Subgroups — #44408093

Order 29056.

472 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300278	C₂	1	2	yes	maximal
2	#28408931	C₂	1	1	yes	maximal, characteristic
3	#28408930	C₂	1	1	yes	maximal, characteristic
4	#28408929	C₂	1	2	yes	maximal
5	#28408928	C₂	1	1	yes	maximal, characteristic
6	#28408927	C₂	1	2	yes	maximal
7	#28408901	C₂	1	1	yes	maximal, characteristic
8	#28408901	C₂	1	1	yes	maximal, characteristic
9	#28408896	C₂	1	2	yes	maximal
10	C₄²×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
11	#389×C₂₂₇	C₂	1	1	yes	maximal, characteristic
12	#20300278	—	2	8	no	—
13	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
16	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
18	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
28	C₂₂₇⋊#95	C₂²	1	2	yes	—
29	C₂₂₇⋊#95	C₂²	1	2	yes	—
30	C₂₂₇⋊#95	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
34	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
35	#95×C₂₂₇	C₂²	1	2	yes	—
36	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
37	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
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	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₂₂₇	—	2	4	no	—
43	C₄⋊Dic₂₂₇	—	2	2	no	—
44	C₄⋊Dic₂₂₇	—	2	2	no	—
45	Dic₂₂₇⋊C₄	—	2	4	no	—
46	Dic₂₂₇⋊C₄	—	2	4	no	—
47	Dic₂₂₇⋊C₄	—	2	4	no	—
48	Dic₂₂₇⋊C₄	—	2	4	no	—
49	Dic₂₂₇⋊C₄	—	2	4	no	—
50	Dic₂₂₇⋊C₄	—	2	4	no	—
51	C₄×Dic₂₂₇	—	2	2	no	—
52	C₄×Dic₂₂₇	—	2	2	no	—
53	C₄×Dic₂₂₇	—	2	2	no	—
54	C₄×Dic₂₂₇	D₄	1	4	yes	—
55	C₄×Dic₂₂₇	Q₈	1	4	yes	—
56	C₄×Dic₂₂₇	Q₈	1	4	yes	—
57	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₂₂₇	—	2	4	no	—
63	C₄⋊C₄×C₂₂₇	—	2	2	no	—
64	C₄⋊C₄×C₂₂₇	—	2	2	no	—
65	C₄²×C₂₂₇	—	2	2	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×D₄	1	2	yes	—
77	C₂×Dic₂₂₇	C₂×D₄	1	2	yes	—
78	C₂×Dic₂₂₇	C₂×Q₈	1	2	yes	—
79	C₂×Dic₂₂₇	C₂×Q₈	1	2	yes	—
80	C₂×Dic₂₂₇	C₂×Q₈	1	2	yes	—
81	C₂×Dic₂₂₇	C₂×Q₈	1	2	yes	—
82	C₂×Dic₂₂₇	#41	1	2	yes	—
83	C₂×Dic₂₂₇	#41	1	2	yes	—
84	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₂₂₇	—	2	1	no	—
86	C₂×C₄×C₂₂₇	—	2	1	no	—
87	C₂×C₄×C₂₂₇	—	2	1	no	—
88	C₂×C₄×C₂₂₇	—	2	1	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	—	2	2	no	—
93	C₂×C₄×C₂₂₇	—	2	2	no	—
94	C₂×C₄×C₂₂₇	—	2	2	no	—
95	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
97	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₂₂₇	#41	1	2	yes	—
99	C₂×C₄×C₂₂₇	#41	1	2	yes	—
100	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
104	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
105	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
106	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
107	C₄×C₂₂₇	—	2	2	no	—
108	C₄×C₂₂₇	—	2	2	no	—
109	C₄×C₂₂₇	—	4	2	no	—
110	C₄×C₂₂₇	—	4	2	no	—
111	C₄×C₂₂₇	#125	1	4	yes	complemented
112	Dic₂₂₇	—	2	2	no	—
113	Dic₂₂₇	—	2	2	no	—
114	Dic₂₂₇	—	4	2	no	—
115	Dic₂₂₇	—	4	2	no	—
116	Dic₂₂₇	#125	1	4	yes	complemented
117	Dic₂₂₇	C₄⋊Q₈	1	4	yes	—
118	C₂×C₂₂₇	C₂×#125	1	1	yes	characteristic
119	C₂×C₂₂₇	C₂×#125	1	1	yes	characteristic
120	C₂×C₂₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₂₂₇	#532	1	1	yes	characteristic
122	C₂×C₂₂₇	#532	1	1	yes	characteristic
123	C₂×C₂₂₇	#533	1	1	yes	characteristic
124	C₂×C₂₂₇	#533	1	1	yes	characteristic
125	C₂₂₇	#2859	1	1	yes	characteristic, complemented, Sylow
126	#2859	—	227	1	no	maximal, Sylow
127	C₂×#125	—	227	2	no	—
128	#389	—	227	2	no	—
129	#389	—	227	1	no	—
130	#389	—	227	1	no	—
