Subgroups — #26372999

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20255131	C₂	1	2	yes	maximal, complemented
4	#20245649	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245479	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245474	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245451	C₂	1	2	yes	maximal, complemented
8	#20245330	C₂	1	2	yes	maximal, complemented
9	#20245307	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	2	yes	—
13	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
14	#15689782	—	2	2	no	—
15	#15689555	—	2	4	no	—
16	C₂×Dic₃₇₆	—	2	4	no	—
17	C₂×Dic₃₇₆	C₂²	1	2	yes	—
18	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
20	C₂×#13093969	C₂²	1	2	yes	—
21	C₂×#13093969	C₂²	1	2	yes	—
22	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
23	#15689507	C₂²	1	2	yes	—
24	#15689364	C₂²	1	2	yes	—
25	#15689359	C₂²	1	2	yes	—
26	#15689346	—	2	2	no	—
27	#15689287	—	2	2	no	—
28	#15689264	—	2	2	no	—
29	C₄×Dic₁₈₈	C₂²	1	2	yes	—
30	#15688949	C₂²	1	1	yes	characteristic
31	#15688928	C₂²	1	2	yes	—
32	#15688928	C₂²	1	2	yes	—
33	#15688925	C₂²	1	2	yes	—
34	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄₇×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₄₇×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#10625350	—	2	2	no	—
42	C₂²×Dic₉₄	—	2	2	no	—
43	#13094046	—	2	4	no	—
44	#13094046	—	4	2	no	—
45	C₂×Dic₁₈₈	—	2	4	no	—
46	C₂×Dic₁₈₈	—	2	2	no	—
47	C₂×Dic₁₈₈	—	4	2	no	—
48	C₂×Dic₁₈₈	C₂³	1	2	yes	—
49	C₂×Dic₁₈₈	C₂³	1	2	yes	—
50	C₂×#10625325	—	2	2	no	—
51	C₂×#10625325	—	2	2	no	—
52	C₂×D₃₇₆	C₂³	1	2	yes	—
53	#13094012	—	2	2	no	—
54	#13094010	—	4	2	no	—
55	#13094003	—	2	2	no	—
56	#13094002	—	4	2	no	—
57	#13093998	—	2	2	no	—
58	#13093996	—	4	2	no	—
59	C₄×Dic₉₄	—	2	2	no	—
60	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
61	Dic₃₇₆	—	2	4	no	—
62	Dic₃₇₆	—	2	2	no	—
63	Dic₃₇₆	D₄	1	4	yes	complemented
64	#13093969	—	2	4	no	—
65	#13093969	—	4	2	no	—
66	D₇₅₂	—	2	2	no	—
67	#13093958	C₂³	1	2	yes	—
68	#13093949	—	2	2	no	—
69	#13093949	—	2	2	no	—
70	#13093946	—	2	2	no	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
74	C₂×C₁₆×C₄₇	—	2	2	no	—
75	C₂×C₁₆×C₄₇	—	2	2	no	—
76	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
81	C₂²⋊C₈×C₄₇	D₄	1	1	yes	characteristic
82	C₂²⋊C₈×C₄₇	D₄	1	1	yes	characteristic
83	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
84	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₄₇	—	4	2	no	—
88	C₂×C₂²⋊D₄₇	—	4	2	no	—
89	C₂²×Dic₄₇	—	4	2	no	—
90	#10625350	—	4	4	no	—
91	#10625350	—	8	2	no	—
92	C₂×D₁₈₈	—	2	2	no	—
93	C₂×C₄×D₄₇	—	4	2	no	—
94	C₂×Dic₉₄	—	2	2	no	—
95	C₂×Dic₉₄	—	2	2	no	—
96	C₂×Dic₉₄	—	4	4	no	—
97	C₂×Dic₉₄	—	4	2	no	—
98	C₂×Dic₉₄	—	8	2	no	—
