Subgroups — #26373012

Order 12032.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689553	C₂	1	2	yes	maximal, complemented
2	#20255148	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245637	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245472	C₂	1	2	yes	maximal, complemented
5	#20245428	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245425	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245311	C₂	1	2	yes	maximal, complemented
8	#20245309	C₂	1	2	yes	maximal, complemented
9	C₄×#13093970	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#3287	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	2	yes	—
13	#15689794	—	2	1	no	—
14	#15689553	—	2	8	no	—
15	C₂×#13093970	C₂²	1	1	yes	characteristic
16	C₂×#13093968	C₂²	1	2	yes	—
17	C₂×#13093967	C₂²	1	2	yes	—
18	C₂×#13093965	C₂²	1	2	yes	—
19	C₂×#13093965	C₂²	1	2	yes	—
20	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
21	#15689500	C₂²	1	1	yes	characteristic
22	#15689359	C₂²	1	2	yes	—
23	#15689324	—	2	1	no	—
24	#15689270	—	2	2	no	—
25	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
26	#15688948	C₂²	1	1	yes	characteristic
27	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
28	#15688927	C₂²	1	1	yes	characteristic
29	#15688926	C₂²	1	1	yes	characteristic
30	#15688924	C₂²	1	2	yes	—
31	#15688923	C₂²	1	2	yes	—
32	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
33	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₂×#135×C₄₇	C₂²	1	2	yes	—
35	C₄₇×#550	—	2	1	no	—
36	C₄₇×#493	C₂²	1	1	yes	characteristic
37	C₄₇×#466	C₂²	1	2	yes	—
38	C₄₇×#460	—	2	1	no	—
39	C₄₇×#449	—	2	2	no	—
40	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
42	C₂×#10625350	—	2	2	no	—
43	#13094046	—	2	4	no	—
44	#13094046	—	4	2	no	—
45	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
46	C₂×#10625325	—	2	2	no	—
47	C₂×D₃₇₆	—	2	1	no	—
48	C₂×D₃₇₆	C₂³	1	2	yes	—
49	#13094021	—	2	1	no	—
50	#13094020	—	4	1	no	—
51	#13093998	—	2	2	no	—
52	#13093997	—	4	2	no	—
53	C₄×Dic₉₄	—	2	1	no	—
54	#13093986	—	4	1	no	—
55	#13093970	D₄	1	4	yes	complemented
56	#13093968	—	2	4	no	—
57	#13093967	—	2	4	no	—
58	#13093965	—	2	4	no	—
59	#13093965	—	4	2	no	—
60	C₂×C₄₇⋊C₁₆	—	2	4	no	—
61	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
62	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
64	#13093958	C₂³	1	1	yes	characteristic
65	#13093949	—	2	1	no	—
66	#13093946	—	2	2	no	—
67	C₂×#41×C₄₇	—	2	2	no	—
68	#135×C₄₇	—	2	4	no	—
69	#135×C₄₇	—	4	2	no	—
70	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×SD₁₆×C₄₇	—	2	2	no	—
72	C₂×D₈×C₄₇	—	2	1	no	—
73	C₂×D₈×C₄₇	C₂³	1	2	yes	—
74	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
75	#127×C₄₇	—	2	1	no	—
76	#127×C₄₇	—	4	1	no	—
77	#121×C₄₇	—	2	2	no	—
78	#121×C₄₇	—	4	2	no	—
79	C₄×Q₈×C₄₇	—	2	1	no	—
80	C₄×D₄×C₄₇	—	4	1	no	—
81	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
82	#107×C₄₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
84	#103×C₄₇	—	2	1	no	—
85	#102×C₄₇	—	2	2	no	—
86	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
