Subgroups — #44402136

Order 29056.

656 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300244	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
3	#28408901	C₂	1	2	yes	maximal, complemented
4	#28408896	C₂	1	2	yes	maximal, complemented
5	#28408892	C₂	1	1	yes	maximal, characteristic
6	#28408890	C₂	1	2	yes	maximal
7	C₄⋊C₄×Dic₂₂₇	C₂	1	2	yes	maximal, complemented
8	#28408883	C₂	1	2	yes	maximal, complemented
9	#380×C₂₂₇	C₂	1	1	yes	maximal, characteristic
10	C₂³×Dic₂₂₇	C₂²	1	1	yes	characteristic
11	C₂³×Dic₂₂₇	C₂²	1	1	yes	characteristic
12	#20300244	—	2	8	no	—
13	C₂×C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
14	C₂×Dic₂₂₇⋊C₄	C₂²	1	4	yes	—
15	C₂×Dic₂₂₇⋊C₄	C₂²	1	4	yes	—
16	C₂×C₂²⋊Dic₂₂₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₂₂₇	C₄	1	8	yes	complemented
18	C₂×C₄×Dic₂₂₇	C₂²	1	4	yes	—
19	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
23	C₂₂₇⋊#95	C₂²	1	2	yes	—
24	C₂₂₇⋊#95	C₂²	1	4	yes	—
25	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	2	yes	—
28	#95×C₂₂₇	C₂²	1	2	yes	—
29	C₂²×Dic₂₂₇	—	2	4	no	—
30	C₂²×Dic₂₂₇	C₂×C₄	1	4	yes	—
31	C₂²×Dic₂₂₇	C₂×C₄	1	4	yes	—
32	C₂²×Dic₂₂₇	D₄	1	4	yes	—
33	C₂²×Dic₂₂₇	Q₈	1	4	yes	—
34	C₂²×Dic₂₂₇	C₂³	1	4	yes	—
35	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
36	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
38	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₂₂₇	—	2	4	no	—
40	Dic₂₂₇⋊C₄	—	2	8	no	—
41	Dic₂₂₇⋊C₄	—	2	8	no	—
42	C₂²⋊Dic₂₂₇	—	2	4	no	—
43	C₄×Dic₂₂₇	—	2	8	no	—
44	C₄×Dic₂₂₇	C₂×C₄	1	4	yes	—
45	C₄×Dic₂₂₇	C₂×C₄	1	4	yes	—
46	C₄×Dic₂₂₇	C₂×C₄	1	4	yes	—
47	C₄×Dic₂₂₇	C₂×C₄	1	4	yes	—
48	C₂⁴×C₂₂₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₂₇	—	2	4	no	—
50	C₂²×C₄×C₂₂₇	C₂×C₄	1	4	yes	—
51	C₂²×C₄×C₂₂₇	C₂³	1	4	yes	—
52	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₂₇	—	2	4	no	—
55	C₂²⋊C₄×C₂₂₇	—	2	4	no	—
56	C₂×Dic₂₂₇	—	2	2	no	—
57	C₂×Dic₂₂₇	—	2	8	no	—
58	C₂×Dic₂₂₇	—	2	4	no	—
59	C₂×Dic₂₂₇	—	2	4	no	—
60	C₂×Dic₂₂₇	—	2	4	no	—
61	C₂×Dic₂₂₇	C₄⋊C₄	1	16	yes	complemented
62	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
63	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
64	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
65	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
66	C₂×Dic₂₂₇	C₂×D₄	1	2	yes	—
67	C₂×Dic₂₂₇	C₂×D₄	1	2	yes	—
68	C₂×Dic₂₂₇	C₂×Q₈	1	2	yes	—
69	C₂×Dic₂₂₇	C₂×Q₈	1	2	yes	—
70	C₂×Dic₂₂₇	#41	1	4	yes	—
