Subgroups — #44408367

Order 29056.

484 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28409089	C₂	1	4	yes	maximal, complemented
2	#28409085	C₂	1	4	yes	maximal, complemented
3	#28409056	C₂	1	2	yes	maximal, complemented
4	#28408996	C₂	1	2	yes	maximal
5	C₄×#16813868	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₂₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#16813868	C₂²	1	4	yes	—
8	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	8	yes	complemented
9	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	8	yes	—
10	#20300199	C₂²	1	4	yes	complemented
11	C₄×C₂₂₇⋊C₈	C₂²	1	4	yes	complemented
12	C₂³×C₄×C₂₂₇	C₄	1	2	yes	—
13	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₂₇	C₄	1	2	yes	—
15	C₂×C₄²×C₂₂₇	C₄	1	2	yes	—
16	C₂×C₄²×C₂₂₇	C₄	1	2	yes	—
17	C₂×C₄²×C₂₂₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₂₇	C₂²	1	2	yes	—
21	#16813868	—	2	8	no	—
22	C₂×C₂₂₇⋊C₈	C₂³	1	8	yes	—
23	C₂⁴×C₂₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₂₇	—	2	8	no	—
25	C₂²×C₄×C₂₂₇	—	2	4	no	—
26	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
36	C₄²×C₂₂₇	C₂×C₄	1	2	yes	—
37	C₄²×C₂₂₇	C₂×C₄	1	2	yes	—
38	C₄²×C₂₂₇	C₂×C₄	1	2	yes	—
39	C₄²×C₂₂₇	C₂×C₄	1	2	yes	—
40	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₂₂₇	C₂×C₄	1	2	yes	—
42	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
44	C₄²×C₂₂₇	C₂³	1	2	yes	—
45	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
46	C₂₂₇⋊C₈	—	2	8	no	—
47	C₂³×C₂₂₇	—	2	2	no	—
48	C₂³×C₂₂₇	—	2	4	no	—
49	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₂₂₇	—	2	2	no	—
52	C₂×C₄×C₂₂₇	—	2	4	no	—
53	C₂×C₄×C₂₂₇	—	2	4	no	—
54	C₂×C₄×C₂₂₇	—	2	4	no	—
55	C₂×C₄×C₂₂₇	—	2	4	no	—
56	C₂×C₄×C₂₂₇	—	2	4	no	—
57	C₂×C₄×C₂₂₇	—	2	4	no	—
58	C₂×C₄×C₂₂₇	—	2	4	no	—
59	C₂×C₄×C₂₂₇	M₁₆	1	8	yes	complemented
60	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₂₇	#41	1	4	yes	—
67	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
68	C₂²×C₂₂₇	—	2	8	no	—
69	C₂²×C₂₂₇	—	2	2	no	—
70	C₂²×C₂₂₇	—	2	2	no	—
71	C₂²×C₂₂₇	—	2	2	no	—
72	C₂²×C₂₂₇	#117	1	2	yes	—
73	C₂²×C₂₂₇	C₂×M₁₆	1	4	yes	—
74	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
75	C₄×C₂₂₇	—	2	2	no	—
76	C₄×C₂₂₇	—	2	2	no	—
77	C₄×C₂₂₇	—	2	2	no	—
78	C₄×C₂₂₇	#117	1	2	yes	—
79	C₄×C₂₂₇	C₂×M₁₆	1	4	yes	—
80	C₄×C₂₂₇	C₂×M₁₆	1	4	yes	—
81	C₄×C₂₂₇	C₂×#41	1	2	yes	—
82	C₂×C₂₂₇	—	2	2	no	—
83	C₂×C₂₂₇	—	2	4	no	—
84	C₂×C₂₂₇	C₂×#117	1	1	yes	characteristic
85	C₂×C₂₂₇	C₂²×M₁₆	1	2	yes	—
86	C₂₂₇	#2920	1	1	yes	characteristic, complemented, Sylow
87	#2920	—	227	1	no	maximal, Sylow
88	C₂²×C₄²	D₂₂₇	1	1	yes	characteristic
89	#432	—	227	4	no	—
90	#431	—	227	4	no	—
91	#423	—	227	2	no	—
92	#407	—	227	2	no	—
93	C₄×M₁₆	—	227	2	no	—
94	C₂³×C₄	Dic₂₂₇	1	2	yes	—
95	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
96	C₂×M₁₆	—	227	4	no	—
97	C₂×C₄²	Dic₂₂₇	1	2	yes	—
98	C₂×C₄²	Dic₂₂₇	1	2	yes	—
99	C₂×C₄²	Dic₂₂₇	1	2	yes	—
100	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₂₂₇	1	2	yes	—
102	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
