Subgroups — #44408442

Order 29056.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300296	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20300295	C₂	1	1	yes	maximal, characteristic, complemented
3	#28409578	C₂	1	1	yes	maximal, characteristic
4	#28409107	C₂	1	4	yes	maximal, complemented
5	(C₂²⋊D₂₂₇)⋊C₈	C₂	1	4	yes	maximal, complemented
6	#28409055	C₂	1	1	yes	maximal, characteristic, complemented
7	#28408998	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#408×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16813884	C₂²	1	1	yes	characteristic
11	#20300296	—	2	4	no	—
12	#20300295	—	2	4	no	—
13	C₂×#16813868	C₂²	1	2	yes	—
14	C₂×#16813868	C₂²	1	1	yes	characteristic
15	C₂×#16813867	C₂²	1	2	yes	—
16	C₂²×C₂₂₇⋊C₈	C₂²	1	2	yes	—
17	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₂₂₇	C₂²	1	2	yes	—
19	#20300286	C₄	1	1	yes	characteristic
20	#20300284	C₄	1	1	yes	characteristic
21	#20300277	C₄	1	1	yes	characteristic
22	#20300275	C₄	1	1	yes	characteristic
23	#20300260	C₄	1	4	yes	—
24	C₄×C₂²⋊D₂₂₇	C₂²	1	4	yes	—
25	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
26	Dic₂₂₇⋊C₈	C₂²	1	4	yes	—
27	(C₂×D₂₂₇)⋊C₈	C₂²	1	4	yes	—
28	C₈×Dic₂₂₇	C₂²	1	2	yes	—
29	C₈×Dic₂₂₇	C₂²	1	2	yes	—
30	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
33	C₂²⋊C₈×C₂₂₇	C₂²	1	4	yes	—
34	#16813884	—	2	2	no	—
35	#16813884	—	2	2	no	—
36	C₂×Dic₄₅₄	C₂×C₄	1	1	yes	characteristic
37	C₂×D₉₀₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	4	yes	—
40	C₂²×Dic₂₂₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₂₂₇	C₂³	1	2	yes	—
42	#16813868	—	2	4	no	—
43	#16813868	—	2	2	no	—
44	#16813867	—	2	4	no	—
45	C₂×C₂₂₇⋊C₈	—	2	4	no	—
46	C₂×C₂₂₇⋊C₈	D₄	1	4	yes	complemented
47	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
48	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₈×D₂₂₇	—	2	4	no	—
51	Dic₂₂₇⋊C₄	C₂×C₄	1	4	yes	—
52	C₂²⋊Dic₂₂₇	C₂×C₄	1	4	yes	—
53	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	4	yes	—
54	C₄×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₂₂₇	C₂³	1	2	yes	—
57	#41×C₂₂₇	—	2	2	no	—
58	#41×C₂₂₇	—	2	2	no	—
59	C₂×Q₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
60	C₂×D₄×C₂₂₇	C₂×C₄	1	2	yes	—
61	C₂×D₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
63	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
64	M₁₆×C₂₂₇	—	2	2	no	—
65	C₂×C₈×C₂₂₇	—	2	2	no	—
66	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
67	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
68	C₂²×D₂₂₇	C₂²×C₄	1	2	yes	—
69	Dic₄₅₄	—	2	2	no	—
70	D₉₀₈	—	2	2	no	—
71	C₂²⋊D₂₂₇	—	2	4	no	—
72	C₂²⋊D₂₂₇	—	2	8	no	—
73	C₂×Dic₂₂₇	—	2	4	no	—
74	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
75	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
77	C₄×D₂₂₇	—	2	2	no	—
78	C₄×D₂₂₇	—	2	2	no	—
79	C₂₂₇⋊C₈	—	2	2	no	—
80	C₂₂₇⋊C₈	C₂×D₄	1	2	yes	—
81	C₂₂₇⋊C₈	C₂×D₄	1	2	yes	—
82	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
83	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
84	Q₈×C₂₂₇	—	2	2	no	—
85	D₄×C₂₂₇	—	2	4	no	—
86	D₄×C₂₂₇	—	2	2	no	—
87	C₂×C₄×C₂₂₇	—	2	2	no	—
88	C₂×C₄×C₂₂₇	—	2	2	no	—
89	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₂₇	#41	1	2	yes	—
93	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₂₇	—	2	2	no	—
95	C₈×C₂₂₇	—	2	2	no	—
96	C₂²×C₂₂₇	—	2	2	no	—
97	C₂²×C₂₂₇	—	2	2	no	—
98	C₂²×C₂₂₇	—	2	4	no	—
99	C₂²×C₂₂₇	C₄×D₄	1	2	yes	—
100	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₂₂₇	—	2	2	no	—
102	C₂×D₂₂₇	—	2	4	no	—
103	C₄×C₂₂₇	—	2	2	no	—
104	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
105	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
107	Dic₂₂₇	—	2	2	no	—
108	Dic₂₂₇	—	2	2	no	—
109	Dic₂₂₇	#131	1	4	yes	complemented
110	C₂×C₂₂₇	—	2	2	no	—
111	C₂×C₂₂₇	—	4	2	no	—
112	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₂₂₇	C₂×#131	1	2	yes	—
114	D₂₂₇	—	4	2	no	—
