Subgroups — #44408682

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	#28409586	C₂	1	1	yes	maximal, characteristic
2	#28409160	C₂	1	2	yes	maximal, complemented
3	#28409159	C₂	1	1	yes	maximal, characteristic, complemented
4	#28409159	C₂	1	2	yes	maximal, complemented
5	Q₈×C₂₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₄₅₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#28409057	C₂	1	2	yes	maximal
8	C₄⋊C₈×D₂₂₇	C₂	1	1	yes	maximal, characteristic
9	#28408989	C₂	1	2	yes	maximal
10	C₄×#16813867	C₂	1	1	yes	maximal, characteristic
11	#446×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16813867	C₂²	1	1	yes	characteristic
13	C₂×#16813867	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₂₂₇	C₂²	1	2	yes	—
15	#20300287	C₄	1	1	yes	characteristic
16	#20300286	C₄	1	1	yes	characteristic
17	#20300278	C₄	1	2	yes	—
18	#20300268	C₄	1	1	yes	characteristic
19	#20300266	C₄	1	2	yes	—
20	#20300264	C₄	1	1	yes	characteristic
21	C₄×Dic₄₅₄	C₂²	1	1	yes	characteristic
22	C₄×Dic₄₅₄	C₂²	1	2	yes	—
23	#20300245	C₂²	1	2	yes	—
24	C₄²×D₂₂₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
27	Dic₂₂₇⋊C₈	C₂²	1	2	yes	—
28	Dic₂₂₇⋊C₈	C₂²	1	2	yes	—
29	Dic₂₂₇⋊C₈	C₂²	1	2	yes	—
30	#20300198	C₂²	1	1	yes	characteristic
31	#20300198	C₂²	1	2	yes	—
32	#20300198	C₂²	1	1	yes	characteristic
33	C₄×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
34	C₄×C₂₂₇⋊C₈	C₂²	1	2	yes	—
35	C₈×Dic₂₂₇	C₂²	1	2	yes	—
36	C₄×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₂₂₇	C₂²	1	2	yes	—
39	#97×C₂₂₇	C₂²	1	2	yes	—
40	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₄₅₄	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₄₅₄	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₂₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₂₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₂₂₇	C₂³	1	1	yes	characteristic
46	#16813867	—	2	2	no	—
47	#16813867	Q₈	1	4	yes	complemented
48	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
49	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₈×D₂₂₇	—	2	4	no	—
52	C₄⋊Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₂₂₇	C₂×C₄	1	2	yes	—
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	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
58	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₂₂₇	C₂³	1	2	yes	—
64	C₄×Dic₂₂₇	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
67	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
70	C₄⋊C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
72	C₄²×C₂₂₇	C₂³	1	2	yes	—
73	C₂²×D₂₂₇	C₂²×C₄	1	1	yes	characteristic
74	Dic₄₅₄	—	2	2	no	—
75	Dic₄₅₄	—	4	2	no	—
76	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₄×D₂₂₇	—	2	2	no	—
82	C₄×D₂₂₇	—	4	1	no	—
83	C₄×D₂₂₇	C₂×Q₈	1	2	yes	—
84	C₄×D₂₂₇	#41	1	2	yes	—
85	C₂₂₇⋊C₈	—	2	2	no	—
86	C₂₂₇⋊C₈	—	2	1	no	—
87	C₂₂₇⋊C₈	C₂×Q₈	1	2	yes	—
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	2	no	—
95	C₈×C₂₂₇	—	2	1	no	—
96	C₈×C₂₂₇	C₂×Q₈	1	2	yes	—
97	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₂₂₇	—	2	1	no	—
99	C₂×D₂₂₇	—	2	1	no	—
100	C₂×D₂₂₇	C₄×Q₈	1	2	yes	—
101	C₄×C₂₂₇	—	2	2	no	—
102	C₄×C₂₂₇	—	2	1	no	—
103	C₄×C₂₂₇	—	2	2	no	—
104	C₄×C₂₂₇	#131	1	2	yes	—
105	C₄×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₂₂₇	C₂×#41	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	C₄×Q₈	1	2	yes	—
