Subgroups — #44408438

Order 29056.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300295	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#16813868	C₂	1	1	yes	maximal, characteristic, complemented
3	#28409110	C₂	1	4	yes	maximal, complemented
4	#28409104	C₂	1	4	yes	maximal, complemented
5	#28409059	C₂	1	1	yes	maximal, characteristic, complemented
6	#28408999	C₂	1	1	yes	maximal, characteristic, complemented
7	#28408996	C₂	1	1	yes	maximal, characteristic, complemented
8	#28408988	C₂	1	1	yes	maximal, characteristic, complemented
9	#525×C₂₂₇	C₂	1	1	yes	maximal, characteristic
10	#20300295	—	2	4	no	—
11	C₂×#16813868	—	2	4	no	—
12	C₂×#16813868	C₂²	1	2	yes	—
13	C₂×#16813868	C₂²	1	2	yes	—
14	C₂×#16813868	C₂²	1	1	yes	characteristic
15	C₂×#16813868	C₂²	1	1	yes	characteristic
16	C₂×#16813868	C₂²	1	2	yes	complemented
17	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₂₂₇⋊C₈	C₂²	1	2	yes	complemented
20	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	4	yes	complemented
21	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	4	yes	—
22	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	4	yes	complemented
23	#20300199	C₂²	1	2	yes	complemented
24	C₄×C₂₂₇⋊C₈	C₂²	1	2	yes	complemented
25	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
27	#123×C₂₂₇	C₄	1	2	yes	—
28	C₂²⋊Q₈×C₂₂₇	C₄	1	2	yes	—
29	#121×C₂₂₇	C₄	1	2	yes	—
30	#120×C₂₂₇	C₄	1	2	yes	—
31	C₄×D₄×C₂₂₇	C₂²	1	4	yes	—
32	#117×C₂₂₇	C₂²	1	1	yes	characteristic
33	#16813868	—	2	4	no	—
34	#16813868	—	2	4	no	—
35	#16813868	—	2	4	no	—
36	#16813868	—	2	2	no	—
37	#16813868	—	2	2	no	—
38	C₂×C₂₂₇⋊C₈	—	2	4	no	—
39	C₂×C₂₂₇⋊C₈	—	2	2	no	—
40	C₂×C₂₂₇⋊C₈	D₄	1	4	yes	complemented
41	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
42	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
43	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
44	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂⁴×C₂₂₇	C₂×C₄	1	1	yes	characteristic
47	#41×C₂₂₇	—	2	2	no	—
48	#41×C₂₂₇	—	2	2	no	—
49	C₂×Q₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₂₂₇	C₂×C₄	1	2	yes	—
51	C₂×D₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
52	C₂×D₄×C₂₂₇	C₂×C₄	1	4	yes	—
53	C₂²×C₄×C₂₂₇	—	2	2	no	—
54	C₂²×C₄×C₂₂₇	—	2	2	no	—
55	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
56	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	—
62	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	—
64	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	—
65	C₄²×C₂₂₇	C₂³	1	2	yes	—
66	C₂₂₇⋊C₈	—	2	2	no	—
67	C₂₂₇⋊C₈	—	2	2	no	—
68	C₂₂₇⋊C₈	—	2	2	no	—
69	C₂₂₇⋊C₈	C₂×D₄	1	2	yes	—
70	C₂₂₇⋊C₈	C₂×D₄	1	2	yes	—
71	C₂³×C₂₂₇	—	2	2	no	—
72	C₂³×C₂₂₇	—	4	2	no	—
73	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
74	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
75	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₂₂₇	—	2	2	no	—
77	D₄×C₂₂₇	—	2	4	no	—
78	D₄×C₂₂₇	—	2	2	no	—
79	D₄×C₂₂₇	—	4	4	no	—
80	C₂×C₄×C₂₂₇	—	2	2	no	—
81	C₂×C₄×C₂₂₇	—	2	4	no	—
82	C₂×C₄×C₂₂₇	—	2	2	no	—
83	C₂×C₄×C₂₂₇	—	2	2	no	—
84	C₂×C₄×C₂₂₇	—	2	2	no	—
85	C₂×C₄×C₂₂₇	—	2	2	no	—
86	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
87	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₂₂₇	#41	1	2	yes	—
92	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
93	C₂²×C₂₂₇	—	2	2	no	—
94	C₂²×C₂₂₇	—	2	2	no	—
95	C₂²×C₂₂₇	—	2	2	no	—
96	C₂²×C₂₂₇	—	2	2	no	—
97	C₂²×C₂₂₇	—	2	4	no	—
98	C₂²×C₂₂₇	—	2	2	no	—
99	C₂²×C₂₂₇	—	4	4	no	—
100	C₂²×C₂₂₇	C₄×D₄	1	2	yes	—
101	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₂₂₇	—	2	2	no	—
103	C₄×C₂₂₇	—	2	2	no	—
104	C₄×C₂₂₇	—	2	4	no	—
105	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
106	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₂₂₇	—	2	2	no	—
109	C₂×C₂₂₇	—	4	2	no	—
110	C₂×C₂₂₇	—	4	2	no	—
111	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₂₂₇	#568	1	2	yes	—