87	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
88	#41×C₄₇	—	4	4	no	—
89	#41×C₄₇	—	8	2	no	—
90	C₂×Q₈×C₄₇	—	2	1	no	—
91	C₂×D₄×C₄₇	—	2	2	no	—
92	C₂×D₄×C₄₇	—	4	2	no	—
93	C₂×D₄×C₄₇	—	4	2	no	—
94	C₂×D₄×C₄₇	—	4	1	no	—
95	C₂×D₄×C₄₇	—	4	1	no	—
96	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	—	4	2	no	—
98	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
99	C₂×C₂²⋊D₄₇	—	4	2	no	—
100	C₂×C₂²⋊D₄₇	—	4	2	no	—
101	#10625350	—	4	4	no	—
102	#10625350	—	8	2	no	—
103	C₂×D₁₈₈	—	2	2	no	—
104	C₂×D₁₈₈	—	4	1	no	—
105	C₂×D₁₈₈	—	4	1	no	—
106	C₂×C₄×D₄₇	—	4	2	no	—
107	C₂×Dic₉₄	—	2	1	no	—
108	Q₁₆×C₄₇	—	2	1	no	—
109	Q₁₆×C₄₇	C₂×D₄	1	2	yes	—
110	SD₁₆×C₄₇	—	4	4	no	—
111	D₈×C₄₇	—	2	2	no	—
112	D₈×C₄₇	—	2	2	no	—
113	D₈×C₄₇	—	8	1	no	—
114	C₂×C₈×C₄₇	—	2	2	no	—
115	C₂×C₈×C₄₇	—	2	2	no	—
116	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₄₇	—	2	1	no	—
120	C₄⋊C₄×C₄₇	—	4	1	no	—
121	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
122	C₂²⋊C₄×C₄₇	—	4	2	no	—
123	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
124	C₄²×C₄₇	—	4	1	no	—
125	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
126	(C₂×D₄₇)⋊C₄	—	4	2	no	—
127	C₄⋊Dic₄₇	—	2	1	no	—
128	Dic₄₇⋊C₄	—	4	1	no	—
129	C₄×Dic₄₇	—	4	1	no	—
130	Dic₁₈₈	—	2	1	no	—
131	Dic₁₈₈	C₂×D₄	1	2	yes	—
132	D₃₇₆	—	2	2	no	—
133	D₃₇₆	—	2	2	no	—
134	D₃₇₆	—	8	1	no	—
135	#10625325	—	4	4	no	—
136	C₄₇⋊C₁₆	—	2	2	no	—
137	C₄₇⋊C₁₆	—	2	1	no	—
138	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
139	C₂³×C₄₇	—	4	2	no	—
140	C₂³×C₄₇	C₂×D₈	1	2	yes	—
141	C₂²×D₄₇	—	4	2	no	—
142	Q₈×C₄₇	—	2	2	no	—
143	Q₈×C₄₇	—	4	1	no	—
144	D₄×C₄₇	—	2	2	no	—
145	D₄×C₄₇	—	4	2	no	—
146	D₄×C₄₇	—	4	2	no	—
147	D₄×C₄₇	—	8	4	no	—
148	D₄×C₄₇	—	8	2	no	—
149	D₄×C₄₇	—	16	1	no	—
150	D₄×C₄₇	—	16	2	no	—
151	C₂×C₄×C₄₇	—	2	2	no	—
152	C₂×C₄×C₄₇	—	2	2	no	—
153	C₂×C₄×C₄₇	—	4	1	no	—
154	C₂×C₄×C₄₇	—	4	1	no	—
155	C₂×C₄×C₄₇	—	8	4	no	—
156	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
158	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
160	C₂²⋊D₄₇	—	8	4	no	—
161	C₂²⋊D₄₇	—	16	2	no	—
162	C₂×Dic₄₇	—	4	1	no	—
163	C₂×Dic₄₇	—	4	1	no	—
164	D₁₈₈	—	4	2	no	—
165	D₁₈₈	—	4	2	no	—
166	D₁₈₈	—	8	2	no	—
167	D₁₈₈	—	16	1	no	—
168	C₄×D₄₇	—	8	4	no	—
169	Dic₉₄	—	2	2	no	—
170	Dic₉₄	—	4	1	no	—
171	C₈×C₄₇	—	2	2	no	—
172	C₈×C₄₇	—	2	1	no	—
173	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
175	#33853	—	47	1	no	maximal, Sylow
176	C₂²×C₄₇	—	2	2	no	—
177	C₂²×C₄₇	—	2	2	no	—
178	C₂²×C₄₇	—	2	2	no	—
179	C₂²×C₄₇	—	8	2	no	—
180	C₂²×C₄₇	—	8	2	no	—
181	C₂²×C₄₇	—	8	2	no	—
182	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
183	C₂×D₄₇	—	8	2	no	—
184	C₂×D₄₇	—	8	2	no	—
185	C₂×D₄₇	—	8	2	no	—
186	C₄×C₄₇	—	2	1	no	—
187	C₄×C₄₇	—	2	2	no	—
188	C₄×C₄₇	—	4	2	no	—
189	C₄×C₄₇	—	8	1	no	—
190	C₄×C₄₇	#508	1	2	yes	complemented
191	C₄×C₄₇	D₄²	1	1	yes	characteristic
192	C₄×C₄₇	#576	1	1	yes	characteristic
193	Dic₄₇	—	4	2	no	—
194	Dic₄₇	—	8	1	no	—
195	C₂×#508	—	47	2	no	—
196	#3287	—	47	1	no	—
197	#3287	D₄₇	1	1	yes	characteristic, complemented
198	#2246	—	47	1	no	—
199	#2215	—	47	2	no	—
200	#2207	—	47	1	no	—
201	#2207	—	47	1	no	—
