Subgroups — #46845272

Order 34688.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774694	C₂	1	1	yes	maximal, characteristic
2	#31774662	C₂	1	2	yes	maximal, complemented
3	#31774020	C₂	1	1	yes	maximal, characteristic, complemented
4	#31774016	C₂	1	2	yes	maximal, complemented
5	#31774014	C₂	1	1	yes	maximal, characteristic, complemented
6	#31774013	C₂	1	1	yes	maximal, characteristic, complemented
7	#31774012	C₂	1	2	yes	maximal, complemented
8	#31773944	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₂₇₁	C₂	1	2	yes	maximal, complemented
10	#31773849	C₂	1	1	yes	maximal, characteristic, complemented
11	#444×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₂₇₁	C₂²	1	2	yes	—
13	#22576192	C₄	1	2	yes	complemented
14	#22576189	C₄	1	2	yes	complemented
15	#22576187	C₄	1	2	yes	complemented
16	#22576184	C₄	1	2	yes	complemented
17	#22576182	C₂²	1	1	yes	characteristic
18	#22576180	C₂²	1	2	yes	—
19	#22576168	—	2	2	no	—
20	#22576168	C₂²	1	2	yes	complemented
21	#22576167	C₂²	1	2	yes	complemented
22	#22576166	—	2	2	no	—
23	#22576166	C₂²	1	2	yes	complemented
24	C₂×#16864637	—	2	2	no	—
25	C₂×#16864637	C₂²	1	1	yes	characteristic
26	C₂×#16864637	C₂²	1	1	yes	characteristic
27	M₁₆×D₂₇₁	—	2	2	no	—
28	M₁₆×D₂₇₁	C₂²	1	2	yes	—
29	C₄²×D₂₇₁	C₂²	1	1	yes	characteristic
30	#22576096	C₂²	1	1	yes	characteristic, complemented
31	#22576095	C₂²	1	2	yes	complemented
32	#22576088	C₂²	1	2	yes	complemented
33	#22576087	C₂²	1	2	yes	complemented
34	#22576086	C₂²	1	2	yes	complemented
35	#22576071	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic, complemented
38	#135×C₂₇₁	C₂²	1	1	yes	characteristic
39	#131×C₂₇₁	C₂²	1	2	yes	complemented
40	#108×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
41	#104×C₂₇₁	C₂²	1	2	yes	complemented
42	#97×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
43	#16864655	C₂×C₄	1	2	yes	—
44	#16864654	—	2	2	no	—
45	#16864654	C₂³	1	2	yes	—
46	#16864653	—	2	2	no	—
47	#16864653	C₂×C₄	1	2	yes	—
48	Q₈×D₂₇₁	C₂×C₄	1	2	yes	—
49	D₄×D₂₇₁	—	2	2	no	—
50	D₄×D₂₇₁	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₂₇₁	—	2	2	no	—
52	C₂×C₄×D₂₇₁	—	2	1	no	—
53	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
54	Dic₁₀₈₄	C₂×C₄	1	1	yes	characteristic, complemented
55	#16864644	C₂×C₄	1	2	yes	complemented
56	#16864643	C₂×C₄	1	2	yes	complemented
57	#16864642	C₂×C₄	1	2	yes	complemented
58	#16864641	C₂×C₄	1	2	yes	complemented
59	D₂₁₆₈	C₂×C₄	1	1	yes	characteristic, complemented
60	#16864639	C₂×C₄	1	2	yes	complemented
61	#16864638	—	2	2	no	—
62	#16864638	C₂³	1	2	yes	—
63	#16864637	—	2	2	no	—
