Subgroups — #26370858

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 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	C₂×#15689539	C₂	1	1	yes	maximal, characteristic, complemented
3	#20253109	C₂	1	2	yes	maximal, complemented
4	#20245519	C₂	1	4	yes	maximal, complemented
5	#20245514	C₂	1	2	yes	maximal, complemented
6	#20245509	C₂	1	2	yes	maximal, complemented
7	#20245135	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₇×#368	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689923	—	2	2	no	—
10	C₂×#13094046	C₂²	1	2	yes	—
11	C₂×#13094030	C₂²	1	1	yes	characteristic
12	#15689555	—	2	1	no	—
13	#15689555	—	2	1	no	—
14	#15689555	—	2	1	no	—
15	#15689555	—	2	1	no	—
16	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic, complemented
17	C₂×#13093969	C₂²	1	2	yes	complemented
18	C₂×D₇₅₂	C₂²	1	1	yes	characteristic, complemented
19	#15689539	—	2	1	no	—
20	#15689539	—	2	1	no	—
21	#15689539	C₂²	1	2	yes	—
22	#15689539	C₂²	1	2	yes	—
23	C₂×#13093964	C₂²	1	1	yes	characteristic
24	C₂×#13093963	—	2	1	no	—
25	C₂×C₁₆×D₄₇	—	2	1	no	—
26	#15689399	C₂²	1	4	yes	—
27	#15689394	C₂²	1	4	yes	—
28	#15689387	—	2	4	no	—
29	#15689105	—	2	2	no	—
30	#15688938	C₂²	1	4	yes	complemented
31	#15688935	C₂²	1	4	yes	complemented
32	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#108×C₄₇	C₂²	1	2	yes	—
34	C₄₇×#368	C₂²	1	2	yes	complemented
35	C₄₇×#368	C₂²	1	2	yes	complemented
36	C₂×#10625350	—	2	2	no	—
37	C₂×#10625350	D₄	1	1	yes	characteristic
38	#13094057	—	2	4	no	—
39	#13094057	—	4	2	no	—
40	#13094052	—	2	4	no	—
41	#13094052	—	4	2	no	—
42	#13094046	—	2	2	no	—
43	#13094046	—	2	2	no	—
44	#13094046	—	2	2	no	—
45	#13094046	—	2	2	no	—
46	#13094046	—	4	4	no	—
47	C₂×Dic₁₈₈	C₂³	1	2	yes	—
48	C₂×#10625325	—	2	2	no	—
49	C₂×D₃₇₆	C₂³	1	2	yes	—
50	#13094030	—	2	1	no	—
51	#13094030	—	2	1	no	—
52	#13094030	D₄	1	2	yes	—
53	#13094030	C₂³	1	2	yes	—
54	C₂×#10625329	D₄	1	1	yes	characteristic
55	C₂×#10625324	—	2	1	no	—
56	C₂×C₈×D₄₇	—	2	1	no	—
57	Dic₃₇₆	—	2	1	no	—
58	Dic₃₇₆	—	2	1	no	—
59	#13093969	—	2	2	no	—
60	#13093969	—	2	2	no	—
61	D₇₅₂	—	2	1	no	—
62	D₇₅₂	—	2	1	no	—
63	#13093964	—	2	1	no	—
64	#13093964	C₂³	1	2	yes	—
65	#13093963	—	2	2	no	—
66	#13093963	—	2	2	no	—
67	C₂×C₄₇⋊C₁₆	—	2	1	no	—
68	C₁₆×D₄₇	—	2	2	no	—
69	C₁₆×D₄₇	—	2	2	no	—
70	#13093942	—	2	4	no	—
71	#13093940	—	2	4	no	—
72	C₂×M₁₆×C₄₇	—	2	2	no	—
73	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
75	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
76	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
77	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
78	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
80	#108×C₄₇	—	2	2	no	—
81	#108×C₄₇	C₂³	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₂²⋊D₄₇	—	2	1	no	—
84	C₂×C₂²⋊D₄₇	—	4	2	no	—
85	#10625350	—	2	1	no	—
