Subgroups — #44408400

Order 29056.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28409093	C₂	1	4	yes	maximal, complemented
2	#28409089	C₂	1	2	yes	maximal, complemented
3	#28409085	C₂	1	2	yes	maximal, complemented
4	#28409056	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₂₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#28408999	C₂	1	2	yes	maximal
7	#28408988	C₂	1	2	yes	maximal, complemented
8	C₄×#41×C₂₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#16813868	C₂²	1	2	yes	—
10	C₂²×C₂₂₇⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
13	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
14	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
15	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	4	yes	—
17	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	4	yes	—
18	#20300199	C₂²	1	4	yes	—
19	C₄×C₂₂₇⋊C₈	C₂²	1	4	yes	—
20	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₂₂₇	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₂₂₇	C₄	1	1	yes	characteristic
23	C₄×D₄×C₂₂₇	C₄	1	2	yes	—
24	C₄×D₄×C₂₂₇	C₄	1	1	yes	characteristic
25	C₄×D₄×C₂₂₇	C₄	1	2	yes	—
26	C₄×D₄×C₂₂₇	C₄	1	1	yes	characteristic
27	#117×C₂₂₇	C₂²	1	2	yes	—
28	#117×C₂₂₇	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₂₂₇	C₂²	1	2	yes	—
30	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
31	#16813868	—	2	4	no	—
32	C₂×C₂₂₇⋊C₈	—	2	4	no	—
33	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
34	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
35	C₂×C₂₂₇⋊C₈	C₂³	1	4	yes	—
36	#41×C₂₂₇	—	2	2	no	—
37	#41×C₂₂₇	—	2	2	no	—
38	C₂×Q₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₂₂₇	C₂×C₄	1	2	yes	—
40	C₂×D₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
54	C₄²×C₂₂₇	C₂×C₄	1	2	yes	—
55	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₂₂₇	C₂×C₄	1	2	yes	—
57	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
59	C₄²×C₂₂₇	C₂³	1	2	yes	—
60	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
61	C₂₂₇⋊C₈	—	2	2	no	—
62	C₂₂₇⋊C₈	—	2	2	no	—
63	C₂₂₇⋊C₈	—	2	4	no	—
64	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
65	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
66	Q₈×C₂₂₇	—	2	2	no	—
67	D₄×C₂₂₇	—	2	4	no	—
68	D₄×C₂₂₇	—	2	2	no	—
69	C₂×C₄×C₂₂₇	—	2	2	no	—
70	C₂×C₄×C₂₂₇	—	2	4	no	—
71	C₂×C₄×C₂₂₇	—	2	2	no	—
72	C₂×C₄×C₂₂₇	—	2	2	no	—
73	C₂×C₄×C₂₂₇	—	4	1	no	—
74	C₂×C₄×C₂₂₇	—	4	2	no	—
75	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₂₇	#41	1	2	yes	—
84	C₂×C₄×C₂₂₇	#41	1	2	yes	—
85	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
86	C₂²×C₂₂₇	—	2	1	no	—
87	C₂²×C₂₂₇	—	2	1	no	—
88	C₂²×C₂₂₇	—	2	2	no	—
89	C₂²×C₂₂₇	—	2	2	no	—
90	C₂²×C₂₂₇	—	2	2	no	—
91	C₂²×C₂₂₇	#117	1	2	yes	—
92	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₂₂₇	—	2	2	no	—
94	C₄×C₂₂₇	—	2	1	no	—
95	C₄×C₂₂₇	—	2	2	no	—
96	C₄×C₂₂₇	—	2	1	no	—
97	C₄×C₂₂₇	—	2	1	no	—
98	C₄×C₂₂₇	—	2	2	no	—
99	C₄×C₂₂₇	#117	1	2	yes	—
100	C₄×C₂₂₇	#131	1	2	yes	—
101	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
102	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
103	C₂×C₂₂₇	—	2	2	no	—
104	C₂×C₂₂₇	—	4	2	no	—
105	C₂×C₂₂₇	C₂×#117	1	1	yes	characteristic
106	C₂×C₂₂₇	C₂×#131	1	1	yes	characteristic
107	C₂×C₂₂₇	#568	1	1	yes	characteristic
108	C₂₂₇	#2925	1	1	yes	characteristic, complemented, Sylow
109	#2925	—	227	1	no	maximal, Sylow
110	C₄×#41	D₂₂₇	1	1	yes	characteristic
111	#433	—	227	4	no	—
112	#432	—	227	2	no	—
113	#431	—	227	2	no	—
114	#423	—	227	1	no	—
