Subgroups — #2330×C₂₂₇ #44396286

Order 29056.

444 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₂₂₇	C₂	1	1	yes	maximal, characteristic
2	#389×C₂₂₇	C₂	1	4	yes	maximal
3	#389×C₂₂₇	C₂	1	2	yes	maximal
4	#387×C₂₂₇	C₂	1	4	yes	maximal
5	#384×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊C₄×C₂₂₇	C₂	1	2	yes	maximal
7	#376×C₂₂₇	C₂	1	1	yes	maximal, characteristic
8	#125×C₂₂₇	C₄	1	8	yes	complemented
9	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	4	yes	—
14	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₂₇	C₂²	1	4	yes	—
16	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
18	#95×C₂₂₇	C₂²	1	2	yes	—
19	#95×C₂₂₇	C₂²	1	2	yes	—
20	#95×C₂₂₇	C₂²	1	4	yes	—
21	#95×C₂₂₇	C₂²	1	4	yes	—
22	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₂₂₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₂₂₇	C₂³	1	4	yes	—
26	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₂₇	—	2	4	no	—
30	C₄⋊C₄×C₂₂₇	—	4	4	no	—
31	C₄⋊C₄×C₂₂₇	—	4	4	no	—
32	C₄⋊C₄×C₂₂₇	C₂×C₄	1	16	yes	—
33	C₄⋊C₄×C₂₂₇	C₂×C₄	1	8	yes	—
34	C₄²×C₂₂₇	—	2	2	no	—
35	C₄²×C₂₂₇	—	4	4	no	—
36	C₄²×C₂₂₇	C₂×C₄	1	4	yes	—
37	C₄²×C₂₂₇	Q₈	1	4	yes	—
38	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₂₂₇	—	2	2	no	—
40	C₂×C₄×C₂₂₇	—	2	4	no	—
41	C₂×C₄×C₂₂₇	—	2	2	no	—
42	C₂×C₄×C₂₂₇	—	2	4	no	—
43	C₂×C₄×C₂₂₇	—	2	2	no	—
44	C₂×C₄×C₂₂₇	—	2	2	no	—
45	C₂×C₄×C₂₂₇	—	2	1	no	—
46	C₂×C₄×C₂₂₇	—	2	1	no	—
47	C₂×C₄×C₂₂₇	—	2	4	no	—
48	C₂×C₄×C₂₂₇	—	2	4	no	—
49	C₂×C₄×C₂₂₇	—	2	2	no	—
50	C₂×C₄×C₂₂₇	—	2	4	no	—
51	C₂×C₄×C₂₂₇	—	2	2	no	—
52	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₂₂₇	C₂²×C₄	1	8	yes	—
54	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₂₇	C₂×Q₈	1	4	yes	—
56	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₂₂₇	#41	1	2	yes	—
58	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
59	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
60	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
61	C₂²×C₂₂₇	H(5,𝔽₂)	1	2	yes	—
62	C₂²×C₂₂₇	2₋¹⁺⁴	1	2	yes	—
63	C₄×C₂₂₇	—	2	8	no	—
64	C₄×C₂₂₇	—	2	4	no	—
65	C₄×C₂₂₇	—	2	4	no	—
66	C₄×C₂₂₇	—	4	1	no	—
67	C₄×C₂₂₇	—	4	4	no	—
68	C₄×C₂₂₇	—	4	1	no	—
69	C₄×C₂₂₇	C₄×Q₈	1	4	yes	—
70	C₂×C₂₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₂₂₇	#520	1	2	yes	—
72	C₂×C₂₂₇	#540	1	1	yes	characteristic
73	C₂×C₂₂₇	#541	1	1	yes	characteristic
74	C₂×C₂₂₇	#544	1	2	yes	—
75	C₂₂₇	#2330	1	1	yes	characteristic, direct product factor, Sylow
76	#2330	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×#125	C₂×C₂₂₇	1	1	yes	characteristic
78	#389	C₂×C₂₂₇	1	4	yes	—
79	#389	C₂×C₂₂₇	1	2	yes	—
