Subgroups — #44408816

Order 29056.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₂₂₇	C₂	1	1	yes	maximal, characteristic
2	#28409163	C₂	1	1	yes	maximal, characteristic, complemented
3	#28409162	C₂	1	1	yes	maximal, characteristic, complemented
4	#28409161	C₂	1	2	yes	maximal, complemented
5	#28409160	C₂	1	2	yes	maximal, complemented
6	#28409159	C₂	1	1	yes	maximal, characteristic, complemented
7	#28409088	C₂	1	2	yes	maximal
8	#28409087	C₂	1	2	yes	maximal
9	#28409084	C₂	1	1	yes	maximal, characteristic
10	#28409083	C₂	1	1	yes	maximal, characteristic
11	#446×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₂₇	C₄	1	1	yes	characteristic
13	Q₈×Dic₂₂₇	C₄	1	1	yes	characteristic
14	C₄×Dic₄₅₄	C₂²	1	1	yes	characteristic
15	C₄×Dic₄₅₄	C₂²	1	2	yes	—
16	#20300253	C₄	1	2	yes	—
17	#20300253	C₄	1	1	yes	characteristic
18	C₄⋊C₄×D₂₂₇	C₄	1	2	yes	—
19	C₄⋊C₄×D₂₂₇	C₄	1	1	yes	characteristic
20	C₄²×D₂₂₇	C₂²	1	2	yes	—
21	C₄²×D₂₂₇	C₂²	1	1	yes	characteristic
22	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
23	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	1	yes	characteristic
24	Dic₂₂₇⋊C₈	C₂²	1	2	yes	—
25	Dic₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₂₂₇⋊C₈	C₂²	1	2	yes	—
27	Dic₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
28	(C₂×D₂₂₇)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₂₂₇)⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₂₂₇)⋊C₈	C₂²	1	1	yes	characteristic
31	#20300199	C₂²	1	2	yes	—
32	#20300198	C₂²	1	2	yes	—
33	#20300198	C₂²	1	2	yes	—
34	C₄×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₂₂₇	C₂²	1	2	yes	—
36	C₄×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₂₂₇	C₂²	1	2	yes	—
38	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
39	#97×C₂₂₇	C₂²	1	2	yes	—
40	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₄₅₄	C₂×C₄	1	2	yes	—
42	C₂×Dic₄₅₄	C₂×C₄	1	1	yes	characteristic
43	Q₈×D₂₂₇	—	2	4	no	—
44	C₂×C₄×D₂₂₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₂₂₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₂₂₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₂₂₇	C₂³	1	1	yes	characteristic
49	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
50	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
52	C₄⋊Dic₂₂₇	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
54	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
57	C₄×Dic₂₂₇	C₂×C₄	1	2	yes	—
58	C₄×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₂₂₇	C₂×C₄	1	2	yes	—
60	C₄×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₂₂₇	C₂³	1	1	yes	characteristic
62	C₄×Dic₂₂₇	C₂³	1	2	yes	—
63	C₂×Q₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
67	C₄⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
68	C₄⋊C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
70	C₄²×C₂₂₇	C₂³	1	2	yes	—
71	C₂²×D₂₂₇	C₂²×C₄	1	1	yes	characteristic
72	Dic₄₅₄	—	2	2	no	—
73	Dic₄₅₄	—	2	4	no	—
74	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
77	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₄×D₂₂₇	—	2	4	no	—
80	C₄×D₂₂₇	—	2	1	no	—
81	C₄×D₂₂₇	—	2	1	no	—
82	C₄×D₂₂₇	—	4	2	no	—
83	C₄×D₂₂₇	—	4	1	no	—
84	C₄×D₂₂₇	M₁₆	1	4	yes	—
85	C₂₂₇⋊C₈	—	2	1	no	—
86	C₂₂₇⋊C₈	—	2	2	no	—
87	C₂₂₇⋊C₈	—	2	1	no	—
88	Q₈×C₂₂₇	—	2	2	no	—
89	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₂₇	—	2	1	no	—
95	C₈×C₂₂₇	—	2	2	no	—
96	C₈×C₂₂₇	—	2	1	no	—
97	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₂₂₇	—	2	1	no	—
99	C₂×D₂₂₇	—	2	1	no	—
100	C₂×D₂₂₇	C₂×M₁₆	1	2	yes	—
101	C₄×C₂₂₇	—	2	2	no	—
102	C₄×C₂₂₇	—	2	1	no	—
103	C₄×C₂₂₇	—	2	2	no	—
104	C₄×C₂₂₇	C₂×M₁₆	1	2	yes	—
105	C₄×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
106	C₄×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
107	Dic₂₂₇	—	2	1	no	—
108	Dic₂₂₇	—	2	2	no	—
109	Dic₂₂₇	—	2	1	no	—
