Subgroups — #44408113

Order 29056.

408 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28408939	C₂	1	1	yes	maximal, characteristic
2	#28408939	C₂	1	1	yes	maximal, characteristic
3	#28408931	C₂	1	1	yes	maximal, characteristic
4	#28408930	C₂	1	1	yes	maximal, characteristic
5	#28408928	C₂	1	1	yes	maximal, characteristic
6	#28408919	C₂	1	2	yes	maximal
7	#28408917	C₂	1	2	yes	maximal
8	#28408914	C₂	1	2	yes	maximal
9	#28408899	C₂	1	2	yes	maximal
10	C₄²×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
11	#391×C₂₂₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
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	2	yes	—
25	C₂₂₇⋊#95	C₂²	1	2	yes	—
26	C₂₂₇⋊#95	C₂²	1	2	yes	—
27	C₂₂₇⋊#95	C₂²	1	2	yes	—
28	C₂₂₇⋊#95	C₂²	1	2	yes	—
29	C₂₂₇⋊#95	C₂²	1	2	yes	—
30	C₂²×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
33	#95×C₂₂₇	C₂²	1	2	yes	—
34	#95×C₂₂₇	C₂²	1	2	yes	—
35	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
36	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
37	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
38	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
41	C₄⋊Dic₂₂₇	—	2	4	no	—
42	C₄⋊Dic₂₂₇	—	2	2	no	—
43	C₄⋊Dic₂₂₇	—	2	2	no	—
44	Dic₂₂₇⋊C₄	—	2	4	no	—
45	Dic₂₂₇⋊C₄	—	2	4	no	—
46	C₄×Dic₂₂₇	—	2	2	no	—
47	C₄×Dic₂₂₇	—	2	2	no	—
48	C₄×Dic₂₂₇	—	2	2	no	—
49	C₄×Dic₂₂₇	—	2	2	no	—
50	C₄×Dic₂₂₇	—	2	2	no	—
51	C₄×Dic₂₂₇	Q₈	1	4	yes	complemented
52	C₂×Q₈×C₂₂₇	—	2	2	no	—
53	C₂×Q₈×C₂₂₇	—	2	2	no	—
54	C₂×Q₈×C₂₂₇	—	2	2	no	—
55	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₂₇	—	2	2	no	—
61	C₄²×C₂₂₇	—	2	2	no	—
62	C₂×Dic₂₂₇	—	2	1	no	—
63	C₂×Dic₂₂₇	—	2	1	no	—
64	C₂×Dic₂₂₇	—	2	1	no	—
65	C₂×Dic₂₂₇	—	2	1	no	—
66	C₂×Dic₂₂₇	—	2	1	no	—
67	C₂×Dic₂₂₇	—	2	1	no	—
68	C₂×Dic₂₂₇	—	2	1	no	—
69	C₂×Dic₂₂₇	—	2	1	no	—
70	C₂×Dic₂₂₇	—	2	2	no	—
71	C₂×Dic₂₂₇	—	2	2	no	—
72	C₂×Dic₂₂₇	—	2	2	no	—
73	C₂×Dic₂₂₇	—	2	2	no	—
74	C₂×Dic₂₂₇	—	2	2	no	—
75	C₂×Dic₂₂₇	—	2	2	no	—
76	C₂×Dic₂₂₇	C₂×Q₈	1	2	yes	—
77	C₂×Dic₂₂₇	C₂×Q₈	1	2	yes	—
78	C₂×Dic₂₂₇	#41	1	2	yes	—
79	C₂×Dic₂₂₇	#41	1	2	yes	—
80	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
81	Q₈×C₂₂₇	—	4	4	no	—
82	C₂×C₄×C₂₂₇	—	2	1	no	—
83	C₂×C₄×C₂₂₇	—	2	1	no	—
84	C₂×C₄×C₂₂₇	—	2	1	no	—
85	C₂×C₄×C₂₂₇	—	2	1	no	—
86	C₂×C₄×C₂₂₇	—	2	2	no	—
87	C₂×C₄×C₂₂₇	—	2	2	no	—
88	C₂×C₄×C₂₂₇	—	2	2	no	—
89	C₂×C₄×C₂₂₇	—	2	2	no	—
90	C₂×C₄×C₂₂₇	—	2	2	no	—
91	C₂×C₄×C₂₂₇	—	2	2	no	—
92	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
93	C₂×C₄×C₂₂₇	#41	1	2	yes	—
94	C₂×C₄×C₂₂₇	#41	1	2	yes	—
95	C₂×C₄×C₂₂₇	#41	1	2	yes	—
96	C₂×C₄×C₂₂₇	#41	1	2	yes	—
97	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
99	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
101	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
102	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
103	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
104	C₄×C₂₂₇	—	2	2	no	—
105	C₄×C₂₂₇	—	2	2	no	—
106	C₄×C₂₂₇	—	4	2	no	—
107	C₄×C₂₂₇	—	4	2	no	—
108	C₄×C₂₂₇	#126	1	4	yes	—
109	Dic₂₂₇	—	2	2	no	—
110	Dic₂₂₇	—	2	2	no	—
111	Dic₂₂₇	—	2	2	no	—
112	Dic₂₂₇	—	2	2	no	—
113	Dic₂₂₇	—	4	2	no	—
114	Dic₂₂₇	—	4	2	no	—
115	C₂×C₂₂₇	C₂×#126	1	1	yes	characteristic
116	C₂×C₂₂₇	#529	1	1	yes	characteristic
117	C₂×C₂₂₇	#529	1	1	yes	characteristic
118	C₂×C₂₂₇	#529	1	1	yes	characteristic
119	C₂×C₂₂₇	#533	1	1	yes	characteristic
120	C₂×C₂₂₇	#533	1	1	yes	characteristic
121	C₂×C₂₂₇	#533	1	1	yes	characteristic
122	C₂₂₇	#2863	1	1	yes	characteristic, complemented, Sylow
123	#2863	—	227	1	no	maximal, Sylow
124	#391	—	227	1	no	—
