Subgroups — #25254349

Order 11008.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147584	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136832	C₂	1	2	yes	maximal
4	#19136823	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136743	C₂	1	1	yes	maximal, characteristic
6	#19136738	C₂	1	2	yes	maximal, complemented
7	#19136658	C₂	1	2	yes	maximal
8	#19136646	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136467	C₂	1	2	yes	maximal, complemented
10	C₄×#13063270	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₈₆	—	2	1	no	—
13	#14583622	—	2	1	no	—
14	C₂×Dic₃₄₄	C₂²	1	2	yes	—
15	C₂×#13063270	C₂²	1	1	yes	characteristic
16	#14583304	C₂²	1	1	yes	characteristic
17	#14583304	C₂²	1	2	yes	—
18	#14583283	C₂²	1	2	yes	—
19	#14583279	C₂²	1	1	yes	characteristic
20	#14583195	—	2	1	no	—
21	#14583195	—	2	2	no	—
22	#14583122	—	2	2	no	—
23	#14583105	—	2	1	no	—
24	C₄×Dic₁₇₂	C₂²	1	2	yes	—
25	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₇₂	C₂²	1	2	yes	—
27	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
28	#14582727	C₂²	1	1	yes	characteristic
29	#14582727	C₂²	1	2	yes	—
30	#14582704	C₂²	1	2	yes	—
31	#14582704	C₂²	1	2	yes	—
32	#14582704	C₂²	1	2	yes	—
33	#14582704	C₂²	1	1	yes	characteristic
34	#14582701	C₂²	1	1	yes	characteristic
35	#14582701	C₂²	1	2	yes	—
36	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₄₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₇₂	—	2	1	no	—
42	C₂×Dic₁₇₂	C₂³	1	2	yes	—
43	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
44	C₂×#10621873	—	2	2	no	—
45	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
46	#13063325	—	2	1	no	—
47	#13063325	—	2	2	no	—
48	#13063325	—	2	1	no	—
49	#13063324	—	4	1	no	—
50	#13063324	—	4	2	no	—
51	#13063320	—	4	2	no	—
52	#13063313	—	2	2	no	—
53	#13063301	—	4	1	no	—
54	Q₈×Dic₄₃	—	4	1	no	—
55	C₄×Dic₈₆	—	2	2	no	—
56	C₄×Dic₈₆	—	2	1	no	—
57	C₄×Dic₈₆	—	2	2	no	—
58	C₄×D₁₇₂	—	2	1	no	—
59	#13063283	—	4	1	no	—
60	Dic₃₄₄	—	2	4	no	—
61	#13063270	—	2	2	no	—
62	#13063259	—	2	1	no	—
63	#13063259	—	2	2	no	—
64	#13063259	C₂³	1	2	yes	—
65	#13063259	C₂³	1	1	yes	characteristic
66	#13063259	C₂³	1	1	yes	characteristic
67	#13063250	—	2	2	no	—
68	#13063250	—	2	1	no	—
69	#13063250	—	2	2	no	—
70	#13063247	—	2	1	no	—
71	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
77	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₄₃	C₂³	1	2	yes	—
79	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
80	C₂×D₁₇₂	—	2	1	no	—
81	C₂×C₄×D₄₃	—	4	1	no	—
82	C₂×Dic₈₆	—	2	1	no	—
83	C₂×Dic₈₆	—	2	2	no	—
84	C₂×Dic₈₆	—	4	2	no	—
85	C₂×Dic₈₆	—	4	1	no	—
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
91	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂×D₄	1	2	yes	—
93	(C₂×D₄₃)⋊C₄	—	4	2	no	—
94	(C₂×D₄₃)⋊C₄	—	4	1	no	—
95	(C₂×D₄₃)⋊C₄	—	4	2	no	—
96	C₄⋊Dic₄₃	—	2	1	no	—
97	C₄⋊Dic₄₃	—	2	1	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	C₄⋊Dic₄₃	—	4	1	no	—
100	C₄⋊Dic₄₃	—	4	2	no	—
101	C₄⋊Dic₄₃	—	4	1	no	—
102	C₄⋊Dic₄₃	—	4	1	no	—
103	Dic₄₃⋊C₄	—	4	2	no	—
104	Dic₄₃⋊C₄	—	4	2	no	—
105	Dic₄₃⋊C₄	—	4	1	no	—
106	C₄×Dic₄₃	—	4	2	no	—
107	C₄×Dic₄₃	—	4	2	no	—
108	C₄×Dic₄₃	—	4	1	no	—
109	Dic₁₇₂	—	2	2	no	—
110	Dic₁₇₂	—	2	1	no	—