86	#10625350	—	2	1	no	—
87	#10625350	—	4	2	no	—
88	#10625350	—	4	2	no	—
89	#10625350	—	4	2	no	—
90	#10625350	—	4	2	no	—
91	#10625350	D₈	1	2	yes	—
92	#10625350	C₂×D₄	1	2	yes	—
93	C₂×D₁₈₈	—	2	2	no	—
94	C₂×D₁₈₈	D₈	1	1	yes	characteristic
95	C₂×C₄×D₄₇	—	2	1	no	—
96	C₂×C₄×D₄₇	—	4	2	no	—
97	C₂×Dic₉₄	—	2	2	no	—
98	C₂×Dic₉₄	D₈	1	1	yes	characteristic
99	M₁₆×C₄₇	—	2	4	no	—
100	M₁₆×C₄₇	—	4	2	no	—
101	C₂×C₈×C₄₇	—	4	2	no	—
102	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
106	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
108	#10625329	—	2	1	no	—
109	#10625329	C₂×D₄	1	2	yes	—
110	C₂×C₄₇⋊C₈	—	2	1	no	—
111	Dic₁₈₈	—	2	2	no	—
112	Dic₁₈₈	—	2	2	no	—
113	Dic₁₈₈	—	4	4	no	—
114	D₃₇₆	—	2	2	no	—
115	D₃₇₆	—	2	2	no	—
116	D₃₇₆	—	4	4	no	—
117	#10625325	—	4	4	no	—
118	#10625325	—	4	4	no	—
119	#10625325	—	4	2	no	—
120	#10625325	—	4	2	no	—
121	#10625324	—	2	2	no	—
122	#10625324	—	2	2	no	—
123	C₈×D₄₇	—	2	2	no	—
124	C₈×D₄₇	—	2	2	no	—
125	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
129	C₄₇⋊C₁₆	—	2	2	no	—
130	C₂³×C₄₇	#135	1	1	yes	characteristic
131	C₂²×D₄₇	—	2	1	no	—
132	C₂²×D₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₄₇	—	2	2	no	—
140	C₂²⋊D₄₇	—	2	2	no	—
141	C₂²⋊D₄₇	—	8	2	no	—
142	C₂²⋊D₄₇	—	8	2	no	—
143	C₂×Dic₄₇	—	2	1	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	D₁₈₈	—	2	1	no	—
146	D₁₈₈	—	4	2	no	—
147	D₁₈₈	—	4	2	no	—
148	D₁₈₈	C₂×D₈	1	2	yes	—
149	C₄×D₄₇	—	2	2	no	—
150	C₄×D₄₇	—	2	2	no	—
151	C₄×D₄₇	—	8	2	no	—
152	C₄×D₄₇	—	8	2	no	—
153	Dic₉₄	—	2	1	no	—
154	Dic₉₄	—	4	2	no	—
155	Dic₉₄	—	4	2	no	—
156	Dic₉₄	C₂×D₈	1	2	yes	—
157	C₈×C₄₇	—	4	4	no	—
158	C₈×C₄₇	#121	1	1	yes	characteristic
159	C₈×C₄₇	#121	1	1	yes	characteristic
160	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
162	C₄₇⋊C₈	—	2	2	no	—
163	#33617	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	—	4	1	no	—
165	C₂²×C₄₇	#466	1	1	yes	characteristic
166	C₂²×C₄₇	#466	1	1	yes	characteristic
167	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
168	C₂×D₄₇	—	2	2	no	—
169	C₂×D₄₇	—	4	2	no	—
170	C₂×D₄₇	—	8	2	no	—
171	C₂×D₄₇	—	16	2	no	—
172	C₄×C₄₇	#466	1	1	yes	characteristic
173	C₄×C₄₇	#466	1	1	yes	characteristic
174	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
175	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
176	Dic₄₇	—	2	2	no	—
177	Dic₄₇	—	8	2	no	—
178	C₂×#508	—	47	1	no	—
179	C₂×#504	—	47	1	no	—
180	#3067	—	47	2	no	—
181	#2223	—	47	4	no	—
182	#2222	—	47	2	no	—
183	#2221	—	47	2	no	—
184	C₂×#368	D₄₇	1	1	yes	characteristic, complemented
185	#2148	—	47	2	no	—
186	C₂×C₄₇	—	2	1	no	—
187	C₂×C₄₇	—	2	1	no	—
188	C₂×C₄₇	—	2	1	no	—
189	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
190	D₄₇	—	4	2	no	—
191	D₄₇	—	16	2	no	—
192	#575	—	94	2	no	—
193	C₂×#135	—	47	2	no	—
194	C₂×#131	—	47	1	no	—
195	#508	—	94	1	no	—