112	C₂×C₂₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂×C₂₂₇	C₂×#131	1	1	yes	characteristic
114	C₂×C₂₂₇	#568	1	1	yes	characteristic
115	D₂₂₇	—	2	2	no	—
116	C₂₂₇	#2962	1	1	yes	characteristic, complemented, Sylow
117	#2962	—	227	1	no	maximal, Sylow
118	#533	—	227	1	no	—
119	#446	—	227	1	no	—
120	#446	—	227	2	no	—
121	#446	—	227	2	no	—
122	#446	D₂₂₇	1	1	yes	characteristic, complemented
123	C₈×Q₈	—	227	1	no	—
124	C₈×Q₈	—	227	1	no	—
125	#424	—	227	2	no	—
126	C₂×C₄⋊C₈	—	227	1	no	—
127	#405	—	227	2	no	—
128	C₄×M₁₆	—	227	1	no	—
129	C₂×M₁₆	—	227	1	no	—
130	C₂×M₁₆	—	227	1	no	—
131	C₂²×C₈	—	227	2	no	—
132	C₄⋊Q₈	—	227	1	no	—
133	C₄⋊Q₈	—	227	1	no	—
134	#125	—	227	2	no	—
135	C₂²⋊Q₈	—	227	1	no	—
136	C₂²⋊Q₈	—	227	2	no	—
137	C₂²⋊Q₈	—	227	1	no	—
138	C₄×Q₈	—	227	1	no	—
139	C₄×Q₈	—	227	2	no	—
140	C₄×Q₈	C₂×D₂₂₇	1	1	yes	characteristic
141	#117	—	227	2	no	—
142	C₂×C₄²	—	227	1	no	—
143	C₄⋊C₈	—	227	2	no	—
144	C₄⋊C₈	—	227	2	no	—
145	C₄⋊C₈	—	227	2	no	—
146	C₄⋊C₈	—	227	1	no	—
147	C₄⋊C₈	—	227	2	no	—
148	C₄⋊C₈	C₂×D₂₂₇	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₂₂₇	1	2	yes	—
150	#97	—	227	1	no	—
151	#97	—	227	1	no	—
152	#97	—	227	2	no	—
153	#97	C₂×D₂₂₇	1	2	yes	—
154	C₄×C₈	—	227	2	no	—
155	C₄×C₈	—	227	1	no	—
156	C₄×C₈	—	227	2	no	—
157	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
158	C₂×Q₈	—	227	2	no	—
159	C₂×Q₈	—	227	1	no	—
160	C₂×Q₈	C₄×D₂₂₇	1	1	yes	characteristic
161	C₂²×C₄	—	227	1	no	—
162	C₂²×C₄	—	227	1	no	—
163	C₂²×C₄	—	227	1	no	—
164	M₁₆	—	454	2	no	—
165	M₁₆	—	227	4	no	—
166	C₂×C₈	—	454	4	no	—
167	C₂×C₈	—	227	2	no	—
168	C₂×C₈	—	227	1	no	—
169	C₂×C₈	—	227	1	no	—
170	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
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	2	no	—
178	C₄⋊C₄	—	227	2	no	—
179	C₄⋊C₄	—	227	1	no	—
180	C₄⋊C₄	C₄×D₂₂₇	1	2	yes	—
181	C₄⋊C₄	C₄×D₂₂₇	1	1	yes	characteristic
182	C₂²⋊C₄	—	227	2	no	—
183	C₂²⋊C₄	—	227	2	no	—
184	C₄²	—	227	1	no	—
185	C₄²	—	227	1	no	—
186	C₄²	—	227	2	no	—
187	C₄²	—	227	1	no	—
188	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
189	C₄²	C₂²×D₂₂₇	1	2	yes	—
190	C₂³	—	227	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	908	2	no	—
193	Q₈	—	454	2	no	—
194	C₂×C₄	—	908	1	no	—
195	C₂×C₄	—	454	2	no	—
196	C₂×C₄	—	227	1	no	—
197	C₂×C₄	—	227	1	no	—
198	C₂×C₄	—	227	2	no	—
199	C₂×C₄	—	227	2	no	—
200	C₂×C₄	—	227	1	no	—
201	C₂×C₄	—	227	2	no	—
202	C₂×C₄	—	227	2	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	—	454	2	no	—
211	C₈	—	454	1	no	—
212	C₈	—	227	2	no	—
213	C₈	Q₈×D₂₂₇	1	2	yes	—
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	#20300296	1	2	yes	—
227	C₄	C₂×Q₈×D₂₂₇	1	1	yes	characteristic
228	C₄	#41×D₂₂₇	1	1	yes	characteristic
229	C₂	—	454	2	no	—
230	C₂	C₄×Q₈×D₂₂₇	1	1	yes	characteristic
231	C₂	C₂×#20300296	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	#2962
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. #28409160	C₂	semidirect
3. #28409159	C₂	semidirect
4. #28409159	C₂	semidirect
5. Q₈×C₂₂₇⋊C₈	C₂	semidirect
6. C₈×Dic₄₅₄	C₂	semidirect
11. #446×C₂₂₇	C₂	semidirect
47. #16813867	Q₈	semidirect
116. C₂₂₇	#2962	semidirect
122. #446	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
#28409586	2
#28409160	2
#28409159	2
Q₈×C₂₂₇⋊C₈	2
C₈×Dic₄₅₄	2
#28409057	2
C₄⋊C₈×D₂₂₇	2
#28408989	2
C₄×#16813867	2
#446×C₂₂₇	2
#2962	227