71	C₂³×C₂₂₇	—	2	2	no	—
72	C₂³×C₂₂₇	—	2	2	no	—
73	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
74	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
75	C₂³×C₂₂₇	C₂×Q₈	1	2	yes	—
76	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₄×C₂₂₇	—	2	2	no	—
78	C₂×C₄×C₂₂₇	—	2	8	no	—
79	C₂×C₄×C₂₂₇	—	2	4	no	—
80	C₂×C₄×C₂₂₇	—	2	4	no	—
81	C₂×C₄×C₂₂₇	—	2	4	no	—
82	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₂₂₇	#41	1	4	yes	—
85	C₂²×C₂₂₇	—	2	2	no	—
86	C₂²×C₂₂₇	—	2	2	no	—
87	C₂²×C₂₂₇	—	2	4	no	—
88	C₂²×C₂₂₇	—	2	4	no	—
89	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
91	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
93	C₂²×C₂₂₇	C₂×#41	1	2	yes	—
94	C₂²×C₂₂₇	C₂×#41	1	2	yes	—
95	C₄×C₂₂₇	—	2	2	no	—
96	C₄×C₂₂₇	—	4	4	no	—
97	C₄×C₂₂₇	#117	1	4	yes	complemented
98	Dic₂₂₇	—	2	2	no	—
99	Dic₂₂₇	—	4	4	no	—
100	Dic₂₂₇	C₂×C₄⋊C₄	1	4	yes	—
101	Dic₂₂₇	C₂×C₄⋊C₄	1	4	yes	—
102	Dic₂₂₇	#117	1	4	yes	complemented
103	C₂×C₂₂₇	—	2	4	no	—
104	C₂×C₂₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₂₂₇	C₂×#117	1	1	yes	characteristic
106	C₂×C₂₂₇	C₂×#117	1	1	yes	characteristic
107	C₂×C₂₂₇	#532	1	2	yes	—
108	C₂×C₂₂₇	#533	1	2	yes	—
109	C₂₂₇	#2285	1	1	yes	characteristic, complemented, Sylow
110	#2285	—	227	1	no	maximal, Sylow
111	C₂×#117	—	227	2	no	—
112	C₂²×C₄²	—	227	1	no	—
113	#384	—	227	2	no	—
114	#382	—	227	2	no	—
115	#380	—	227	1	no	—
116	#380	—	227	2	no	—
117	#380	D₂₂₇	1	1	yes	characteristic
118	C₄×C₄⋊C₄	—	227	2	no	—
119	C₄×C₄⋊C₄	—	227	2	no	—
120	C₂³×C₄	—	227	1	no	—
121	C₂³×C₄	—	227	1	no	—
122	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
123	#117	—	454	8	no	—
124	C₂×C₄⋊C₄	—	227	4	no	—
125	C₂×C₄⋊C₄	—	227	4	no	—
126	C₂×C₄⋊C₄	—	227	2	no	—
127	C₂×C₄⋊C₄	C₂×D₂₂₇	1	2	yes	—
128	C₂×C₂²⋊C₄	—	227	2	no	—
129	C₂×C₂²⋊C₄	C₂×D₂₂₇	1	2	yes	—
130	C₂×C₄²	—	227	4	no	—
131	C₂×C₄²	—	227	8	no	—
132	C₂×C₄²	—	227	1	no	—
133	C₂×C₄²	—	227	1	no	—
134	C₂×C₄²	—	227	1	no	—
135	C₂×C₄²	—	227	1	no	—
136	#95	—	227	2	no	—
137	#95	—	227	4	no	—
138	#95	C₂×D₂₂₇	1	2	yes	—
139	C₂⁴	C₂²×D₂₂₇	1	1	yes	characteristic
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	454	4	no	—
142	C₂²×C₄	—	227	4	no	—
143	C₂²×C₄	—	227	4	no	—
144	C₂²×C₄	—	227	1	no	—
145	C₂²×C₄	—	227	4	no	—
146	C₂²×C₄	—	227	4	no	—
147	C₂²×C₄	—	227	4	no	—
148	C₂²×C₄	—	227	1	no	—
149	C₂²×C₄	—	227	1	no	—
150	C₂²×C₄	—	227	1	no	—
151	C₂²×C₄	C₄×D₂₂₇	1	4	yes	—
152	C₂²×C₄	C₂²×D₂₂₇	1	4	yes	—