99	C₂×C₈×C₄₇	—	2	1	no	—
100	C₂×C₈×C₄₇	—	2	1	no	—
101	C₂×C₈×C₄₇	—	2	1	no	—
102	C₂×C₈×C₄₇	—	2	1	no	—
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₄₇	D₈	1	1	yes	characteristic
109	C₂²⋊C₄×C₄₇	D₈	1	1	yes	characteristic
110	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₂²⋊Dic₄₇	—	4	2	no	—
112	(C₂×D₄₇)⋊C₄	—	4	2	no	—
113	(C₂×D₄₇)⋊C₄	—	4	2	no	—
114	(C₂×D₄₇)⋊C₄	—	4	2	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	Dic₄₇⋊C₄	—	4	2	no	—
117	Dic₄₇⋊C₄	—	4	2	no	—
118	C₄×Dic₄₇	—	4	2	no	—
119	Dic₁₈₈	—	2	2	no	—
120	Dic₁₈₈	—	2	2	no	—
121	Dic₁₈₈	—	2	2	no	—
122	Dic₁₈₈	—	4	4	no	—
123	Dic₁₈₈	C₂×D₄	1	4	yes	—
124	D₃₇₆	—	2	2	no	—
125	D₃₇₆	—	2	2	no	—
126	#10625325	—	4	4	no	—
127	#10625325	—	8	2	no	—
128	C₁₆×C₄₇	—	2	1	no	—
129	C₁₆×C₄₇	—	2	1	no	—
130	C₁₆×C₄₇	—	2	1	no	—
131	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
132	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
133	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₄₇	—	4	2	no	—
135	D₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	1	no	—
137	C₂×C₄×C₄₇	—	2	1	no	—
138	C₂×C₄×C₄₇	—	2	1	no	—
139	C₂×C₄×C₄₇	—	2	1	no	—
140	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₄₇	—	8	4	no	—
146	C₂²⋊D₄₇	—	8	4	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	8	4	no	—
151	D₁₈₈	—	4	2	no	—
152	D₁₈₈	—	4	2	no	—
153	C₄×D₄₇	—	8	4	no	—
154	Dic₉₄	—	2	4	no	—
155	Dic₉₄	—	4	2	no	—
156	Dic₉₄	—	4	2	no	—
157	Dic₉₄	—	4	2	no	—
158	Dic₉₄	—	8	4	no	—
159	C₈×C₄₇	—	2	1	no	—
160	C₈×C₄₇	—	2	1	no	—
161	C₈×C₄₇	—	2	1	no	—
162	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
164	#33851	—	47	1	no	maximal, Sylow
165	C₂²×C₄₇	—	2	1	no	—
166	C₂²×C₄₇	—	2	1	no	—
167	C₂²×C₄₇	—	2	1	no	—
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	—	2	1	no	—
170	C₂²×C₄₇	#508	1	2	yes	complemented
171	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
172	C₂×D₄₇	—	8	2	no	—
173	C₂×D₄₇	—	8	2	no	—
174	C₂×D₄₇	—	8	2	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	D₄²	1	1	yes	characteristic
180	C₄×C₄₇	#576	1	1	yes	characteristic
181	Dic₄₇	—	4	4	no	—
182	Dic₄₇	—	8	2	no	—
183	Dic₄₇	—	8	2	no	—
184	C₂×#508	—	47	1	no	—
185	C₂²×Q₃₂	—	47	1	no	—
186	#3285	—	47	2	no	—
187	#2249	—	47	1	no	—
188	#2216	—	47	1	no	—
189	#2215	—	47	1	no	—
190	#2211	—	47	2	no	—
191	#2189	—	47	2	no	—
192	#2185	—	47	2	no	—
193	C₄×Q₃₂	—	47	1	no	—
194	D₄×C₁₆	D₄₇	1	1	yes	characteristic, complemented
195	C₂×C₄₇	—	2	2	no	—
196	C₂×C₄₇	—	4	1	no	—
197	C₂×C₄₇	D₄×D₈	1	1	yes	characteristic
198	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
199	C₂×C₄₇	#3417	1	1	yes	characteristic
200	D₄₇	—	16	2	no	—
201	C₂×#135	—	47	2	no	—
202	C₂²×Q₁₆	—	47	2	no	—
203	#548	—	94	2	no	—
204	#508	—	94	4	no	—
205	C₂×Q₃₂	—	47	1	no	—
206	C₂×Q₃₂	—	47	2	no	—