196	#508	—	94	1	no	—
197	#508	—	94	1	no	—
198	#508	—	94	1	no	—
199	C₂×Q₃₂	—	47	1	no	—
200	C₂×SD₃₂	—	47	2	no	—
201	C₂×D₁₆	—	47	1	no	—
202	#504	—	47	2	no	—
203	#504	—	47	2	no	—
204	#504	—	94	1	no	—
205	#504	—	94	1	no	—
206	C₂×M₃₂	—	47	1	no	—
207	C₂×M₃₂	—	94	1	no	—
208	C₂²×C₁₆	—	94	1	no	—
209	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
210	#473	—	47	4	no	—
211	#472	—	47	4	no	—
212	#471	—	94	4	no	—
213	C₂×#108	C₂×D₄₇	1	2	yes	—
214	#413	—	94	2	no	—
215	#368	C₂×D₄₇	1	2	yes	complemented
216	#368	C₂×D₄₇	1	2	yes	complemented
217	#362	—	47	4	no	—
218	#361	—	47	4	no	—
219	C₄₇	#33617	1	1	yes	characteristic, complemented, Sylow
220	C₂×#41	—	47	1	no	—
221	C₂×#41	—	94	2	no	—
222	#137	—	94	4	no	—
223	#137	—	188	2	no	—
224	AGL(1,ℤ/8)	—	94	4	no	—
225	AGL(1,ℤ/8)	—	188	2	no	—
226	#135	—	94	2	no	—
227	#135	—	94	2	no	—
228	#135	—	94	2	no	—
229	#135	—	94	2	no	—
230	#135	—	188	4	no	—
231	C₂×Q₁₆	—	47	2	no	—
232	C₂×SD₁₆	—	94	2	no	—
233	C₂×D₈	—	47	2	no	—
234	#131	—	47	2	no	—
235	#131	—	47	2	no	—
236	#131	—	94	1	no	—
237	#131	—	94	1	no	—
238	C₂×M₁₆	—	2	2	no	—
239	C₂×M₁₆	—	47	1	no	—
240	C₂×M₁₆	—	94	1	no	—
241	C₂²×C₈	—	94	1	no	—
242	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
243	Q₃₂	—	94	1	no	—
244	Q₃₂	—	94	1	no	—
245	SD₃₂	—	94	2	no	—
246	SD₃₂	—	94	2	no	—
247	D₁₆	—	94	1	no	—
248	D₁₆	—	94	1	no	—
249	M₃₂	—	47	2	no	—
250	M₃₂	—	94	1	no	—
251	M₃₂	—	94	2	no	—
252	M₃₂	—	94	2	no	—
253	C₂×C₁₆	—	94	1	no	—
254	C₂×C₁₆	—	94	2	no	—
255	C₂×C₁₆	—	94	2	no	—
256	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
257	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
258	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
259	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
261	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
262	#108	—	2	2	no	—
263	#108	C₂²×D₄₇	1	4	yes	—
264	#101	—	94	4	no	—
265	#100	—	94	4	no	—
266	#41	—	47	2	no	—
267	#41	—	47	2	no	—
268	#41	—	94	1	no	—
269	#41	—	94	1	no	—
270	#41	—	188	2	no	—
271	#41	—	188	2	no	—
272	#41	—	188	2	no	—
273	#41	—	188	2	no	—
274	C₂×Q₈	—	47	1	no	—
275	C₂×Q₈	—	94	2	no	—
276	C₂×D₄	—	47	1	no	—
277	C₂×D₄	—	94	1	no	—
278	C₂×D₄	—	94	2	no	—
279	C₂×D₄	—	188	2	no	—
280	C₂²×C₄	—	94	1	no	—
281	C₂²×C₄	—	188	2	no	—
282	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
283	Q₁₆	—	94	2	no	—
284	Q₁₆	—	94	2	no	—
285	Q₁₆	—	188	4	no	—
286	SD₁₆	—	188	4	no	—
287	SD₁₆	—	188	4	no	—
288	SD₁₆	—	188	2	no	—
289	SD₁₆	—	188	2	no	—
290	D₈	—	94	2	no	—
291	D₈	—	94	2	no	—
292	D₈	—	188	4	no	—
293	M₁₆	—	2	4	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	47	2	no	—
296	M₁₆	—	94	1	no	—
297	M₁₆	—	94	2	no	—
298	M₁₆	—	94	2	no	—
299	C₂×C₈	—	4	2	no	—
300	C₂×C₈	—	94	1	no	—
301	C₂×C₈	—	94	2	no	—
302	C₂×C₈	—	94	2	no	—
303	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