111	Dic₁₇₂	—	2	2	no	—
112	Dic₁₇₂	—	8	1	no	—
113	Dic₁₇₂	#41	1	2	yes	—
114	D₃₄₄	—	2	1	no	—
115	D₃₄₄	—	2	1	no	—
116	#10621873	—	8	2	no	—
117	C₁₆×C₄₃	—	2	1	no	—
118	C₁₆×C₄₃	—	2	2	no	—
119	C₁₆×C₄₃	—	2	1	no	—
120	C₂²×D₄₃	—	4	1	no	—
121	Q₈×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₄₃	—	4	1	no	—
128	C₂×Dic₄₃	—	4	1	no	—
129	C₂×Dic₄₃	—	4	2	no	—
130	C₂×Dic₄₃	—	4	1	no	—
131	C₂×Dic₄₃	—	4	2	no	—
132	D₁₇₂	—	4	1	no	—
133	D₁₇₂	—	4	1	no	—
134	C₄×D₄₃	—	8	2	no	—
135	Dic₈₆	—	2	2	no	—
136	Dic₈₆	—	4	1	no	—
137	Dic₈₆	—	4	2	no	—
138	Dic₈₆	—	4	2	no	—
139	Dic₈₆	—	16	2	no	—
140	Dic₈₆	—	16	1	no	—
141	C₈×C₄₃	—	2	2	no	—
142	C₈×C₄₃	—	2	1	no	—
143	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
145	#33936	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₄₃	—	8	1	no	—
148	C₂×D₄₃	—	8	1	no	—
149	C₂×D₄₃	—	8	1	no	—
150	C₄×C₄₃	—	2	2	no	—
151	C₄×C₄₃	—	2	2	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	#510	1	2	yes	—
154	C₄×C₄₃	#548	1	1	yes	characteristic
155	C₄×C₄₃	#577	1	1	yes	characteristic
156	Dic₄₃	—	4	2	no	—
157	Dic₄₃	—	8	2	no	—
158	Dic₄₃	—	8	1	no	—
159	Dic₄₃	—	8	1	no	—
160	Dic₄₃	—	8	2	no	—
161	#3393	—	43	1	no	—
162	Q₈×Q₁₆	—	43	1	no	—
163	#2243	—	43	2	no	—
164	#2241	—	43	1	no	—
165	#2226	—	43	1	no	—
166	#2225	—	43	2	no	—
167	#2211	—	43	2	no	—
168	#2209	—	43	1	no	—
169	#2183	D₄₃	1	1	yes	characteristic, complemented
170	#2176	—	43	2	no	—
171	C₄×SD₃₂	—	43	1	no	—
172	C₂×C₄₃	#3358	1	1	yes	characteristic
173	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
174	C₂×C₄₃	#3416	1	1	yes	characteristic
175	D₄₃	—	16	1	no	—
176	Q₈²	—	86	1	no	—
177	#556	—	86	1	no	—
178	C₂×Q₃₂	—	43	2	no	—
179	C₂×SD₃₂	—	43	1	no	—
180	#500	—	43	2	no	—
181	#500	—	43	1	no	—
182	#495	—	43	2	no	—
183	#494	—	43	1	no	—
184	#477	—	86	1	no	—
185	#477	—	86	2	no	—
186	#464	—	86	2	no	—
187	#461	—	86	1	no	—
188	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
189	C₄×Q₁₆	—	43	1	no	—
190	C₄×Q₁₆	—	43	2	no	—
191	C₄×Q₁₆	—	43	2	no	—
192	C₄×D₈	—	43	1	no	—
193	#367	—	43	1	no	—
194	#367	—	43	2	no	—
195	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
196	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
197	#358	—	43	1	no	—
198	#358	—	43	2	no	—
199	#358	—	43	2	no	—
200	#358	—	43	2	no	—
201	#357	—	43	1	no	—
202	#357	—	43	2	no	—
203	#346	C₂×D₄₃	1	2	yes	—
204	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
205	C₄₃	#33936	1	1	yes	characteristic, complemented, Sylow
206	C₂×Q₁₆	—	43	1	no	—
207	C₂×Q₁₆	—	43	2	no	—
208	C₂×Q₁₆	—	86	1	no	—
209	C₂×SD₁₆	—	86	2	no	—
210	C₂×D₈	—	43	1	no	—
211	C₄⋊Q₈	—	86	1	no	—
212	C₄⋊Q₈	—	86	2	no	—
213	C₄⋊Q₈	—	86	1	no	—
214	C₄⋊Q₈	—	172	1	no	—
215	C₄⋊Q₈	—	172	2	no	—
216	#126	—	172	2	no	—
217	#124	—	86	2	no	—
218	C₂²⋊Q₈	—	172	1	no	—
219	C₄×Q₈	—	86	2	no	—
220	C₄×Q₈	—	86	1	no	—
221	C₄×Q₈	—	86	2	no	—
222	C₄×Q₈	—	172	1	no	—
223	C₄×Q₈	D₁₇₂	1	1	yes	characteristic
224	C₄×D₄	—	86	1	no	—
225	#117	—	172	1	no	—
226	Q₃₂	—	86	4	no	—
227	SD₃₂	—	86	2	no	—
228	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
230	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
231	#107	—	43	1	no	—
232	#107	—	43	2	no	—
233	#107	—	43	1	no	—
234	#107	—	86	1	no	—
235	#107	—	86	2	no	—