64	#16864637	—	2	2	no	—
65	#16864637	—	2	2	no	—
66	#16864637	C₂×C₄	1	2	yes	—
67	#16864637	C₂×C₄	1	2	yes	—
68	#16864637	D₄	1	2	yes	—
69	#16864637	D₄	1	2	yes	—
70	C₂×C₂₇₁⋊C₈	—	2	2	no	—
71	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×D₂₇₁	—	2	2	no	—
74	C₈×D₂₇₁	—	2	2	no	—
75	C₈×D₂₇₁	—	2	2	no	—
76	C₄×Dic₂₇₁	—	2	1	no	—
77	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
78	#41×C₂₇₁	C₂³	1	2	yes	—
79	Q₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic, complemented
80	SD₁₆×C₂₇₁	C₂×C₄	1	2	yes	complemented
81	D₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic, complemented
82	M₁₆×C₂₇₁	—	2	2	no	—
83	M₁₆×C₂₇₁	C₂³	1	2	yes	—
84	C₂×C₈×C₂₇₁	—	2	2	no	—
85	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
87	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
88	C₂²×D₂₇₁	—	2	2	no	—
89	C₂²×D₂₇₁	#41	1	1	yes	characteristic
90	Dic₅₄₂	—	2	2	no	—
91	Dic₅₄₂	C₂²×C₄	1	2	yes	—
92	D₁₀₈₄	—	2	2	no	—
93	D₁₀₈₄	C₂²×C₄	1	2	yes	—
94	C₂²⋊D₂₇₁	—	2	2	no	—
95	C₂²⋊D₂₇₁	—	2	2	no	—
96	C₂²⋊D₂₇₁	—	2	2	no	—
97	C₂×Dic₂₇₁	—	2	1	no	—
98	C₂×Dic₂₇₁	—	2	2	no	—
99	C₂×Dic₂₇₁	#41	1	1	yes	characteristic
100	C₄×D₂₇₁	—	2	2	no	—
101	C₄×D₂₇₁	—	2	2	no	—
102	C₄×D₂₇₁	—	2	2	no	—
103	C₄×D₂₇₁	—	4	2	no	—
104	C₄×D₂₇₁	C₂²×C₄	1	2	yes	—
105	C₄×D₂₇₁	C₂×D₄	1	2	yes	—
106	C₂₇₁⋊C₈	—	2	2	no	—
107	C₂₇₁⋊C₈	C₂²×C₄	1	2	yes	—
108	C₂₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₂₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₂₇₁	C₂²×C₄	1	2	yes	—
111	D₄×C₂₇₁	—	2	2	no	—
112	D₄×C₂₇₁	C₂²×C₄	1	2	yes	—
113	C₂×C₄×C₂₇₁	—	2	2	no	—
114	C₂×C₄×C₂₇₁	—	2	1	no	—
115	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
116	C₈×C₂₇₁	—	2	2	no	—
117	C₈×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
119	C₈×C₂₇₁	C₂×D₄	1	2	yes	—
120	C₂²×C₂₇₁	—	2	2	no	—
121	C₂²×C₂₇₁	C₂×#41	1	1	yes	characteristic
122	C₂×D₂₇₁	—	2	2	no	—
123	C₂×D₂₇₁	—	4	1	no	—
124	C₂×D₂₇₁	—	8	2	no	—
125	C₂×D₂₇₁	C₄×D₄	1	2	yes	—
126	C₄×C₂₇₁	—	2	2	no	—
127	C₄×C₂₇₁	—	4	1	no	—
128	C₄×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
130	Dic₂₇₁	—	2	2	no	—
131	Dic₂₇₁	—	2	2	no	—
132	Dic₂₇₁	C₄×D₄	1	2	yes	—
133	C₂×C₂₇₁	—	2	1	no	—
134	C₂×C₂₇₁	—	4	2	no	—
135	C₂×C₂₇₁	C₂×C₄×D₄	1	1	yes	characteristic
136	D₂₇₁	—	2	2	no	—
137	D₂₇₁	—	4	2	no	—
138	C₂₇₁	#2956	1	1	yes	characteristic, complemented, Sylow
139	#2956	—	271	1	no	maximal, Sylow
140	#575	—	271	1	no	—
141	#568	—	271	2	no	—
142	#444	—	271	2	no	—
143	#444	—	271	1	no	—
144	#444	D₂₇₁	1	1	yes	characteristic, complemented
145	#443	—	271	1	no	—
146	#443	—	271	2	no	—
147	#443	—	271	1	no	—
148	#430	—	271	1	no	—
149	C₂×#104	—	271	2	no	—