80	#387	C₂×C₂₂₇	1	4	yes	—
81	#384	C₂×C₂₂₇	1	1	yes	characteristic
82	C₄×C₄⋊C₄	C₂×C₂₂₇	1	2	yes	—
83	#376	C₂×C₂₂₇	1	1	yes	characteristic
84	#125	C₄×C₂₂₇	1	8	yes	complemented
85	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
86	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	4	yes	—
87	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	4	yes	—
88	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	4	yes	—
90	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
91	C₂×C₄²	C₂²×C₂₂₇	1	4	yes	—
92	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
93	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
94	#95	C₂²×C₂₂₇	1	2	yes	—
95	#95	C₂²×C₂₂₇	1	2	yes	—
96	#95	C₂²×C₂₂₇	1	4	yes	—
97	#95	C₂²×C₂₂₇	1	4	yes	—
98	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
99	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
100	C₂²×C₄	C₂³×C₂₂₇	1	4	yes	—
101	C₂²×C₄	C₂³×C₂₂₇	1	4	yes	—
102	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
103	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
104	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	4	no	—
108	C₄⋊C₄	C₂×C₄×C₂₂₇	1	16	yes	—
109	C₄⋊C₄	C₂×C₄×C₂₂₇	1	8	yes	—
110	C₄²	—	2	2	no	—
111	C₄²	—	4	4	no	—
112	C₄²	C₂×C₄×C₂₂₇	1	4	yes	—
113	C₄²	Q₈×C₂₂₇	1	4	yes	—
114	C₂³	C₂⁴×C₂₂₇	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₂²×C₄×C₂₂₇	1	4	yes	—
129	C₂×C₄	C₂²×C₄×C₂₂₇	1	8	yes	—
130	C₂×C₄	C₂²×C₄×C₂₂₇	1	2	yes	—
131	C₂×C₄	C₂×Q₈×C₂₂₇	1	4	yes	—
132	C₂×C₄	C₂×Q₈×C₂₂₇	1	2	yes	—
133	C₂×C₄	#41×C₂₂₇	1	2	yes	—
134	C₂²	C₂³×C₄×C₂₂₇	1	1	yes	characteristic
135	C₂²	C₂²×Q₈×C₂₂₇	1	1	yes	characteristic
136	C₂²	C₂×#41×C₂₂₇	1	1	yes	characteristic
137	C₂²	H(5,𝔽₂)×C₂₂₇	1	2	yes	—
138	C₂²	2₋¹⁺⁴×C₂₂₇	1	2	yes	—
139	C₄	—	2	8	no	—
140	C₄	—	2	4	no	—
141	C₄	—	2	4	no	—
142	C₄	—	4	1	no	—
143	C₄	—	4	4	no	—
144	C₄	—	4	1	no	—
145	C₄	C₄×Q₈×C₂₂₇	1	4	yes	—
146	C₂	C₂×C₄×Q₈×C₂₂₇	1	1	yes	characteristic
147	C₂	#520×C₂₂₇	1	2	yes	—
148	C₂	#540×C₂₂₇	1	1	yes	characteristic
149	C₂	#541×C₂₂₇	1	1	yes	characteristic
150	C₂	#544×C₂₂₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₂₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₂₇
Frattini, cosocle	C₂³	C₂⁴×C₂₂₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. #125×C₂₂₇	C₄	semidirect
75. C₂₂₇	#2330	direct (N × H)
76. #2330	C₂₂₇	direct (N × H)
84. #125	C₄×C₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#125×C₂₂₇	2
#389×C₂₂₇	2
#387×C₂₂₇	2
#384×C₂₂₇	2
C₄×C₄⋊C₄×C₂₂₇	2
#376×C₂₂₇	2
#2330	227

Subgroups — #2330×C227 #44396286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2330×C₂₂₇ #44396286