Subgroups — C₂×#28409082 #44408347

Order 29056.

528 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
2	#28409082	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₂₂₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊(C₂₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₂₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₂₇	C₄	1	1	yes	characteristic
8	C₂³×Dic₂₂₇	C₄	1	1	yes	characteristic
9	C₂²×C₂₂₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₂₇	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₂₂₇	C₄	1	4	yes	—
12	C₂×C₄×Dic₂₂₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₂₂₇	C₂²	1	4	yes	—
16	Dic₂₂₇⋊C₈	C₂²	1	8	yes	complemented
17	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	4	yes	—
18	C₈×Dic₂₂₇	C₂²	1	8	yes	complemented
19	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	—
21	C₂²⋊C₈×C₂₂₇	C₂²	1	4	yes	—
22	C₂²×Dic₂₂₇	—	2	4	no	—
23	C₂²×Dic₂₂₇	C₈	1	8	yes	complemented
24	C₂²×Dic₂₂₇	C₂×C₄	1	4	yes	—
25	C₂²×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₂₂₇	C₂×C₄	1	4	yes	—
29	C₂²×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂₂₇⋊C₈	—	2	4	no	—
31	C₂×C₂₂₇⋊C₈	C₂³	1	4	yes	—
32	C₄×Dic₂₂₇	C₂×C₄	1	4	yes	—
33	C₄×Dic₂₂₇	C₂×C₄	1	4	yes	—
34	C₄×Dic₂₂₇	C₂×C₄	1	2	yes	—
35	C₄×Dic₂₂₇	C₂×C₄	1	2	yes	—
36	C₄×Dic₂₂₇	C₂³	1	2	yes	—
37	C₄×Dic₂₂₇	C₂³	1	2	yes	—
38	C₂⁴×C₂₂₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₂₇	—	2	4	no	—
40	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₂₂₇	—	2	4	no	—
45	C₂×C₈×C₂₂₇	C₂³	1	4	yes	—
46	C₂×Dic₂₂₇	—	2	2	no	—
47	C₂×Dic₂₂₇	—	2	8	no	—
48	C₂×Dic₂₂₇	C₂×C₈	1	4	yes	—
49	C₂×Dic₂₂₇	C₂×C₈	1	4	yes	—
50	C₂×Dic₂₂₇	C₂×C₈	1	16	yes	complemented
51	C₂×Dic₂₂₇	M₁₆	1	4	yes	—
52	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
53	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
54	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
55	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
56	C₂₂₇⋊C₈	—	2	4	no	—
57	C₂³×C₂₂₇	—	2	2	no	—
58	C₂³×C₂₂₇	—	2	2	no	—
59	C₂³×C₂₂₇	C₂×C₈	1	4	yes	—
60	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
61	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₂₇	—	2	2	no	—
63	C₂×C₄×C₂₂₇	—	2	4	no	—
64	C₂×C₄×C₂₂₇	—	2	4	no	—
65	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₂₇	#41	1	4	yes	—
68	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
69	C₈×C₂₂₇	—	2	4	no	—
70	C₂²×C₂₂₇	—	2	2	no	—
71	C₂²×C₂₂₇	—	2	2	no	—
72	C₂²×C₂₂₇	—	2	8	no	—
73	C₂²×C₂₂₇	#117	1	2	yes	—
74	C₂²×C₂₂₇	C₂²×C₈	1	4	yes	—
75	C₂²×C₂₂₇	C₂²×C₈	1	2	yes	—
76	C₂²×C₂₂₇	C₂×M₁₆	1	2	yes	—
77	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₂₂₇	—	2	2	no	—
79	C₄×C₂₂₇	#117	1	2	yes	—
80	C₄×C₂₂₇	C₂×#41	1	2	yes	—
81	Dic₂₂₇	—	2	2	no	—
82	Dic₂₂₇	C₂²×C₈	1	4	yes	—
83	Dic₂₂₇	C₂²×C₈	1	4	yes	—
84	Dic₂₂₇	C₂×M₁₆	1	4	yes	—
85	C₂×C₂₂₇	—	2	4	no	—
86	C₂×C₂₂₇	#431	1	4	yes	complemented
87	C₂×C₂₂₇	C₂×#117	1	1	yes	characteristic
88	C₂×C₂₂₇	C₂³×C₈	1	1	yes	characteristic
89	C₂×C₂₂₇	C₂²×M₁₆	1	1	yes	characteristic
90	C₂₂₇	C₂×#431	1	1	yes	characteristic, complemented, Sylow
91	C₂×#431	—	227	1	no	maximal, Sylow
92	C₂²×C₄²	—	227	1	no	—
93	#431	—	227	8	no	—
94	C₂×C₄⋊C₈	—	227	2	no	—
95	C₂×C₂²⋊C₈	—	227	1	no	—
96	C₂×C₂²⋊C₈	D₂₂₇	1	1	yes	characteristic