207	C₂×Q₃₂	—	47	1	no	—
208	C₂×Q₃₂	—	94	4	no	—
209	C₂×SD₃₂	—	47	2	no	—
210	C₂×SD₃₂	—	47	2	no	—
211	C₂×D₁₆	—	47	1	no	—
212	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
213	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
214	#495	—	47	2	no	—
215	#467	—	47	2	no	—
216	#466	—	47	2	no	—
217	#464	—	94	2	no	—
218	#452	—	94	2	no	—
219	#448	—	94	2	no	—
220	C₄×Q₁₆	—	47	2	no	—
221	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
222	#366	—	47	1	no	—
223	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
224	#358	—	47	2	no	—
225	#358	—	47	2	no	—
226	#357	—	47	2	no	—
227	C₂²⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
228	C₂²⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
229	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
230	C₄₇	#33851	1	1	yes	characteristic, complemented, Sylow
231	C₂×#41	—	94	2	no	—
232	C₂²×Q₈	—	94	2	no	—
233	#135	—	94	4	no	—
234	#135	—	188	2	no	—
235	C₂×Q₁₆	—	47	2	no	—
236	C₂×Q₁₆	—	47	2	no	—
237	C₂×Q₁₆	—	94	4	no	—
238	C₂×Q₁₆	—	94	2	no	—
239	C₂×Q₁₆	—	188	2	no	—
240	C₂×SD₁₆	—	94	2	no	—
241	C₂×SD₁₆	—	94	2	no	—
242	C₂×D₈	—	47	2	no	—
243	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
244	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
245	#124	—	94	2	no	—
246	#124	—	188	2	no	—
247	C₂²⋊Q₈	—	94	2	no	—
248	C₂²⋊Q₈	—	188	2	no	—
249	#121	—	94	2	no	—
250	#121	—	188	2	no	—
251	C₄×Q₈	—	94	2	no	—
252	C₄×D₄	—	188	2	no	—
253	C₄×D₄	D₁₈₈	1	1	yes	characteristic
254	Q₃₂	—	47	4	no	—
255	Q₃₂	—	94	4	no	—
256	Q₃₂	—	94	2	no	—
257	SD₃₂	—	94	4	no	—
258	SD₃₂	—	188	2	no	—
259	D₁₆	—	94	2	no	—
260	C₂×C₁₆	—	2	2	no	—
261	C₂×C₁₆	—	2	2	no	—
262	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
263	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
264	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
265	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
266	#107	—	47	2	no	—
267	C₄⋊C₈	D₁₈₈	1	1	yes	characteristic
268	#103	—	94	2	no	—
269	#103	—	94	2	no	—
270	#102	—	94	2	no	—
271	C₂²⋊C₈	D₁₈₈	1	1	yes	characteristic
272	C₂²⋊C₈	D₁₈₈	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
274	#41	—	376	2	no	—
275	#41	—	188	4	no	—
276	C₂×Q₈	—	94	2	no	—
277	C₂×Q₈	—	94	2	no	—
278	C₂×Q₈	—	376	2	no	—
279	C₂×Q₈	—	188	4	no	—
280	C₂×Q₈	—	188	2	no	—
281	C₂×D₄	—	94	2	no	—
282	C₂×D₄	—	188	2	no	—
283	C₂×D₄	—	188	2	no	—
284	C₂×D₄	D₃₇₆	1	1	yes	characteristic
285	C₂²×C₄	—	188	2	no	—
286	C₂²×C₄	—	188	2	no	—
287	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
289	Q₁₆	—	47	4	no	—
290	Q₁₆	—	94	2	no	—
291	Q₁₆	—	94	2	no	—
292	Q₁₆	—	94	2	no	—
293	Q₁₆	—	188	4	no	—
294	SD₁₆	—	376	2	no	—
295	SD₁₆	—	188	4	no	—
296	D₈	—	94	2	no	—
297	D₈	—	94	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