113	C₂₂₇	#2930	1	1	yes	characteristic, complemented, Sylow
114	#2930	—	227	1	no	maximal, Sylow
115	C₂×#131	—	227	1	no	—
116	C₂²×M₁₆	—	227	1	no	—
117	#525	D₂₂₇	1	1	yes	characteristic
118	#436	—	227	4	no	—
119	#435	—	227	4	no	—
120	#424	—	227	1	no	—
121	#408	—	227	1	no	—
122	#407	—	227	1	no	—
123	#405	—	227	1	no	—
124	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
125	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
126	#131	—	454	4	no	—
127	C₂×M₁₆	—	454	4	no	—
128	C₂×M₁₆	—	227	2	no	—
129	C₂×M₁₆	—	227	2	no	—
130	C₂×M₁₆	—	227	1	no	—
131	C₂×M₁₆	—	227	1	no	—
132	C₂×M₁₆	—	227	2	no	—
133	C₂²×C₈	—	227	1	no	—
134	C₂²×C₈	—	227	2	no	—
135	C₂²×C₈	—	227	1	no	—
136	#123	Dic₂₂₇	1	2	yes	—
137	C₂²⋊Q₈	Dic₂₂₇	1	2	yes	—
138	#121	Dic₂₂₇	1	2	yes	—
139	#120	Dic₂₂₇	1	2	yes	—
140	C₄×D₄	C₂×D₂₂₇	1	4	yes	—
141	#117	C₂×D₂₂₇	1	1	yes	characteristic
142	C₄⋊C₈	—	227	4	no	—
143	C₂²⋊C₈	—	227	4	no	—
144	C₂²⋊C₈	—	227	4	no	—
145	#97	—	227	2	no	—
146	C₄×C₈	—	227	2	no	—
147	C₂⁴	C₂×Dic₂₂₇	1	1	yes	characteristic
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₂×Dic₂₂₇	1	1	yes	characteristic
151	C₂×D₄	C₂×Dic₂₂₇	1	2	yes	—
152	C₂×D₄	C₂×Dic₂₂₇	1	1	yes	characteristic
153	C₂×D₄	C₂×Dic₂₂₇	1	4	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
157	C₂²×C₄	C₂×Dic₂₂₇	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
159	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
161	M₁₆	—	454	2	no	—
162	M₁₆	—	454	2	no	—
163	M₁₆	—	454	4	no	—
164	M₁₆	—	454	4	no	—
165	M₁₆	—	454	4	no	—
166	C₂×C₈	—	454	2	no	—
167	C₂×C₈	—	454	4	no	—
168	C₂×C₈	—	227	2	no	—
169	C₂×C₈	—	227	2	no	—
170	C₂×C₈	—	227	2	no	—
171	C₂×C₈	—	227	4	no	—
172	C₂×C₈	—	227	1	no	—
173	C₂×C₈	—	227	1	no	—
174	C₄⋊C₄	C₂×Dic₂₂₇	1	2	yes	—
175	C₄⋊C₄	C₂×Dic₂₂₇	1	4	yes	—
176	C₂²⋊C₄	C₂×Dic₂₂₇	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₂₂₇	1	4	yes	—
178	C₂²⋊C₄	C₂×Dic₂₂₇	1	4	yes	—
179	C₄²	C₂²×D₂₂₇	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	4	2	no	—
182	C₂³	C₂²×Dic₂₂₇	1	2	yes	—
183	C₂³	C₂²×Dic₂₂₇	1	2	yes	—
184	C₂³	C₂²×Dic₂₂₇	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	4	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	C₂²×Dic₂₂₇	1	4	yes	—
196	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
197	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
198	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
199	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
200	C₂×C₄	#16813883	1	2	yes	—
201	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
202	C₈	—	454	2	no	—
203	C₈	—	454	2	no	—
204	C₈	—	454	2	no	—
205	C₈	—	227	2	no	—
206	C₈	—	227	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	D₄×Dic₂₂₇	1	2	yes	—
215	C₂²	C₂³×Dic₂₂₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	D₄×Dic₂₂₇	1	2	yes	—
220	C₄	C₂×D₄×D₂₂₇	1	1	yes	characteristic
221	C₄	C₂×#16813883	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂×D₄×Dic₂₂₇	1	1	yes	characteristic
226	C₂	#28409799	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂³×C₄
Fitting, prime radical	#525×C₂₂₇	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#525	D₂₂₇
hypocenter	C₂₂₇	#2930
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#20300295	C₂	semidirect
2. C₂²×#16813868	C₂	semidirect
3. #28409110	C₂	semidirect
4. #28409104	C₂	semidirect
5. #28409059	C₂	semidirect
6. #28408999	C₂	semidirect
7. #28408996	C₂	semidirect
8. #28408988	C₂	semidirect
16. C₂×#16813868	C₂²	semidirect
19. C₂²×C₂₂₇⋊C₈	C₂²	semidirect
20. C₄⋊(C₂₂₇⋊C₈)	C₂²	semidirect
22. C₂²⋊(C₂₂₇⋊C₈)	C₂²	semidirect
23. #20300199	C₂²	semidirect
24. C₄×C₂₂₇⋊C₈	C₂²	semidirect
40. C₂×C₂₂₇⋊C₈	D₄	semidirect
113. C₂₂₇	#2930	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300295	2
C₂²×#16813868	2
#28409110	2
#28409104	2
#28409059	2
#28408999	2
#28408996	2
#28408988	2
#525×C₂₂₇	2
#2930	227