236	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
237	C₄⋊C₈	D₁₇₂	1	2	yes	—
238	#103	—	86	2	no	—
239	#103	—	86	1	no	—
240	#103	—	86	2	no	—
241	#102	—	86	1	no	—
242	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₄₃	1	2	yes	—
244	C₂×Q₈	—	86	1	no	—
245	C₂×Q₈	—	86	2	no	—
246	C₂×Q₈	—	172	2	no	—
247	C₂×Q₈	—	172	1	no	—
248	C₂×Q₈	D₃₄₄	1	1	yes	characteristic
249	C₂×D₄	—	86	1	no	—
250	C₂²×C₄	—	172	1	no	—
251	Q₁₆	—	43	2	no	—
252	Q₁₆	—	86	2	no	—
253	Q₁₆	—	86	1	no	—
254	Q₁₆	—	86	2	no	—
255	Q₁₆	—	344	1	no	—
256	SD₁₆	—	344	2	no	—
257	D₈	—	86	1	no	—
258	D₈	—	86	1	no	—
259	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
260	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
261	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
262	C₄⋊C₄	—	86	1	no	—
263	C₄⋊C₄	—	86	1	no	—
264	C₄⋊C₄	—	86	2	no	—
265	C₄⋊C₄	—	172	1	no	—
266	C₄⋊C₄	—	172	2	no	—
267	C₄⋊C₄	—	172	2	no	—
268	C₄⋊C₄	—	172	1	no	—
269	C₄⋊C₄	—	172	2	no	—
270	C₄⋊C₄	—	172	1	no	—
271	C₄⋊C₄	—	172	1	no	—
272	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
273	C₄⋊C₄	D₃₄₄	1	2	yes	—
274	C₂²⋊C₄	—	172	2	no	—
275	C₂²⋊C₄	—	172	1	no	—
276	C₂²⋊C₄	—	172	2	no	—
277	C₄²	—	172	2	no	—
278	C₄²	—	172	2	no	—
279	C₄²	—	172	1	no	—
280	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
281	C₄²	C₂×D₁₇₂	1	2	yes	—
282	C₁₆	—	2	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₂³	—	172	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	86	2	no	—
288	Q₈	—	172	2	no	—
289	Q₈	—	172	2	no	—
290	Q₈	—	172	1	no	—
291	Q₈	—	688	2	no	—
292	Q₈	—	688	1	no	—
293	D₄	—	172	1	no	—
294	D₄	—	172	1	no	—
295	C₂×C₄	—	344	2	no	—
296	C₂×C₄	—	172	1	no	—
297	C₂×C₄	—	172	2	no	—
298	C₂×C₄	—	172	2	no	—
299	C₂×C₄	—	172	1	no	—
300	C₂×C₄	—	172	1	no	—
301	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
302	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
303	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
304	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	#41×D₄₃	1	1	yes	characteristic
309	C₈	#13063373	1	1	yes	characteristic
310	C₂²	—	344	1	no	—
311	C₂²	—	344	1	no	—
312	C₂²	—	344	1	no	—
313	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	344	1	no	—
318	C₄	—	344	2	no	—
319	C₄	—	344	2	no	—
320	C₄	—	344	1	no	—
321	C₄	—	172	2	no	—
322	C₄	#14583342	1	2	yes	—
323	C₄	#14583561	1	1	yes	characteristic
324	C₄	#14583711	1	1	yes	characteristic
325	C₂	—	688	1	no	—
326	C₂	#19147231	1	1	yes	characteristic
327	C₂	C₂×#14583342	1	1	yes	characteristic
328	C₂	#19147743	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2183	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×D₃₄₄	C₂³
hypercenter	#2183	D₄₃
hypocenter	C₄₃	#33936
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₁₇₂	C₂	semidirect
4. #19136823	C₂	semidirect
6. #19136738	C₂	semidirect
8. #19136646	C₂	semidirect
9. #19136467	C₂	semidirect
11. C₄₃×#2183	C₂	semidirect
169. #2183	D₄₃	semidirect
205. C₄₃	#33936	semidirect

Maximal subgroups

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

subgroup	index
#19147584	2
Q₈×Dic₁₇₂	2
#19136832	2
#19136823	2
#19136743	2
#19136738	2
#19136658	2
#19136646	2
#19136467	2
C₄×#13063270	2
C₄₃×#2183	2
#33936	43