Subgroups — #46848151

Order 34688.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774714	C₂	1	1	yes	maximal, characteristic, complemented
2	#31774704	C₂	1	2	yes	maximal, complemented
3	#31774702	C₂	1	1	yes	maximal, characteristic, complemented
4	#31774281	C₂	1	1	yes	maximal, characteristic, complemented
5	#31774172	C₂	1	1	yes	maximal, characteristic, complemented
6	#31774167	C₂	1	1	yes	maximal, characteristic, complemented
7	#31774161	C₂	1	2	yes	maximal, complemented
8	#31774076	C₂	1	2	yes	maximal, complemented
9	#31774071	C₂	1	2	yes	maximal, complemented
10	#31774015	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#22576207	C₂²	1	2	yes	complemented
13	#22576206	C₂²	1	2	yes	—
14	#22576193	—	2	2	no	—
15	#22576193	C₂²	1	2	yes	complemented
16	#22576192	—	2	2	no	—
17	#22576190	C₂²	1	2	yes	—
18	#22576188	—	2	2	no	—
19	#22576188	C₂²	1	2	yes	—
20	#22576185	C₂²	1	2	yes	complemented
21	#22576184	—	2	2	no	—
22	#22576182	—	2	1	no	—
23	#22576182	—	2	1	no	—
24	#22576182	C₂²	1	1	yes	characteristic
25	#22576181	—	2	2	no	—
26	C₂×Dic₁₀₈₄	—	2	1	no	—
27	C₂×Dic₁₀₈₄	C₂²	1	1	yes	characteristic, complemented
28	C₂×#16864643	C₂²	1	2	yes	complemented
29	SD₁₆×D₂₇₁	—	2	2	no	—
30	C₂×D₂₁₆₈	—	2	1	no	—
31	C₂×D₂₁₆₈	C₂²	1	1	yes	characteristic
32	#22576166	C₂²	1	2	yes	complemented
33	#22576148	C₂²	1	2	yes	complemented
34	#22576096	C₂²	1	1	yes	characteristic, complemented
35	#22576088	C₂²	1	2	yes	complemented
36	#22576078	C₂²	1	2	yes	complemented
37	#22576076	C₂²	1	2	yes	complemented
38	#135×C₂₇₁	C₂²	1	1	yes	characteristic
39	#131×C₂₇₁	C₂²	1	1	yes	characteristic
40	#131×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
41	#108×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
42	#104×C₂₇₁	C₂²	1	1	yes	characteristic
43	#104×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
45	#16864655	—	2	2	no	—
46	#16864654	—	2	2	no	—
47	#16864654	—	2	2	no	—
48	#16864654	C₂³	1	2	yes	—
49	#16864653	—	2	2	no	—
50	#16864653	—	4	2	no	—
51	C₂×Dic₅₄₂	—	2	2	no	—
52	C₂×Dic₅₄₂	C₂³	1	2	yes	—
53	Q₈×D₂₇₁	—	2	2	no	—
54	C₂×D₁₀₈₄	—	2	2	no	—
55	C₂×D₁₀₈₄	C₂³	1	2	yes	—
56	D₄×D₂₇₁	—	2	2	no	—
57	D₄×D₂₇₁	—	4	2	no	—
58	Dic₁₀₈₄	—	2	2	no	—
59	Dic₁₀₈₄	—	2	1	no	—
60	Dic₁₀₈₄	—	2	1	no	—
61	Dic₁₀₈₄	D₄	1	1	yes	characteristic
62	#16864644	—	2	2	no	—
63	#16864643	—	2	2	no	—
64	#16864643	—	2	2	no	—
65	#16864643	—	2	2	no	—
66	#16864643	—	2	2	no	—
67	#16864643	D₄	1	2	yes	—
68	#16864642	—	2	2	no	—
69	#16864641	—	2	2	no	—
70	D₂₁₆₈	—	2	2	no	—
71	D₂₁₆₈	—	2	1	no	—
