈₈	1	4	yes	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	94	4	no	—
285	C₂×C₈	—	94	4	no	—
286	C₂×C₈	—	94	4	no	—
287	C₂×C₈	—	188	8	no	—
288	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
292	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
293	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
294	C₂×C₈	#10625350	1	1	yes	characteristic
295	C₂×C₈	#10625350	1	1	yes	characteristic
296	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
297	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
298	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
299	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	2	no	—
304	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
305	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
306	C₂³	—	94	2	no	—
307	C₂³	#13094071	1	1	yes	characteristic
308	Q₈	—	47	4	no	—
309	Q₈	—	94	2	no	—
310	D₄	—	47	4	no	—
311	D₄	—	94	4	no	—
312	D₄	—	94	4	no	—
313	D₄	—	94	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	94	2	no	—
319	C₂×C₄	—	94	4	no	—
320	C₂×C₄	—	94	4	no	—
321	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
324	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
325	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
326	C₂×C₄	#13094070	1	1	yes	characteristic
327	C₈	—	2	4	no	—
328	C₈	—	94	4	no	—
329	C₈	—	188	4	no	—
330	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
331	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
332	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
333	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
334	C₈	C₄×D₁₈₈	1	1	yes	characteristic
335	C₈	C₄×D₁₈₈	1	1	yes	characteristic
336	C₈	C₄×D₁₈₈	1	1	yes	characteristic
337	C₈	C₄×D₁₈₈	1	1	yes	characteristic
338	C₂²	—	4	1	no	—
339	C₂²	—	94	4	no	—
340	C₂²	—	376	2	no	—
341	C₂²	#15689603	1	1	yes	characteristic
342	C₂²	#15689778	1	1	yes	characteristic
343	C₂²	Q₈×D₁₈₈	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	94	4	no	—
348	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
349	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
350	C₄	D₄×D₁₈₈	1	1	yes	characteristic
351	C₄	#15689833	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	188	4	no	—
356	C₂	C₄⋊C₄×D₁₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1772	C₂
socle	C₂×C₄₇	D₄×C₄⋊C₄
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1772	D₄₇
hypocenter	C₄₇	#22694
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #20251996	C₂	semidirect
4. #20244178	C₂	semidirect
5. #20243629	C₂	semidirect
6. #20243560	C₂	semidirect
7. #20243439	C₂	semidirect
8. C₄₇×#1772	C₂	semidirect
14. #15689241	C₄	semidirect
15. #15689241	C₄	semidirect
16. #15689241	C₄	semidirect
17. #15689241	C₄	semidirect
181. #1772	D₄₇	semidirect
211. C₄₇	#22694	semidirect

Maximal subgroups

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

subgroup	index
#20252306	2
C₂×#15689241	2
#20251996	2
#20244178	2
#20243629	2
#20243560	2
#20243439	2
C₄₇×#1772	2
#22694	47