304	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
305	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
306	C₂×C₈	#10625354	1	1	yes	characteristic
307	C₂×C₈	#10625354	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
309	C₁₆	—	94	2	no	—
310	C₁₆	C₂×D₁₈₈	1	1	yes	characteristic
311	C₁₆	C₂×D₁₈₈	1	1	yes	characteristic
312	C₁₆	C₂×D₁₈₈	1	1	yes	characteristic
313	C₁₆	C₂×D₁₈₈	1	1	yes	characteristic
314	C₂³	—	94	1	no	—
315	C₂³	—	188	2	no	—
316	C₂³	#13094047	1	1	yes	characteristic
317	Q₈	—	47	2	no	—
318	Q₈	—	94	1	no	—
319	Q₈	—	188	2	no	—
320	Q₈	—	188	2	no	—
321	D₄	—	47	2	no	—
322	D₄	—	94	2	no	—
323	D₄	—	94	2	no	—
324	D₄	—	94	1	no	—
325	D₄	—	376	2	no	—
326	D₄	—	376	2	no	—
327	D₄	—	188	2	no	—
328	D₄	—	188	2	no	—
329	C₂×C₄	—	94	1	no	—
330	C₂×C₄	—	94	2	no	—
331	C₂×C₄	—	94	2	no	—
332	C₂×C₄	—	376	2	no	—
333	C₂×C₄	—	376	2	no	—
334	C₂×C₄	—	188	2	no	—
335	C₂×C₄	#13093999	1	1	yes	characteristic
336	C₂×C₄	#13093999	1	1	yes	characteristic
337	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
338	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
339	C₂×C₄	#13094047	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
341	C₈	—	4	4	no	—
342	C₈	—	94	2	no	—
343	C₈	#13093999	1	1	yes	characteristic
344	C₈	#13093999	1	1	yes	characteristic
345	C₈	C₂²×D₁₈₈	1	1	yes	characteristic
346	C₈	C₂×#10625354	1	1	yes	characteristic
347	C₂²	—	4	1	no	—
348	C₂²	—	94	2	no	—
349	C₂²	—	376	2	no	—
350	C₂²	—	188	2	no	—
351	C₂²	—	752	2	no	—
352	C₂²	#15689360	1	1	yes	characteristic
353	C₂²	#15689360	1	1	yes	characteristic
354	C₂²	C₂×#13094047	1	1	yes	characteristic
355	C₄	—	94	2	no	—
356	C₄	—	376	2	no	—
357	C₄	#15689360	1	1	yes	characteristic
358	C₄	#15689360	1	1	yes	characteristic
359	C₄	C₂×#13093999	1	1	yes	characteristic
360	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	188	2	no	—
365	C₂	—	752	2	no	—
366	C₂	C₂×#15689360	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15689360
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#368	C₂
socle	C₂×C₄₇	C₂×#466
hypercenter	C₂×#368	D₄₇
hypocenter	C₄₇	#33617
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂×#15689555	C₂	semidirect
2. C₂×#15689539	C₂	semidirect
3. #20253109	C₂	semidirect
4. #20245519	C₂	semidirect
5. #20245514	C₂	semidirect
6. #20245509	C₂	semidirect
7. #20245135	C₂	semidirect
8. C₂×C₄₇×#368	C₂	semidirect
16. C₂×Dic₃₇₆	C₂²	semidirect
17. C₂×#13093969	C₂²	semidirect
18. C₂×D₇₅₂	C₂²	semidirect
30. #15688938	C₂²	semidirect
31. #15688935	C₂²	semidirect
34. C₄₇×#368	C₂²	semidirect
35. C₄₇×#368	C₂²	semidirect
184. C₂×#368	D₄₇	semidirect
215. #368	C₂×D₄₇	semidirect
216. #368	C₂×D₄₇	semidirect
219. C₄₇	#33617	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
C₂×#15689539	2
#20253109	2
#20245519	2
#20245514	2
#20245509	2
#20245135	2
C₂×C₄₇×#368	2
#33617	47