115	C₂×C₄⋊C₈	—	227	1	no	—
116	#408	—	227	2	no	—
117	#405	—	227	2	no	—
118	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
119	C₂×M₁₆	—	227	2	no	—
120	C₂²×C₈	—	227	2	no	—
121	C₄×Q₈	Dic₂₂₇	1	1	yes	characteristic
122	C₄×Q₈	Dic₂₂₇	1	1	yes	characteristic
123	C₄×D₄	Dic₂₂₇	1	2	yes	—
124	C₄×D₄	Dic₂₂₇	1	1	yes	characteristic
125	C₄×D₄	Dic₂₂₇	1	2	yes	—
126	C₄×D₄	Dic₂₂₇	1	1	yes	characteristic
127	#117	C₂×D₂₂₇	1	1	yes	characteristic
128	#117	C₂×D₂₂₇	1	2	yes	—
129	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
130	C₂×C₄²	C₂×D₂₂₇	1	2	yes	—
131	C₄⋊C₈	—	227	1	no	—
132	C₄⋊C₈	—	227	1	no	—
133	C₄⋊C₈	—	227	2	no	—
134	C₄⋊C₈	—	227	2	no	—
135	C₄⋊C₈	—	227	2	no	—
136	C₂²⋊C₈	—	227	4	no	—
137	C₂²⋊C₈	—	227	4	no	—
138	#97	—	227	4	no	—
139	C₄×C₈	—	227	4	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₂×Dic₂₂₇	1	1	yes	characteristic
143	C₂×D₄	C₂×Dic₂₂₇	1	2	yes	—
144	C₂×D₄	C₂×Dic₂₂₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
146	C₂²×C₄	C₂×Dic₂₂₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
148	C₂²×C₄	C₂×Dic₂₂₇	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
150	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
151	M₁₆	—	454	4	no	—
152	C₂×C₈	—	454	4	no	—
153	C₂×C₈	—	227	2	no	—
154	C₂×C₈	—	227	4	no	—
155	C₂×C₈	—	227	2	no	—
156	C₄⋊C₄	C₂×Dic₂₂₇	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₂₂₇	1	2	yes	—
158	C₄⋊C₄	C₂×Dic₂₂₇	1	1	yes	characteristic
159	C₄⋊C₄	C₂×Dic₂₂₇	1	1	yes	characteristic
160	C₂²⋊C₄	C₂×Dic₂₂₇	1	2	yes	—
161	C₂²⋊C₄	C₂×Dic₂₂₇	1	2	yes	—
162	C₂²⋊C₄	C₂×Dic₂₂₇	1	2	yes	—
163	C₄²	C₂×Dic₂₂₇	1	2	yes	—
164	C₄²	C₂×Dic₂₂₇	1	1	yes	characteristic
165	C₄²	C₂×Dic₂₂₇	1	2	yes	—
166	C₄²	C₂×Dic₂₂₇	1	1	yes	characteristic
167	C₄²	C₂²×D₂₂₇	1	2	yes	—
168	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
169	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
170	C₂³	C₂²×Dic₂₂₇	1	2	yes	—
171	C₂³	C₂²×Dic₂₂₇	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	D₄	—	2	4	no	—
174	D₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
183	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
184	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
186	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
189	C₂×C₄	#16813884	1	2	yes	—
190	C₂×C₄	#16813884	1	2	yes	—
191	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
192	C₈	—	454	2	no	—
193	C₈	—	454	2	no	—
194	C₈	—	454	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	#20300247	1	2	yes	—
201	C₂²	C₂³×Dic₂₂₇	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	#20300247	1	2	yes	—
209	C₄	#20300295	1	2	yes	—
210	C₄	C₂×#16813884	1	1	yes	characteristic
211	C₄	C₂×#16813884	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₂×#20300247	1	1	yes	characteristic
215	C₂	C₂×#20300295	1	1	yes	characteristic
216	C₂	#28409799	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂³×C₄
Fitting	C₄×#41×C₂₂₇	C₂
prime radical	C₂×#41×C₂₂₇	C₂²
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	C₄×#41	D₂₂₇
hypocenter	C₂₂₇	#2925
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #28409093	C₂	semidirect
2. #28409089	C₂	semidirect
3. #28409085	C₂	semidirect
4. #28409056	C₂	semidirect
5. C₂×C₄⋊(C₂₂₇⋊C₈)	C₂	semidirect
7. #28408988	C₂	semidirect
108. C₂₂₇	#2925	semidirect

Maximal subgroups

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

subgroup	index
#28409093	2
#28409089	2
#28409085	2
#28409056	2
C₂×C₄⋊(C₂₂₇⋊C₈)	2
#28408999	2
#28408988	2
C₄×#41×C₂₂₇	2
#2925	227