150	C₄×M₁₆	—	271	1	no	—
151	C₂×#41	—	271	2	no	—
152	#137	—	271	2	no	—
153	#137	—	271	2	no	—
154	AGL(1,ℤ/8)	—	271	2	no	—
155	AGL(1,ℤ/8)	—	271	2	no	—
156	#135	—	271	2	no	—
157	#135	—	271	1	no	—
158	#135	C₂×D₂₇₁	1	1	yes	characteristic
159	#131	—	271	2	no	—
160	#131	—	271	2	no	—
161	#131	—	271	2	no	—
162	#131	—	542	2	no	—
163	#131	—	542	2	no	—
164	#131	C₂×D₂₇₁	1	2	yes	complemented
165	C₂×M₁₆	—	271	1	no	—
166	C₂×M₁₆	—	271	1	no	—
167	C₂×M₁₆	—	271	2	no	—
168	C₂×M₁₆	—	542	2	no	—
169	C₂×M₁₆	—	542	2	no	—
170	C₂×C₄²	—	271	1	no	—
171	#108	—	271	2	no	—
172	#108	—	271	1	no	—
173	#108	C₂×D₂₇₁	1	1	yes	characteristic, complemented
174	#104	—	271	2	no	—
175	#104	—	271	2	no	—
176	#104	—	271	2	no	—
177	#104	C₂×D₂₇₁	1	2	yes	complemented
178	#97	—	271	1	no	—
179	#97	C₂×D₂₇₁	1	1	yes	characteristic, complemented
180	C₄×C₈	—	271	1	no	—
181	C₄×C₈	—	271	1	no	—
182	#41	—	271	2	no	—
183	#41	—	271	2	no	—
184	#41	—	271	2	no	—
185	#41	—	542	2	no	—
186	#41	—	542	2	no	—
187	#41	C₂²×D₂₇₁	1	2	yes	—
188	C₂×Q₈	—	271	2	no	—
189	C₂×D₄	—	271	2	no	—
190	C₂×D₄	—	542	2	no	—
191	C₂²×C₄	—	271	1	no	—
192	C₂²×C₄	—	542	2	no	—
193	C₂²×C₄	—	542	1	no	—
194	Q₁₆	—	271	2	no	—
195	Q₁₆	—	271	1	no	—
196	Q₁₆	C₄×D₂₇₁	1	1	yes	characteristic, complemented
197	SD₁₆	—	271	2	no	—
198	SD₁₆	—	271	2	no	—
199	SD₁₆	—	271	2	no	—
200	SD₁₆	C₄×D₂₇₁	1	2	yes	complemented
201	D₈	—	271	2	no	—
202	D₈	—	271	1	no	—
203	D₈	C₄×D₂₇₁	1	1	yes	characteristic, complemented
204	M₁₆	—	2	2	no	—
205	M₁₆	—	271	2	no	—
206	M₁₆	—	271	2	no	—
207	M₁₆	—	271	2	no	—
208	M₁₆	—	271	2	no	—
209	M₁₆	—	271	2	no	—
210	M₁₆	—	542	2	no	—
211	M₁₆	—	542	2	no	—
212	M₁₆	—	542	2	no	—
213	M₁₆	—	542	2	no	—
214	M₁₆	C₂²×D₂₇₁	1	2	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	271	1	no	—
217	C₂×C₈	—	271	1	no	—
218	C₂×C₈	—	542	2	no	—
219	C₂×C₈	—	542	2	no	—
220	C₂×C₈	—	542	2	no	—
221	C₂×C₈	—	542	2	no	—
222	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
224	C₄²	—	271	1	no	—
225	C₄²	—	542	1	no	—
226	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
227	C₂³	—	271	1	no	—
228	C₂³	—	542	2	no	—
229	Q₈	—	271	2	no	—
230	Q₈	—	542	2	no	—
231	Q₈	C₂×C₄×D₂₇₁	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	271	2	no	—
234	D₄	—	542	2	no	—
235	D₄	—	542	2	no	—
236	D₄	—	542	2	no	—
237	D₄	—	542	2	no	—
238	D₄	C₂×C₄×D₂₇₁	1	2	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	271	1	no	—
242	C₂×C₄	—	271	2	no	—
243	C₂×C₄	—	271	2	no	—
244	C₂×C₄	—	542	2	no	—
245	C₂×C₄	—	542	2	no	—
246	C₂×C₄	—	542	2	no	—
247	C₂×C₄	—	542	1	no	—
248	C₂×C₄	—	542	2	no	—
249	C₂×C₄	—	1084	2	no	—