115	C₂₂₇	#2931	1	1	yes	characteristic, complemented, Sylow
116	#2931	—	227	1	no	maximal, Sylow
117	C₂×#131	—	227	1	no	—
118	C₂×#131	—	227	1	no	—
119	#532	—	227	1	no	—
120	#436	—	227	4	no	—
121	C₈×D₄	—	227	4	no	—
122	#423	—	227	1	no	—
123	#408	—	227	1	no	—
124	#408	D₂₂₇	1	1	yes	characteristic, complemented
125	C₂×C₄×C₈	—	227	1	no	—
126	C₂×#41	—	227	1	no	—
127	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
128	#131	—	454	4	no	—
129	#131	—	454	4	no	—
130	C₂×M₁₆	—	227	1	no	—
131	C₂×M₁₆	—	227	2	no	—
132	C₂×M₁₆	—	227	2	no	—
133	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic
134	C₂²×C₈	—	227	2	no	—
135	C₂²×C₈	—	227	2	no	—
136	C₂²×C₈	—	227	1	no	—
137	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	227	1	no	—
139	#127	—	227	1	no	—
140	#124	—	227	1	no	—
141	#124	—	227	1	no	—
142	#121	—	227	4	no	—
143	C₄×D₄	—	227	4	no	—
144	C₂×C₄²	—	227	1	no	—
145	C₄⋊C₈	—	227	4	no	—
146	C₂²⋊C₈	—	227	4	no	—
147	C₂²⋊C₈	C₂×D₂₂₇	1	4	yes	—
148	C₄×C₈	—	227	2	no	—
149	C₄×C₈	—	227	2	no	—
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	#41	—	454	2	no	—
153	#41	—	454	2	no	—
154	C₂×Q₈	—	227	1	no	—
155	C₂×Q₈	C₄×D₂₂₇	1	1	yes	characteristic
156	C₂×D₄	—	227	4	no	—
157	C₂×D₄	—	227	2	no	—
158	C₂×D₄	—	227	1	no	—
159	C₂×D₄	C₄×D₂₂₇	1	2	yes	—
160	C₂×D₄	C₄×D₂₂₇	1	1	yes	characteristic
161	C₂²×C₄	—	227	2	no	—
162	C₂²×C₄	—	227	2	no	—
163	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
164	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
165	M₁₆	—	2	2	no	—
166	M₁₆	—	454	4	no	—
167	M₁₆	—	454	2	no	—
168	M₁₆	—	454	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	454	4	no	—
171	C₂×C₈	—	454	4	no	—
172	C₂×C₈	—	227	2	no	—
173	C₂×C₈	—	227	4	no	—
174	C₂×C₈	—	227	1	no	—
175	C₂×C₈	—	227	1	no	—
176	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
177	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
178	C₄⋊C₄	—	227	4	no	—
179	C₂²⋊C₄	—	227	4	no	—
180	C₂²⋊C₄	—	227	4	no	—
181	C₄²	—	227	1	no	—
182	C₄²	—	227	1	no	—
183	C₄²	—	227	2	no	—
184	C₂³	—	227	2	no	—
185	C₂³	C₂×C₄×D₂₂₇	1	2	yes	—
186	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
187	Q₈	—	2	2	no	—
188	Q₈	—	454	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	454	4	no	—
192	D₄	—	454	8	no	—
193	D₄	—	454	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	454	4	no	—
197	C₂×C₄	—	454	2	no	—
198	C₂×C₄	—	454	2	no	—
199	C₂×C₄	—	227	2	no	—
200	C₂×C₄	—	227	2	no	—
201	C₂×C₄	—	227	2	no	—
202	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
204	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
205	C₂×C₄	#16813883	1	2	yes	—
206	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	454	2	no	—
210	C₈	—	227	2	no	—
211	C₈	—	227	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	454	2	no	—
216	C₂²	—	454	4	no	—
217	C₂²	(C₂²⋊D₂₂₇)⋊C₄	1	2	yes	—
218	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	454	2	no	—
221	C₄	—	454	2	no	—
222	C₄	—	227	4	no	—
223	C₄	(C₂²⋊D₂₂₇)⋊C₄	1	2	yes	—
224	C₄	C₂×D₄×D₂₂₇	1	1	yes	characteristic
225	C₄	C₂×#16813883	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	908	2	no	—
229	C₂	C₂×(C₂²⋊D₂₂₇)⋊C₄	1	1	yes	characteristic
230	C₂	#131×D₂₂₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂³×C₄
Fitting	#408×C₂₂₇	C₂
prime radical	#28409578	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#408	D₂₂₇
hypocenter	C₂₂₇	#2931
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#20300296	C₂	semidirect
2. C₂×#20300295	C₂	semidirect
4. #28409107	C₂	semidirect
5. (C₂²⋊D₂₂₇)⋊C₈	C₂	semidirect
6. #28409055	C₂	semidirect
7. #28408998	C₂	semidirect
8. C₂×C₈×Dic₂₂₇	C₂	semidirect
9. #408×C₂₂₇	C₂	semidirect
46. C₂×C₂₂₇⋊C₈	D₄	semidirect
109. Dic₂₂₇	#131	semidirect
115. C₂₂₇	#2931	semidirect
124. #408	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300296	2
C₂×#20300295	2
#28409578	2
#28409107	2
(C₂²⋊D₂₂₇)⋊C₈	2
#28409055	2
#28408998	2
C₂×C₈×Dic₂₂₇	2
#408×C₂₂₇	2
#2931	227