202	#2186	—	47	2	no	—
203	#2186	—	47	2	no	—
204	C₄×Q₃₂	—	47	1	no	—
205	D₄×C₁₆	—	47	1	no	—
206	C₂×C₄₇	—	4	2	no	—
207	C₂×C₄₇	—	16	2	no	—
208	C₂×C₄₇	D₄×D₈	1	1	yes	characteristic
209	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
210	C₂×C₄₇	#3415	1	1	yes	characteristic
211	D₄₇	—	16	2	no	—
212	C₂×#135	—	47	2	no	—
213	C₂×#135	C₂×D₄₇	1	2	yes	—
214	#550	—	2	1	no	—
215	#550	—	94	1	no	—
216	#508	—	94	8	no	—
217	C₂×Q₃₂	—	47	1	no	—
218	C₂×SD₃₂	—	47	2	no	—
219	C₂×SD₃₂	—	47	2	no	—
220	C₂×D₁₆	—	47	2	no	—
221	C₂×D₁₆	—	47	2	no	—
222	C₂²×C₁₆	—	47	2	no	—
223	#493	—	47	1	no	—
224	#493	C₂×D₄₇	1	1	yes	characteristic
225	#466	—	47	2	no	—
226	#466	C₂×D₄₇	1	2	yes	—
227	#460	—	2	1	no	—
228	#460	—	94	1	no	—
229	#449	—	2	2	no	—
230	#449	—	94	2	no	—
231	C₄×Q₁₆	—	47	1	no	—
232	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
233	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
234	#366	—	47	1	no	—
235	C₄⋊C₁₆	—	47	1	no	—
236	#358	—	47	1	no	—
237	#358	—	47	1	no	—
238	#357	—	47	2	no	—
239	#357	—	47	2	no	—
240	C₂²⋊C₁₆	—	47	2	no	—
241	C₄×C₁₆	—	47	1	no	—
242	C₄₇	#33853	1	1	yes	characteristic, complemented, Sylow
243	C₂×#41	—	2	2	no	—
244	C₂×#41	—	94	2	no	—
245	#135	—	2	4	no	—
246	#135	—	4	2	no	—
247	#135	—	94	4	no	—
248	#135	—	188	2	no	—
249	C₂×Q₁₆	—	47	1	no	—
250	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
251	C₂×SD₁₆	—	2	2	no	—
252	C₂×SD₁₆	—	94	2	no	—
253	C₂×D₈	—	2	1	no	—
254	C₂×D₈	—	47	2	no	—
255	C₂×D₈	—	94	1	no	—
256	C₂×D₈	C₂²×D₄₇	1	2	yes	—
257	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
258	#127	—	2	1	no	—
259	#127	—	4	1	no	—
260	#127	—	94	1	no	—
261	#127	—	188	1	no	—
262	#121	—	2	2	no	—
263	#121	—	4	2	no	—
264	#121	—	94	2	no	—
265	#121	—	188	2	no	—
266	C₄×Q₈	—	2	1	no	—
267	C₄×Q₈	—	94	1	no	—
268	C₄×D₄	—	4	1	no	—
269	C₄×D₄	—	188	1	no	—
270	C₄×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
271	Q₃₂	—	47	4	no	—
272	SD₃₂	—	94	4	no	—
273	SD₃₂	—	94	4	no	—
274	D₁₆	—	94	4	no	—
275	D₁₆	—	188	2	no	—
276	C₂×C₁₆	—	47	1	no	—
277	C₂×C₁₆	—	47	1	no	—
278	C₂×C₁₆	—	47	2	no	—
279	C₂×C₁₆	—	94	4	no	—
280	#107	—	47	1	no	—
281	#107	C₂²×D₄₇	1	1	yes	characteristic
282	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
283	#103	—	2	1	no	—
284	#103	—	94	1	no	—
285	#102	—	2	2	no	—
286	#102	—	94	2	no	—
287	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
288	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
289	#41	—	4	4	no	—
290	#41	—	8	2	no	—
291	#41	—	376	2	no	—
292	#41	—	188	4	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	94	1	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	4	2	no	—
297	C₂×D₄	—	4	2	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	94	2	no	—
301	C₂×D₄	—	188	2	no	—
302	C₂×D₄	—	188	2	no	—
303	C₂×D₄	—	188	1	no	—
304	C₂×D₄	—	188	1	no	—
305	C₂×D₄	#10625334	1	1	yes	characteristic
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	188	2	no	—
308	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
309	Q₁₆	—	2	1	no	—
310	Q₁₆	—	47	2	no	—
311	Q₁₆	—	94	1	no	—
312	Q₁₆	D₄×D₄₇	1	2	yes	—
313	SD₁₆	—	4	4	no	—
314	SD₁₆	—	188	4	no	—