131	#389	—	227	1	no	—
132	#389	—	227	2	no	—
133	#389	D₂₂₇	1	1	yes	characteristic
134	#384	—	227	1	no	—
135	#384	—	227	1	no	—
136	#382	—	227	2	no	—
137	C₄³	—	227	1	no	—
138	#125	—	454	8	no	—
139	C₂×C₄⋊C₄	—	227	2	no	—
140	C₂×C₄⋊C₄	—	227	2	no	—
141	C₂×C₄⋊C₄	—	227	2	no	—
142	C₂×C₄⋊C₄	—	227	2	no	—
143	C₂×C₄⋊C₄	—	227	2	no	—
144	C₂×C₄⋊C₄	—	227	2	no	—
145	C₂×C₄⋊C₄	—	227	2	no	—
146	C₂×C₄⋊C₄	—	227	1	no	—
147	C₂×C₄⋊C₄	—	227	1	no	—
148	C₂×C₄⋊C₄	C₂×D₂₂₇	1	2	yes	—
149	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
151	C₂×C₄²	—	227	1	no	—
152	C₂×C₄²	—	227	1	no	—
153	C₂×C₄²	—	227	1	no	—
154	C₂×C₄²	—	227	1	no	—
155	C₂×C₄²	—	227	1	no	—
156	C₂×C₄²	—	227	1	no	—
157	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
158	#95	—	227	2	no	—
159	#95	—	227	2	no	—
160	#95	—	227	2	no	—
161	#95	C₂×D₂₂₇	1	2	yes	—
162	C₂²×C₄	—	227	2	no	—
163	C₂²×C₄	—	227	2	no	—
164	C₂²×C₄	—	227	1	no	—
165	C₂²×C₄	—	227	1	no	—
166	C₂²×C₄	—	227	1	no	—
167	C₂²×C₄	—	227	1	no	—
168	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
169	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
170	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	454	4	no	—
177	C₄⋊C₄	—	454	4	no	—
178	C₄⋊C₄	—	454	4	no	—
179	C₄⋊C₄	—	454	4	no	—
180	C₄⋊C₄	—	454	4	no	—
181	C₄⋊C₄	—	454	4	no	—
182	C₄⋊C₄	—	454	4	no	—
183	C₄⋊C₄	—	454	2	no	—
184	C₄⋊C₄	—	454	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	454	2	no	—
187	C₄²	—	454	2	no	—
188	C₄²	—	454	2	no	—
189	C₄²	—	227	4	no	—
190	C₄²	—	227	4	no	—
191	C₄²	—	227	4	no	—
192	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	454	2	no	—
204	C₂×C₄	—	454	2	no	—
205	C₂×C₄	—	454	2	no	—
206	C₂×C₄	—	454	2	no	—
207	C₂×C₄	—	454	2	no	—
208	C₂×C₄	—	454	2	no	—
209	C₂×C₄	—	454	1	no	—
210	C₂×C₄	—	454	1	no	—
211	C₂×C₄	—	454	1	no	—
212	C₂×C₄	—	454	1	no	—
213	C₂×C₄	—	227	2	no	—
214	C₂×C₄	—	227	2	no	—
215	C₂×C₄	—	227	2	no	—
216	C₂×C₄	—	227	2	no	—
217	C₂×C₄	—	227	2	no	—
218	C₂×C₄	—	227	2	no	—
219	C₂×C₄	—	227	2	no	—
220	C₂×C₄	—	227	2	no	—
221	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
222	C₂×C₄	Q₈×D₂₂₇	1	2	yes	—
223	C₂×C₄	Q₈×D₂₂₇	1	2	yes	—
224	C₂×C₄	#16813883	1	2	yes	—
225	C₂×C₄	#16813884	1	2	yes	—
226	C₂²	C₂×D₄×D₂₂₇	1	1	yes	characteristic
227	C₂²	C₂×Q₈×D₂₂₇	1	1	yes	characteristic
228	C₂²	C₂×Q₈×D₂₂₇	1	1	yes	characteristic
229	C₂²	#41×D₂₂₇	1	1	yes	characteristic
230	C₂²	#41×D₂₂₇	1	1	yes	characteristic
231	C₂²	C₂×#16813883	1	1	yes	characteristic
232	C₂²	C₂×#16813884	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	908	2	no	—
238	C₄	—	908	2	no	—
239	C₄	—	454	2	no	—
240	C₄	—	454	2	no	—
241	C₄	—	227	4	no	—
242	C₄	—	227	4	no	—
243	C₄	#20300278	1	4	yes	complemented
244	C₂	#125×D₂₂₇	1	1	yes	characteristic
245	C₂	C₂×#20300278	1	1	yes	characteristic
246	C₂	C₄⋊Q₈×D₂₂₇	1	1	yes	characteristic
247	C₂	#28409577	1	1	yes	characteristic
248	C₂	#28409579	1	1	yes	characteristic
249	C₂	#28409584	1	1	yes	characteristic
250	C₂	#28409586	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₂₇
commutator, socle, prime radical, cosocle	C₂³×C₂₂₇	C₂⁴
Fitting	#389×C₂₂₇	C₂
hypercenter	#389	D₂₂₇
hypocenter	C₂₂₇	#2859
LCS	G ▷ C₂³×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
111. C₄×C₂₂₇	#125	semidirect
116. Dic₂₂₇	#125	semidirect
125. C₂₂₇	#2859	semidirect
243. C₄	#20300278	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300278	2
#28408931	2
#28408930	2
#28408929	2
#28408928	2
#28408927	2
#28408901	2
#28408896	2
C₄²×Dic₂₂₇	2
#389×C₂₂₇	2
#2859	227