125	#391	—	227	1	no	—
126	#391	D₂₂₇	1	1	yes	characteristic
127	#389	—	227	1	no	—
128	#389	—	227	1	no	—
129	#389	—	227	1	no	—
130	#387	—	227	2	no	—
131	#387	—	227	2	no	—
132	#387	—	227	2	no	—
133	#383	—	227	2	no	—
134	C₄³	—	227	1	no	—
135	C₂²×Q₈	C₂×D₂₂₇	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	227	2	no	—
137	C₂×C₄⋊C₄	—	227	2	no	—
138	C₂×C₄⋊C₄	—	227	2	no	—
139	C₂×C₄⋊C₄	—	227	1	no	—
140	C₂×C₄⋊C₄	—	227	1	no	—
141	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
142	C₂×C₄²	—	227	1	no	—
143	C₂×C₄²	—	227	1	no	—
144	C₂×C₄²	—	227	1	no	—
145	C₂×C₄²	—	227	1	no	—
146	C₂×C₄²	—	227	1	no	—
147	C₂×C₄²	—	227	1	no	—
148	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
149	#95	—	227	2	no	—
150	#95	—	227	2	no	—
151	#95	—	227	2	no	—
152	#95	—	227	2	no	—
153	#95	—	227	2	no	—
154	#95	—	227	2	no	—
155	#95	—	227	2	no	—
156	#95	C₂×D₂₂₇	1	2	yes	—
157	#95	C₂×D₂₂₇	1	2	yes	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂²×C₄	—	227	2	no	—
162	C₂²×C₄	—	227	2	no	—
163	C₂²×C₄	—	227	1	no	—
164	C₂²×C₄	—	227	1	no	—
165	C₂²×C₄	—	227	1	no	—
166	C₂²×C₄	—	227	1	no	—
167	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
168	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
169	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	454	4	no	—
174	C₄⋊C₄	—	454	4	no	—
175	C₄⋊C₄	—	454	2	no	—
176	C₄⋊C₄	—	454	2	no	—
177	C₄⋊C₄	—	454	4	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	454	2	no	—
180	C₄²	—	454	2	no	—
181	C₄²	—	454	2	no	—
182	C₄²	—	454	2	no	—
183	C₄²	—	454	2	no	—
184	C₄²	—	227	4	no	—
185	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
186	Q₈	—	4	4	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	454	2	no	—
198	C₂×C₄	—	454	2	no	—
199	C₂×C₄	—	454	2	no	—
200	C₂×C₄	—	454	2	no	—
201	C₂×C₄	—	454	2	no	—
202	C₂×C₄	—	454	2	no	—
203	C₂×C₄	—	454	1	no	—
204	C₂×C₄	—	454	1	no	—
205	C₂×C₄	—	454	1	no	—
206	C₂×C₄	—	454	1	no	—
207	C₂×C₄	—	454	1	no	—
208	C₂×C₄	—	454	1	no	—
209	C₂×C₄	—	454	1	no	—
210	C₂×C₄	—	454	1	no	—
211	C₂×C₄	—	227	2	no	—
212	C₂×C₄	—	227	2	no	—
213	C₂×C₄	—	227	2	no	—
214	C₂×C₄	—	227	2	no	—
215	C₂×C₄	Q₈×D₂₂₇	1	2	yes	—
216	C₂×C₄	#16813883	1	2	yes	—
217	C₂×C₄	#16813883	1	2	yes	—
218	C₂×C₄	#16813884	1	2	yes	—
219	C₂×C₄	#16813885	1	2	yes	—
220	C₂²	C₂×Q₈×D₂₂₇	1	1	yes	characteristic
221	C₂²	#41×D₂₂₇	1	1	yes	characteristic
222	C₂²	#41×D₂₂₇	1	1	yes	characteristic
223	C₂²	C₂×#16813883	1	1	yes	characteristic
224	C₂²	C₂×#16813883	1	1	yes	characteristic
225	C₂²	C₂×#16813884	1	1	yes	characteristic
226	C₂²	C₂×#16813885	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	908	2	no	—
232	C₄	—	908	2	no	—
233	C₄	—	454	2	no	—
234	C₄	—	454	2	no	—
235	C₄	—	454	2	no	—
236	C₄	—	454	2	no	—
237	C₄	#20300281	1	4	yes	—
238	C₂	C₂×#20300281	1	1	yes	characteristic
239	C₂	#28409561	1	1	yes	characteristic
240	C₂	#28409568	1	1	yes	characteristic
241	C₂	#28409569	1	1	yes	characteristic
242	C₂	#28409584	1	1	yes	characteristic
243	C₂	#28409586	1	1	yes	characteristic
244	C₂	#28409587	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₂₇
commutator, socle, prime radical, cosocle	C₂³×C₂₂₇	C₂⁴
Fitting	#391×C₂₂₇	C₂
hypercenter	#391	D₂₂₇
hypocenter	C₂₂₇	#2863
LCS	G ▷ C₂³×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
51. C₄×Dic₂₂₇	Q₈	semidirect
122. C₂₂₇	#2863	semidirect

Maximal subgroups

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

subgroup	index
#28408939	2
#28408931	2
#28408930	2
#28408928	2
#28408919	2
#28408917	2
#28408914	2
#28408899	2
C₄²×Dic₂₂₇	2
#391×C₂₂₇	2
#2863	227