Subgroups — #44397990

Order 29056.

520 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₂₇⋊#95	C₂	1	2	yes	maximal
3	C₂×C₂₂₇⋊#95	C₂	1	2	yes	maximal
4	C₂×C₂₂₇⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×#95×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₂₂₇	C₂²	1	2	yes	—
7	C₂³×Dic₂₂₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₂₂₇	—	2	4	no	—
9	C₂×C₄⋊Dic₂₂₇	—	2	2	no	—
10	C₂₂₇⋊#95	—	2	4	no	—
11	C₂₂₇⋊#95	—	2	4	no	—
12	C₂₂₇⋊#95	C₄	1	4	yes	complemented
13	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
16	#95×C₂₂₇	—	2	2	no	—
17	C₂²×Dic₂₂₇	—	2	2	no	—
18	C₂²×Dic₂₂₇	—	2	2	no	—
19	C₂²×Dic₂₂₇	—	2	2	no	—
20	C₂²×Dic₂₂₇	—	2	2	no	—
21	C₂²×Dic₂₂₇	—	2	2	no	—
22	C₂²×Dic₂₂₇	—	2	2	no	—
23	C₂²×Dic₂₂₇	—	2	2	no	—
24	C₂²×Dic₂₂₇	—	2	2	no	—
25	C₂²×Dic₂₂₇	—	2	2	no	—
26	C₂²×Dic₂₂₇	—	2	2	no	—
27	C₂²×Dic₂₂₇	—	2	2	no	—
28	C₂²×Dic₂₂₇	—	2	2	no	—
29	C₂²×Dic₂₂₇	C₂×C₄	1	4	yes	—
30	C₂²×Dic₂₂₇	D₄	1	4	yes	—
31	C₄⋊Dic₂₂₇	—	2	8	no	—
32	C₂⁴×C₂₂₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₂₇	—	2	2	no	—
34	C₂²×C₄×C₂₂₇	—	2	2	no	—
35	C₂²×C₄×C₂₂₇	—	2	1	no	—
36	C₂²×C₄×C₂₂₇	—	2	1	no	—
37	C₂²×C₄×C₂₂₇	—	2	2	no	—
38	C₂²×C₄×C₂₂₇	—	2	2	no	—
39	C₂²×C₄×C₂₂₇	—	2	2	no	—
40	C₂²×C₄×C₂₂₇	—	2	2	no	—
41	C₂²×C₄×C₂₂₇	—	2	1	no	—
42	C₂²×C₄×C₂₂₇	—	2	1	no	—
43	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₂₂₇	D₄	1	2	yes	—
45	C₂²×C₄×C₂₂₇	D₄	1	2	yes	—
46	C₂²×C₄×C₂₂₇	Q₈	1	2	yes	—
47	C₂²×C₄×C₂₂₇	Q₈	1	2	yes	—
48	C₂×Dic₂₂₇	—	2	4	no	—
49	C₂×Dic₂₂₇	—	2	4	no	—
50	C₂×Dic₂₂₇	—	2	4	no	—
51	C₂×Dic₂₂₇	—	2	4	no	—
52	C₂×Dic₂₂₇	—	2	4	no	—
53	C₂×Dic₂₂₇	—	2	4	no	—
54	C₂×Dic₂₂₇	—	4	2	no	—
55	C₂×Dic₂₂₇	—	4	2	no	—
56	C₂×Dic₂₂₇	—	4	2	no	—
57	C₂×Dic₂₂₇	—	4	2	no	—
58	C₂×Dic₂₂₇	—	4	2	no	—
59	C₂×Dic₂₂₇	—	4	2	no	—
60	C₂×Dic₂₂₇	—	4	2	no	—
61	C₂×Dic₂₂₇	—	4	2	no	—
62	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
63	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
64	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
65	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
67	C₂³×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
68	C₂³×C₂₂₇	#41	1	2	yes	—
69	C₂³×C₂₂₇	#41	1	2	yes	—
70	C₂³×C₂₂₇	#41	1	2	yes	—
71	C₂³×C₂₂₇	#41	1	1	yes	characteristic
72	C₂³×C₂₂₇	#41	1	1	yes	characteristic
73	C₂×C₄×C₂₂₇	—	2	4	no	—
74	C₂×C₄×C₂₂₇	—	2	2	no	—
75	C₂×C₄×C₂₂₇	—	2	4	no	—
76	C₂×C₄×C₂₂₇	—	2	2	no	—
77	C₂×C₄×C₂₂₇	—	2	2	no	—
78	C₂×C₄×C₂₂₇	—	2	4	no	—
79	C₂×C₄×C₂₂₇	—	2	2	no	—
80	C₂×C₄×C₂₂₇	—	2	4	no	—
81	C₂×C₄×C₂₂₇	—	4	1	no	—
82	C₂×C₄×C₂₂₇	—	4	1	no	—