72	D₂₁₆₈	—	2	1	no	—
73	D₂₁₆₈	D₄	1	1	yes	characteristic
74	#16864639	—	2	2	no	—
75	#16864638	C₂³	1	2	yes	—
76	#16864637	—	2	2	no	—
77	C₈×D₂₇₁	—	2	2	no	—
78	(C₂×D₂₇₁)⋊C₄	—	4	2	no	—
79	#41×C₂₇₁	C₂³	1	1	yes	characteristic
80	#41×C₂₇₁	C₂³	1	1	yes	characteristic
81	Q₁₆×C₂₇₁	D₄	1	1	yes	characteristic
82	SD₁₆×C₂₇₁	D₄	1	1	yes	characteristic
83	SD₁₆×C₂₇₁	D₄	1	1	yes	characteristic
84	D₈×C₂₇₁	D₄	1	1	yes	characteristic
85	M₁₆×C₂₇₁	—	2	1	no	—
86	M₁₆×C₂₇₁	—	2	1	no	—
87	M₁₆×C₂₇₁	C₂³	1	1	yes	characteristic
88	M₁₆×C₂₇₁	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₂₇₁	—	2	1	no	—
90	C₂×C₈×C₂₇₁	—	2	1	no	—
91	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
93	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
94	C₂²×D₂₇₁	—	2	2	no	—
95	C₂²×D₂₇₁	—	4	2	no	—
96	Dic₅₄₂	—	2	2	no	—
97	Dic₅₄₂	—	2	2	no	—
98	Dic₅₄₂	—	4	2	no	—
99	Dic₅₄₂	C₂×D₄	1	2	yes	—
100	D₁₀₈₄	—	2	2	no	—
101	D₁₀₈₄	—	2	2	no	—
102	D₁₀₈₄	—	4	2	no	—
103	D₁₀₈₄	C₂×D₄	1	2	yes	—
104	C₂²⋊D₂₇₁	—	2	2	no	—
105	C₂²⋊D₂₇₁	—	4	2	no	—
106	C₂²⋊D₂₇₁	—	4	2	no	—
107	C₂×Dic₂₇₁	—	2	2	no	—
108	C₂×Dic₂₇₁	—	4	2	no	—
109	C₄×D₂₇₁	—	2	2	no	—
110	C₄×D₂₇₁	—	4	2	no	—
111	C₄×D₂₇₁	—	4	2	no	—
112	C₂₇₁⋊C₈	—	2	2	no	—
113	Q₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
115	D₄×C₂₇₁	—	2	1	no	—
116	D₄×C₂₇₁	—	2	1	no	—
117	D₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
118	D₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₇₁	—	2	1	no	—
120	C₂×C₄×C₂₇₁	—	2	1	no	—
121	C₂×C₄×C₂₇₁	—	2	1	no	—
122	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
123	C₈×C₂₇₁	—	2	1	no	—
124	C₈×C₂₇₁	—	2	1	no	—
125	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
126	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
127	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
128	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₂₇₁	—	2	1	no	—
130	C₂²×C₂₇₁	—	2	1	no	—
131	C₂²×C₂₇₁	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×D₂₇₁	—	2	2	no	—
133	C₂×D₂₇₁	—	4	2	no	—
134	C₂×D₂₇₁	—	8	2	no	—
135	C₂×D₂₇₁	—	16	2	no	—
136	C₄×C₂₇₁	—	2	1	no	—
137	C₄×C₂₇₁	—	2	1	no	—
138	C₄×C₂₇₁	—	4	1	no	—
139	C₄×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
141	Dic₂₇₁	—	2	2	no	—
142	Dic₂₇₁	—	4	2	no	—
143	C₂×C₂₇₁	—	2	1	no	—
144	C₂×C₂₇₁	—	4	1	no	—
145	C₂×C₂₇₁	—	4	1	no	—
146	C₂×C₂₇₁	D₄²	1	1	yes	characteristic
147	D₂₇₁	—	4	2	no	—
148	D₂₇₁	—	8	2	no	—
149	C₂₇₁	#3290	1	1	yes	characteristic, complemented, Sylow
150	#3290	—	271	1	no	maximal, Sylow
151	#578	—	271	1	no	—
152	#577	—	271	2	no	—
153	#576	—	271	1	no	—