110	Dic₂₂₇	—	2	2	no	—
111	Dic₂₂₇	C₂×M₁₆	1	2	yes	—
112	C₂×C₂₂₇	#519	1	1	yes	characteristic
113	C₂×C₂₂₇	C₂²×M₁₆	1	1	yes	characteristic
114	C₂×C₂₂₇	#568	1	1	yes	characteristic
115	D₂₂₇	—	2	2	no	—
116	C₂₂₇	#2983	1	1	yes	characteristic, complemented, Sylow
117	#2983	—	227	1	no	maximal, Sylow
118	C₂×C₄×Q₈	—	227	1	no	—
119	#446	—	227	1	no	—
120	#446	—	227	1	no	—
121	#446	—	227	2	no	—
122	#446	—	227	2	no	—
123	#446	—	227	1	no	—
124	#446	D₂₂₇	1	1	yes	characteristic, complemented
125	#432	—	227	2	no	—
126	#432	—	227	2	no	—
127	#431	—	227	1	no	—
128	#431	—	227	1	no	—
129	C₂²×Q₈	—	227	1	no	—
130	C₄×Q₈	—	227	1	no	—
131	C₄×Q₈	—	227	2	no	—
132	C₄×Q₈	—	227	1	no	—
133	C₄×Q₈	—	227	2	no	—
134	C₄×Q₈	—	227	1	no	—
135	C₄×Q₈	C₂×D₂₂₇	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	227	2	no	—
137	C₂×C₄⋊C₄	—	227	1	no	—
138	C₂×C₄²	—	227	1	no	—
139	C₂×C₄²	—	227	2	no	—
140	C₄⋊C₈	—	227	1	no	—
141	C₄⋊C₈	—	227	1	no	—
142	C₄⋊C₈	—	227	2	no	—
143	C₄⋊C₈	—	227	2	no	—
144	C₄⋊C₈	—	227	1	no	—
145	C₄⋊C₈	—	227	2	no	—
146	C₄⋊C₈	C₂×D₂₂₇	1	1	yes	characteristic
147	C₄⋊C₈	C₂×D₂₂₇	1	2	yes	—
148	C₂²⋊C₈	—	227	2	no	—
149	C₂²⋊C₈	—	227	1	no	—
150	C₂²⋊C₈	—	227	1	no	—
151	#97	—	227	2	no	—
152	#97	—	227	2	no	—
153	#97	—	227	2	no	—
154	#97	C₂×D₂₂₇	1	2	yes	—
155	C₄×C₈	—	227	2	no	—
156	C₄×C₈	—	227	1	no	—
157	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
158	C₂×Q₈	—	454	4	no	—
159	C₂×Q₈	—	227	1	no	—
160	C₂×Q₈	—	227	2	no	—
161	C₂×Q₈	C₄×D₂₂₇	1	1	yes	characteristic
162	C₂²×C₄	—	227	1	no	—
163	C₂²×C₄	—	227	2	no	—
164	C₂²×C₄	—	227	2	no	—
165	C₂²×C₄	—	227	1	no	—
166	C₂²×C₄	—	227	1	no	—
167	C₂×C₈	—	227	1	no	—
168	C₂×C₈	—	227	2	no	—
169	C₂×C₈	—	227	1	no	—
170	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
171	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
173	C₄⋊C₄	—	227	2	no	—
174	C₄⋊C₄	—	227	2	no	—
175	C₄⋊C₄	—	227	2	no	—
176	C₄⋊C₄	—	227	2	no	—
177	C₄⋊C₄	—	227	1	no	—
178	C₄⋊C₄	C₄×D₂₂₇	1	2	yes	—
179	C₄⋊C₄	C₄×D₂₂₇	1	1	yes	characteristic
180	C₄²	—	227	2	no	—
181	C₄²	—	227	1	no	—
182	C₄²	—	227	2	no	—
183	C₄²	—	227	1	no	—
184	C₄²	—	227	1	no	—
185	C₄²	—	227	2	no	—
186	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
187	C₄²	C₂²×D₂₂₇	1	2	yes	—
188	C₂³	—	227	1	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	454	4	no	—
191	Q₈	—	454	2	no	—
192	C₂×C₄	—	908	1	no	—
193	C₂×C₄	—	908	2	no	—
194	C₂×C₄	—	454	4	no	—
195	C₂×C₄	—	454	1	no	—
196	C₂×C₄	—	454	1	no	—
197	C₂×C₄	—	227	1	no	—
198	C₂×C₄	—	227	1	no	—
199	C₂×C₄	—	227	2	no	—
200	C₂×C₄	—	227	2	no	—
201	C₂×C₄	—	227	1	no	—
202	C₂×C₄	—	227	4	no	—
203	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	454	1	no	—
212	C₈	—	454	2	no	—
213	C₈	—	454	1	no	—
214	C₂²	—	454	1	no	—
215	C₂²	—	454	1	no	—
216	C₂²	—	227	2	no	—
217	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	454	1	no	—
222	C₄	—	454	2	no	—
223	C₄	—	454	1	no	—
224	C₄	—	454	2	no	—
225	C₄	—	227	2	no	—
226	C₄	M₁₆×D₂₂₇	1	2	yes	—
227	C₄	#20300335	1	1	yes	characteristic
228	C₄	#20300336	1	1	yes	characteristic
229	C₂	—	454	2	no	—
230	C₂	#28409508	1	1	yes	characteristic
231	C₂	C₂×M₁₆×D₂₂₇	1	1	yes	characteristic
232	C₂	#28409797	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂³×C₄
Fitting	#446×C₂₂₇	C₂
prime radical	C₂²×D₂₂₇	C₂²×C₄
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#446	D₂₂₇
hypocenter	C₂₂₇	#2983
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #28409163	C₂	semidirect
3. #28409162	C₂	semidirect
4. #28409161	C₂	semidirect
5. #28409160	C₂	semidirect
6. #28409159	C₂	semidirect
11. #446×C₂₂₇	C₂	semidirect
116. C₂₂₇	#2983	semidirect
124. #446	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₂₂₇	2
#28409163	2
#28409162	2
#28409161	2
#28409160	2
#28409159	2
#28409088	2
#28409087	2
#28409084	2
#28409083	2
#446×C₂₂₇	2
#2983	227