97	C₂×C₄×C₈	—	227	2	no	—
98	C₂³×C₄	—	227	1	no	—
99	C₂³×C₄	—	227	1	no	—
100	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
101	C₂²×C₈	—	227	2	no	—
102	C₂²×C₈	C₂×D₂₂₇	1	2	yes	—
103	C₂×C₄²	—	227	1	no	—
104	C₂×C₄²	—	227	4	no	—
105	C₂×C₄²	—	227	1	no	—
106	C₂×C₄²	—	227	4	no	—
107	C₂×C₄²	—	227	1	no	—
108	C₂×C₄²	—	227	1	no	—
109	C₄⋊C₈	—	227	8	no	—
110	C₂²⋊C₈	—	227	4	no	—
111	C₂²⋊C₈	C₂×D₂₂₇	1	4	yes	—
112	C₄×C₈	—	227	8	no	—
113	C₂⁴	C₄×D₂₂₇	1	1	yes	characteristic
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	454	4	no	—
116	C₂²×C₄	—	227	1	no	—
117	C₂²×C₄	—	227	4	no	—
118	C₂²×C₄	—	227	8	no	—
119	C₂²×C₄	—	227	1	no	—
120	C₂²×C₄	—	227	1	no	—
121	C₂²×C₄	—	227	4	no	—
122	C₂²×C₄	—	227	1	no	—
123	C₂²×C₄	C₄×D₂₂₇	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₂₂₇	1	2	yes	—
125	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
126	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	454	4	no	—
129	C₂×C₈	—	227	4	no	—
130	C₂×C₈	C₂²×D₂₂₇	1	4	yes	—
131	C₄²	—	227	4	no	—
132	C₄²	—	227	2	no	—
133	C₄²	—	227	4	no	—
134	C₄²	—	227	2	no	—
135	C₄²	—	227	2	no	—
136	C₄²	—	227	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	2	2	no	—
139	C₂³	C₈×D₂₂₇	1	4	yes	—
140	C₂³	C₂×C₄×D₂₂₇	1	2	yes	—
141	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	454	2	no	—
146	C₂×C₄	—	454	8	no	—
147	C₂×C₄	—	227	2	no	—
148	C₂×C₄	—	227	4	no	—
149	C₂×C₄	—	227	4	no	—
150	C₂×C₄	—	227	16	no	—
151	C₂×C₄	—	227	4	no	—
152	C₂×C₄	—	227	2	no	—
153	C₂×C₄	—	227	2	no	—
154	C₂×C₄	—	227	2	no	—
155	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
156	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
157	C₂×C₄	#16813883	1	4	yes	—
158	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
159	C₈	—	2	4	no	—
160	C₈	—	454	4	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	8	no	—
164	C₂²	#20300244	1	2	yes	—
165	C₂²	C₂×C₈×D₂₂₇	1	4	yes	—
166	C₂²	C₂×C₈×D₂₂₇	1	2	yes	—
167	C₂²	M₁₆×D₂₂₇	1	2	yes	—
168	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	454	2	no	—
171	C₄	—	227	4	no	—
172	C₄	—	227	4	no	—
173	C₄	—	227	4	no	—
174	C₄	#20300244	1	2	yes	—
175	C₄	C₂×#16813883	1	2	yes	—
176	C₂	—	2	4	no	—
177	C₂	#28409082	1	4	yes	direct product factor
178	C₂	C₂×#20300244	1	1	yes	characteristic
179	C₂	C₂²×C₈×D₂₂₇	1	1	yes	characteristic
180	C₂	C₂×M₁₆×D₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₂₇
commutator	C₂×C₂₂₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting	C₂×C₂²⋊C₈×C₂₂₇	C₂
socle	C₂³×C₂₂₇	C₂²×C₄
prime radical	C₂⁴×C₂₂₇	C₂×C₄
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₂₂₇
hypocenter	C₂₂₇	C₂×#431
LCS	G ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #28409082	C₂	direct (N × H)
3. C₂×Dic₂₂₇⋊C₈	C₂	semidirect
5. C₂×C₈×Dic₂₂₇	C₂	semidirect
16. Dic₂₂₇⋊C₈	C₂²	semidirect
18. C₈×Dic₂₂₇	C₂²	semidirect
23. C₂²×Dic₂₂₇	C₈	semidirect
50. C₂×Dic₂₂₇	C₂×C₈	semidirect
86. C₂×C₂₂₇	#431	semidirect
90. C₂₂₇	C₂×#431	semidirect
177. C₂	#28409082	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Dic₂₂₇	2
#28409082	2
C₂×Dic₂₂₇⋊C₈	2
C₂×C₂²⋊(C₂₂₇⋊C₈)	2
C₂×C₈×Dic₂₂₇	2
C₂×C₂²⋊C₈×C₂₂₇	2
C₂×#431	227

Subgroups — C2×#28409082 #44408347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#28409082 #44408347