303	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
304	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
305	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
306	C₄⋊C₄	—	94	2	no	—
307	C₄⋊C₄	—	188	2	no	—
308	C₄⋊C₄	—	188	2	no	—
309	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
310	C₂²⋊C₄	—	188	2	no	—
311	C₂²⋊C₄	—	188	2	no	—
312	C₂²⋊C₄	—	188	2	no	—
313	C₂²⋊C₄	—	188	2	no	—
314	C₂²⋊C₄	D₃₇₆	1	1	yes	characteristic
315	C₂²⋊C₄	D₃₇₆	1	1	yes	characteristic
316	C₄²	—	188	2	no	—
317	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	—	2	1	no	—
321	C₁₆	D₄×D₄₇	1	2	yes	—
322	C₂³	—	188	2	no	—
323	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
324	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
325	Q₈	—	94	4	no	—
326	Q₈	—	376	4	no	—
327	Q₈	—	188	2	no	—
328	Q₈	—	188	2	no	—
329	Q₈	—	188	2	no	—
330	D₄	—	2	2	no	—
331	D₄	—	376	4	no	—
332	D₄	—	376	4	no	—
333	D₄	—	188	2	no	—
334	D₄	—	188	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	376	4	no	—
340	C₂×C₄	—	376	4	no	—
341	C₂×C₄	—	188	2	no	—
342	C₂×C₄	—	188	2	no	—
343	C₂×C₄	—	188	2	no	—
344	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
345	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
346	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
347	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
353	C₈	#13094070	1	1	yes	characteristic
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	376	2	no	—
360	C₂²	—	376	2	no	—
361	C₂²	—	376	2	no	—
362	C₂²	#15689555	1	2	yes	complemented
363	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	376	2	no	—
369	C₄	—	376	2	no	—
370	C₄	—	188	4	no	—
371	C₄	D₄×D₁₈₈	1	1	yes	characteristic
372	C₄	#15689928	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	4	1	no	—
375	C₂	—	752	2	no	—
376	C₂	D₄×D₃₇₆	1	1	yes	characteristic
377	C₂	C₂×#15689555	1	1	yes	characteristic
378	C₂	#20256541	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	D₄×C₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	D₄×C₁₆	D₄₇
hypocenter	C₄₇	#33851
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15689555	C₂	semidirect
2. C₂²×Dic₃₇₆	C₂	semidirect
3. #20255131	C₂	semidirect
4. #20245649	C₂	semidirect
5. #20245479	C₂	semidirect
6. #20245474	C₂	semidirect
7. #20245451	C₂	semidirect
8. #20245330	C₂	semidirect
9. #20245307	C₂	semidirect
10. C₄×Dic₃₇₆	C₂	semidirect
11. D₄×C₁₆×C₄₇	C₂	semidirect
63. Dic₃₇₆	D₄	semidirect
170. C₂²×C₄₇	#508	semidirect
194. D₄×C₁₆	D₄₇	semidirect
230. C₄₇	#33851	semidirect
362. C₂²	#15689555	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689555	2
C₂²×Dic₃₇₆	2
#20255131	2
#20245649	2
#20245479	2
#20245474	2
#20245451	2
#20245330	2
#20245307	2
C₄×Dic₃₇₆	2
D₄×C₁₆×C₄₇	2
#33851	47