103	C₂×C₄²	C₂×D₂₂₇	1	2	yes	—
104	C₄⋊C₈	—	227	8	no	—
105	C₂²⋊C₈	—	227	8	no	—
106	#97	—	227	4	no	—
107	C₄×C₈	—	227	4	no	—
108	C₂⁴	C₂×Dic₂₂₇	1	1	yes	characteristic
109	C₂²×C₄	—	2	8	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	C₂×Dic₂₂₇	1	1	yes	characteristic
112	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
113	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
114	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
115	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
116	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
117	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
118	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
119	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
120	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
121	M₁₆	—	454	8	no	—
122	C₂×C₈	—	227	8	no	—
123	C₄²	C₂×Dic₂₂₇	1	2	yes	—
124	C₄²	C₂×Dic₂₂₇	1	2	yes	—
125	C₄²	C₂×Dic₂₂₇	1	2	yes	—
126	C₄²	C₂×Dic₂₂₇	1	2	yes	—
127	C₄²	C₂×Dic₂₂₇	1	1	yes	characteristic
128	C₄²	C₂×Dic₂₂₇	1	2	yes	—
129	C₄²	C₂×Dic₂₂₇	1	1	yes	characteristic
130	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
131	C₄²	C₂²×D₂₂₇	1	2	yes	—
132	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
133	C₂³	—	2	4	no	—
134	C₂³	—	2	2	no	—
135	C₂³	C₂²×Dic₂₂₇	1	2	yes	—
136	C₂³	C₂²×Dic₂₂₇	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	#16813868	1	8	yes	complemented
146	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
147	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
148	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
149	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
150	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
151	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
152	C₂×C₄	#16813884	1	4	yes	—
153	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
154	C₈	—	454	8	no	—
155	C₂²	—	2	8	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	#20300247	1	2	yes	—
160	C₂²	C₂×#16813868	1	4	yes	—
161	C₂²	C₂³×Dic₂₂₇	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	#20300247	1	2	yes	—
166	C₄	C₂×#16813868	1	4	yes	—
167	C₄	C₂×#16813868	1	4	yes	—
168	C₄	C₂×#16813884	1	2	yes	—
169	C₂	—	2	4	no	—
170	C₂	—	2	2	no	—
171	C₂	C₂×#20300247	1	1	yes	characteristic
172	C₂	C₂²×#16813868	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting	C₂²×C₄²×C₂₂₇	C₂
prime radical	C₂⁴×C₂₂₇	C₂×C₄
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	C₂²×C₄²	D₂₂₇
hypocenter	C₂₂₇	#2920
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #28409089	C₂	semidirect
2. #28409085	C₂	semidirect
3. #28409056	C₂	semidirect
5. C₄×#16813868	C₂	semidirect
8. C₄⋊(C₂₂₇⋊C₈)	C₂²	semidirect
10. #20300199	C₂²	semidirect
11. C₄×C₂₂₇⋊C₈	C₂²	semidirect
59. C₂×C₄×C₂₂₇	M₁₆	semidirect
86. C₂₂₇	#2920	semidirect
145. C₂×C₄	#16813868	semidirect

Maximal subgroups

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

subgroup	index
#28409089	2
#28409085	2
#28409056	2
#28408996	2
C₄×#16813868	2
C₂²×C₄²×C₂₂₇	2
#2920	227