154	#495	—	271	1	no	—
155	#473	—	271	1	no	—
156	#472	—	271	1	no	—
157	#471	—	271	2	no	—
158	#455	—	271	2	no	—
159	#454	—	271	2	no	—
160	#443	—	271	1	no	—
161	#443	D₂₇₁	1	1	yes	characteristic, complemented
162	2₋¹⁺⁴	—	271	2	no	—
163	H(5,𝔽₂)	—	271	2	no	—
164	#137	—	271	2	no	—
165	#137	—	271	2	no	—
166	#137	—	542	2	no	—
167	#137	—	542	2	no	—
168	AGL(1,ℤ/8)	—	271	2	no	—
169	AGL(1,ℤ/8)	—	271	2	no	—
170	AGL(1,ℤ/8)	—	542	2	no	—
171	AGL(1,ℤ/8)	—	542	2	no	—
172	#135	—	271	1	no	—
173	#135	—	542	1	no	—
174	#135	—	542	1	no	—
175	#135	—	542	2	no	—
176	#135	C₂×D₂₇₁	1	1	yes	characteristic
177	C₂×Q₁₆	—	271	1	no	—
178	C₂×Q₁₆	—	542	1	no	—
179	C₂×SD₁₆	—	271	2	no	—
180	C₂×SD₁₆	—	542	2	no	—
181	C₂×D₈	—	271	1	no	—
182	C₂×D₈	—	542	1	no	—
183	#131	—	271	2	no	—
184	#131	C₂×D₂₇₁	1	1	yes	characteristic
185	#131	C₂×D₂₇₁	1	1	yes	characteristic, complemented
186	#124	—	271	2	no	—
187	#108	—	271	1	no	—
188	#108	C₂×D₂₇₁	1	1	yes	characteristic, complemented
189	#104	—	271	2	no	—
190	#104	C₂×D₂₇₁	1	1	yes	characteristic
191	#104	C₂×D₂₇₁	1	1	yes	characteristic, complemented
192	#101	—	271	2	no	—
193	#100	—	271	2	no	—
194	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic, complemented
195	#41	—	271	2	no	—
196	#41	—	542	2	no	—
197	#41	—	542	2	no	—
198	#41	—	542	2	no	—
199	#41	—	542	2	no	—
200	#41	—	1084	2	no	—
201	#41	C₂²×D₂₇₁	1	1	yes	characteristic
202	#41	C₂²×D₂₇₁	1	1	yes	characteristic
203	C₂×Q₈	—	271	2	no	—
204	C₂×Q₈	—	542	2	no	—
205	C₂×Q₈	—	542	2	no	—
206	C₂×D₄	—	271	2	no	—
207	C₂×D₄	—	542	2	no	—
208	C₂×D₄	—	542	2	no	—
209	C₂×D₄	—	1084	2	no	—
210	Q₁₆	—	271	1	no	—
211	Q₁₆	—	542	2	no	—
212	Q₁₆	—	542	2	no	—
213	Q₁₆	—	542	1	no	—
214	Q₁₆	—	542	1	no	—
215	Q₁₆	D₁₀₈₄	1	1	yes	characteristic
216	SD₁₆	—	271	2	no	—
217	SD₁₆	—	542	2	no	—
218	SD₁₆	—	542	2	no	—
219	SD₁₆	—	542	2	no	—
220	SD₁₆	—	542	2	no	—
221	SD₁₆	—	542	2	no	—
222	SD₁₆	—	542	2	no	—
223	SD₁₆	D₁₀₈₄	1	1	yes	characteristic
224	SD₁₆	D₁₀₈₄	1	1	yes	characteristic
225	D₈	—	271	1	no	—
226	D₈	—	542	2	no	—
227	D₈	—	542	2	no	—
228	D₈	—	542	1	no	—
229	D₈	—	542	1	no	—
230	D₈	D₁₀₈₄	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	271	2	no	—
234	M₁₆	—	542	2	no	—
235	M₁₆	C₂²×D₂₇₁	1	1	yes	characteristic
236	M₁₆	C₂²×D₂₇₁	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	542	2	no	—
240	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
242	C₂²⋊C₄	—	1084	2	no	—
243	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
244	C₂³	—	542	2	no	—
245	C₂³	—	1084	2	no	—
246	Q₈	—	271	2	no	—
247	Q₈	—	542	2	no	—