315	D₈	—	2	2	no	—
316	D₈	—	2	2	no	—
317	D₈	—	8	1	no	—
318	D₈	—	94	2	no	—
319	D₈	—	94	2	no	—
320	D₈	—	376	1	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
324	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
325	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	94	1	no	—
329	C₄⋊C₄	—	188	1	no	—
330	C₄⋊C₄	#10625334	1	1	yes	characteristic
331	C₂²⋊C₄	—	4	2	no	—
332	C₂²⋊C₄	—	188	2	no	—
333	C₂²⋊C₄	#10625334	1	2	yes	—
334	C₄²	—	4	1	no	—
335	C₄²	—	188	1	no	—
336	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
337	C₁₆	—	47	2	no	—
338	C₁₆	—	94	2	no	—
339	C₁₆	—	94	1	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	188	2	no	—
342	C₂³	C₂×#10625334	1	2	yes	—
343	Q₈	—	2	2	no	—
344	Q₈	—	4	1	no	—
345	Q₈	—	94	2	no	—
346	Q₈	—	188	1	no	—
347	D₄	—	2	2	no	—
348	D₄	—	4	2	no	—
349	D₄	—	4	2	no	—
350	D₄	—	8	4	no	—
351	D₄	—	8	2	no	—
352	D₄	—	16	1	no	—
353	D₄	—	16	2	no	—
354	D₄	—	376	4	no	—
355	D₄	—	376	2	no	—
356	D₄	—	188	2	no	—
357	D₄	—	188	2	no	—
358	D₄	—	752	1	no	—
359	D₄	—	752	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	8	4	no	—
365	C₂×C₄	—	376	4	no	—
366	C₂×C₄	—	188	1	no	—
367	C₂×C₄	—	188	1	no	—
368	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
369	C₂×C₄	C₂×#10625334	1	2	yes	—
370	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
371	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
372	C₈	—	2	2	no	—
373	C₈	—	2	1	no	—
374	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
375	C₈	#13094069	1	1	yes	characteristic
376	C₂²	—	2	2	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	8	2	no	—
380	C₂²	—	8	2	no	—
381	C₂²	—	8	2	no	—
382	C₂²	—	376	2	no	—
383	C₂²	—	376	2	no	—
384	C₂²	—	376	2	no	—
385	C₂²	C₂²×#10625334	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	2	no	—
388	C₄	—	4	2	no	—
389	C₄	—	8	1	no	—
390	C₄	—	376	1	no	—
391	C₄	—	188	2	no	—
392	C₄	#15689553	1	2	yes	complemented
393	C₄	D₄×C₂²⋊D₄₇	1	1	yes	characteristic
394	C₄	#15689926	1	1	yes	characteristic
395	C₂	—	4	2	no	—
396	C₂	—	16	2	no	—
397	C₂	—	752	2	no	—
398	C₂	D₄×#10625334	1	1	yes	characteristic
399	C₂	C₂×#15689553	1	1	yes	characteristic
400	C₂	#20256527	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3287	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3287	D₄₇
hypocenter	C₄₇	#33853
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3287

Complements

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

N	H ≅ G/N	type
1. C₂×#15689553	C₂	semidirect
2. #20255148	C₂	semidirect
3. #20245637	C₂	semidirect
4. #20245472	C₂	semidirect
5. #20245428	C₂	semidirect
6. #20245425	C₂	semidirect
7. #20245311	C₂	semidirect
8. #20245309	C₂	semidirect
9. C₄×#13093970	C₂	semidirect
10. D₄×C₄₇⋊C₁₆	C₂	semidirect
11. C₄₇×#3287	C₂	semidirect
55. #13093970	D₄	semidirect
190. C₄×C₄₇	#508	semidirect
197. #3287	D₄₇	semidirect
242. C₄₇	#33853	semidirect
392. C₄	#15689553	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689553	2
#20255148	2
#20245637	2
#20245472	2
#20245428	2
#20245425	2
#20245311	2
#20245309	2
C₄×#13093970	2
D₄×C₄₇⋊C₁₆	2
C₄₇×#3287	2
#33853	47