250	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	271	2	no	—
253	C₈	—	271	1	no	—
254	C₈	—	271	1	no	—
255	C₈	—	542	2	no	—
256	C₈	C₂×C₄×D₂₇₁	1	1	yes	characteristic
257	C₈	C₂×C₄×D₂₇₁	1	1	yes	characteristic
258	C₈	D₄×D₂₇₁	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	271	2	no	—
261	C₂²	—	542	2	no	—
262	C₂²	—	1084	1	no	—
263	C₂²	—	2168	2	no	—
264	C₂²	#41×D₂₇₁	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	271	2	no	—
268	C₄	—	542	2	no	—
269	C₄	—	542	2	no	—
270	C₄	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
271	C₄	C₂×D₄×D₂₇₁	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	542	2	no	—
275	C₂	—	1084	2	no	—
276	C₂	C₄×D₄×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₇₁
commutator	C₄×C₂₇₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	#444×C₂₇₁	C₂
socle	C₂×C₂₇₁	C₂×C₄×D₄
prime radical	#31774694	C₂
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	#444	D₂₇₁
hypocenter	C₂₇₁	#2956
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #31774662	C₂	semidirect
3. #31774020	C₂	semidirect
4. #31774016	C₂	semidirect
5. #31774014	C₂	semidirect
6. #31774013	C₂	semidirect
7. #31774012	C₂	semidirect
8. #31773944	C₂	semidirect
9. #104×D₂₇₁	C₂	semidirect
10. #31773849	C₂	semidirect
11. #444×C₂₇₁	C₂	semidirect
13. #22576192	C₄	semidirect
14. #22576189	C₄	semidirect
15. #22576187	C₄	semidirect
16. #22576184	C₄	semidirect
20. #22576168	C₂²	semidirect
21. #22576167	C₂²	semidirect
23. #22576166	C₂²	semidirect
30. #22576096	C₂²	semidirect
31. #22576095	C₂²	semidirect
32. #22576088	C₂²	semidirect
33. #22576087	C₂²	semidirect
34. #22576086	C₂²	semidirect
35. #22576071	C₂²	semidirect
36. C₈×Dic₂₇₁	C₂²	semidirect
37. C₈×Dic₂₇₁	C₂²	semidirect
39. #131×C₂₇₁	C₂²	semidirect
40. #108×C₂₇₁	C₂²	semidirect
41. #104×C₂₇₁	C₂²	semidirect
42. #97×C₂₇₁	C₂²	semidirect
54. Dic₁₀₈₄	C₂×C₄	semidirect
55. #16864644	C₂×C₄	semidirect
56. #16864643	C₂×C₄	semidirect
57. #16864642	C₂×C₄	semidirect
58. #16864641	C₂×C₄	semidirect
59. D₂₁₆₈	C₂×C₄	semidirect
60. #16864639	C₂×C₄	semidirect
79. Q₁₆×C₂₇₁	C₂×C₄	semidirect
80. SD₁₆×C₂₇₁	C₂×C₄	semidirect
81. D₈×C₂₇₁	C₂×C₄	semidirect
138. C₂₇₁	#2956	semidirect
144. #444	D₂₇₁	semidirect
164. #131	C₂×D₂₇₁	semidirect
173. #108	C₂×D₂₇₁	semidirect
177. #104	C₂×D₂₇₁	semidirect
179. #97	C₂×D₂₇₁	semidirect
196. Q₁₆	C₄×D₂₇₁	semidirect
200. SD₁₆	C₄×D₂₇₁	semidirect
203. D₈	C₄×D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31774694	2
#31774662	2
#31774020	2
#31774016	2
#31774014	2
#31774013	2
#31774012	2
#31773944	2
#104×D₂₇₁	2
#31773849	2
#444×C₂₇₁	2
#2956	271