248	Q₈	—	542	2	no	—
249	Q₈	—	1084	2	no	—
250	Q₈	C₂×D₁₀₈₄	1	1	yes	characteristic
251	Q₈	C₂×D₁₀₈₄	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	271	2	no	—
255	D₄	—	542	2	no	—
256	D₄	—	542	2	no	—
257	D₄	—	542	2	no	—
258	D₄	—	1084	2	no	—
259	D₄	—	1084	2	no	—
260	D₄	—	1084	2	no	—
261	D₄	C₂×D₁₀₈₄	1	1	yes	characteristic
262	D₄	C₂×D₁₀₈₄	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	542	2	no	—
267	C₂×C₄	—	542	2	no	—
268	C₂×C₄	—	1084	2	no	—
269	C₂×C₄	—	1084	2	no	—
270	C₂×C₄	—	1084	2	no	—
271	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	542	2	no	—
275	C₈	D₄×D₂₇₁	1	1	yes	characteristic
276	C₈	D₄×D₂₇₁	1	1	yes	characteristic
277	C₈	C₂×D₁₀₈₄	1	1	yes	characteristic
278	C₈	C₂×D₁₀₈₄	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	542	2	no	—
282	C₂²	—	1084	2	no	—
283	C₂²	—	2168	2	no	—
284	C₂²	—	4336	2	no	—
285	C₂²	#22576206	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	542	2	no	—
290	C₄	—	1084	2	no	—
291	C₄	C₂×D₄×D₂₇₁	1	1	yes	characteristic
292	C₄	C₂²×D₁₀₈₄	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	1084	2	no	—
297	C₂	—	2168	2	no	—
298	C₂	D₄×D₁₀₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₀₈₄
commutator, cosocle	C₂×C₄×C₂₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	#443×C₂₇₁	C₂
socle	C₂×C₂₇₁	D₄²
hypercenter	#443	D₂₇₁
hypocenter	C₂₇₁	#3290
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #31774714	C₂	semidirect
2. #31774704	C₂	semidirect
3. #31774702	C₂	semidirect
4. #31774281	C₂	semidirect
5. #31774172	C₂	semidirect
6. #31774167	C₂	semidirect
7. #31774161	C₂	semidirect
8. #31774076	C₂	semidirect
9. #31774071	C₂	semidirect
10. #31774015	C₂	semidirect
11. #443×C₂₇₁	C₂	semidirect
12. #22576207	C₂²	semidirect
15. #22576193	C₂²	semidirect
20. #22576185	C₂²	semidirect
27. C₂×Dic₁₀₈₄	C₂²	semidirect
28. C₂×#16864643	C₂²	semidirect
32. #22576166	C₂²	semidirect
33. #22576148	C₂²	semidirect
34. #22576096	C₂²	semidirect
35. #22576088	C₂²	semidirect
36. #22576078	C₂²	semidirect
37. #22576076	C₂²	semidirect
40. #131×C₂₇₁	C₂²	semidirect
41. #108×C₂₇₁	C₂²	semidirect
43. #104×C₂₇₁	C₂²	semidirect
44. C₄×C₈×C₂₇₁	C₂²	semidirect
149. C₂₇₁	#3290	semidirect
161. #443	D₂₇₁	semidirect
185. #131	C₂×D₂₇₁	semidirect
188. #108	C₂×D₂₇₁	semidirect
191. #104	C₂×D₂₇₁	semidirect
194. C₄×C₈	C₂×D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31774714	2
#31774704	2
#31774702	2
#31774281	2
#31774172	2
#31774167	2
#31774161	2
#31774076	2
#31774071	2
#31774015	2
#443×C₂₇₁	2
#3290	271