153	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	454	8	no	—
157	C₄⋊C₄	—	454	8	no	—
158	C₄⋊C₄	—	454	4	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	454	4	no	—
161	C₄²	—	454	8	no	—
162	C₄²	—	227	4	no	—
163	C₄²	—	227	4	no	—
164	C₄²	—	227	4	no	—
165	C₄²	—	227	4	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	C₂×C₄×D₂₂₇	1	2	yes	—
169	C₂³	D₄×D₂₂₇	1	2	yes	—
170	C₂³	Q₈×D₂₂₇	1	2	yes	—
171	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	454	4	no	—
178	C₂×C₄	—	454	4	no	—
179	C₂×C₄	—	454	4	no	—
180	C₂×C₄	—	454	2	no	—
181	C₂×C₄	—	454	8	no	—
182	C₂×C₄	—	227	2	no	—
183	C₂×C₄	—	227	2	no	—
184	C₂×C₄	—	227	2	no	—
185	C₂×C₄	—	227	2	no	—
186	C₂×C₄	—	227	2	no	—
187	C₂×C₄	—	227	2	no	—
188	C₂×C₄	—	227	2	no	—
189	C₂×C₄	—	227	2	no	—
190	C₂×C₄	—	227	4	no	—
191	C₂×C₄	—	227	16	no	—
192	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
193	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
194	C₂×C₄	#16813883	1	4	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	4	no	—
199	C₂²	C₄⋊C₄×D₂₂₇	1	4	yes	—
200	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
201	C₂²	C₂×D₄×D₂₂₇	1	1	yes	characteristic
202	C₂²	C₂×Q₈×D₂₂₇	1	1	yes	characteristic
203	C₂²	#41×D₂₂₇	1	2	yes	—
204	C₂²	C₂×#16813883	1	2	yes	—
205	C₄	—	2	2	no	—
206	C₄	—	4	4	no	—
207	C₄	—	908	4	no	—
208	C₄	—	454	2	no	—
209	C₄	—	227	4	no	—
210	C₄	—	227	4	no	—
211	C₄	—	227	4	no	—
212	C₄	#20300244	1	4	yes	complemented
213	C₂	—	2	4	no	—
214	C₂	C₂×C₄⋊C₄×D₂₂₇	1	1	yes	characteristic
215	C₂	#117×D₂₂₇	1	1	yes	characteristic
216	C₂	C₂×#20300244	1	1	yes	characteristic
217	C₂	#28409577	1	2	yes	—
218	C₂	#28409584	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Fitting	#380×C₂₂₇	C₂
socle, cosocle	C₂³×C₂₂₇	C₂⁴
prime radical	C₂⁴×C₂₂₇	C₂³
hypercenter	#380	D₂₂₇
hypocenter	C₂₂₇	#2285
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ #380

Complements

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

N	H ≅ G/N	type
3. #28408901	C₂	semidirect
4. #28408896	C₂	semidirect
7. C₄⋊C₄×Dic₂₂₇	C₂	semidirect
8. #28408883	C₂	semidirect
17. C₂×C₄×Dic₂₂₇	C₄	semidirect
61. C₂×Dic₂₂₇	C₄⋊C₄	semidirect
97. C₄×C₂₂₇	#117	semidirect
102. Dic₂₂₇	#117	semidirect
109. C₂₂₇	#2285	semidirect
212. C₄	#20300244	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300244	2
C₂²×C₄×Dic₂₂₇	2
#28408901	2
#28408896	2
#28408892	2
#28408890	2
C₄⋊C₄×Dic₂₂₇	2
#28408883	2
#380×C₂₂₇	2
#2285	227