83	C₂×C₄×C₂₂₇	—	4	2	no	—
84	C₂×C₄×C₂₂₇	—	4	1	no	—
85	C₂×C₄×C₂₂₇	—	4	1	no	—
86	C₂×C₄×C₂₂₇	—	4	2	no	—
87	C₂×C₄×C₂₂₇	C₄⋊C₄	1	4	yes	—
88	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₂₂₇	#117	1	2	yes	—
90	C₂²×C₂₂₇	C₄×D₄	1	2	yes	—
91	C₂²×C₂₂₇	C₄×D₄	1	1	yes	characteristic
92	C₂²×C₂₂₇	C₄×Q₈	1	1	yes	characteristic
93	C₂²×C₂₂₇	#120	1	1	yes	characteristic
94	C₂²×C₂₂₇	#121	1	1	yes	characteristic
95	C₂²×C₂₂₇	#121	1	2	yes	—
96	C₂²×C₂₂₇	C₂²⋊Q₈	1	2	yes	—
97	C₂²×C₂₂₇	C₂²⋊Q₈	1	2	yes	—
98	C₂²×C₂₂₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂²×C₂₂₇	#123	1	1	yes	characteristic
100	C₂²×C₂₂₇	#123	1	2	yes	—
101	C₂²×C₂₂₇	#123	1	2	yes	—
102	C₂²×C₂₂₇	#124	1	2	yes	—
103	C₂²×C₂₂₇	#124	1	1	yes	characteristic
104	C₂²×C₂₂₇	#124	1	1	yes	characteristic
105	C₂²×C₂₂₇	#125	1	2	yes	—
106	C₂²×C₂₂₇	#125	1	2	yes	—
107	C₂²×C₂₂₇	#125	1	1	yes	characteristic
108	C₂²×C₂₂₇	#126	1	2	yes	—
109	C₂²×C₂₂₇	#126	1	2	yes	—
110	C₂²×C₂₂₇	C₄⋊Q₈	1	1	yes	characteristic
111	C₄×C₂₂₇	—	2	4	no	—
112	C₄×C₂₂₇	—	4	2	no	—
113	C₄×C₂₂₇	—	4	2	no	—
114	Dic₂₂₇	—	4	4	no	—
115	Dic₂₂₇	—	4	4	no	—
116	C₂×C₂₂₇	#382	1	1	yes	characteristic
117	C₂×C₂₂₇	#385	1	1	yes	characteristic
118	C₂×C₂₂₇	#387	1	2	yes	—
119	C₂×C₂₂₇	#388	1	2	yes	—
120	C₂×C₂₂₇	#389	1	1	yes	characteristic
121	C₂×C₂₂₇	#393	1	1	yes	characteristic
122	C₂×C₂₂₇	#394	1	1	yes	characteristic
123	C₂×C₂₂₇	#397	1	1	yes	characteristic
124	C₂×C₂₂₇	#398	1	2	yes	—
125	C₂×C₂₂₇	#400	1	1	yes	characteristic
126	C₂×C₂₂₇	#400	1	2	yes	—
127	C₂₂₇	#1442	1	1	yes	characteristic, complemented, Sylow
128	#1442	—	227	1	no	maximal, Sylow
129	C₂²×C₄⋊C₄	—	227	1	no	—
130	C₂×#95	—	227	2	no	—
131	C₂×#95	—	227	2	no	—
132	C₂×#95	—	227	1	no	—
133	C₂×#95	D₂₂₇	1	1	yes	characteristic
134	C₂³×C₄	—	227	2	no	—
135	C₂³×C₄	—	227	2	no	—
136	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
137	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
138	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	454	4	no	—
140	C₂×C₄⋊C₄	—	454	2	no	—
141	#95	—	2	2	no	—
142	#95	—	454	4	no	—
143	#95	—	454	4	no	—
144	#95	—	227	4	no	—
145	C₂⁴	C₂²×D₂₂₇	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	454	2	no	—
157	C₂²×C₄	—	454	2	no	—
158	C₂²×C₄	—	454	2	no	—
159	C₂²×C₄	—	454	2	no	—
160	C₂²×C₄	—	454	2	no	—
161	C₂²×C₄	—	454	2	no	—
162	C₂²×C₄	—	454	2	no	—
163	C₂²×C₄	—	454	2	no	—
164	C₂²×C₄	—	454	2	no	—
165	C₂²×C₄	—	454	2	no	—
166	C₂²×C₄	—	454	2	no	—
167	C₂²×C₄	—	454	2	no	—
168	C₂²×C₄	—	227	4	no	—
169	C₂²×C₄	—	227	4	no	—
170	C₂²×C₄	C₄×D₂₂₇	1	2	yes	—
171	C₂²×C₄	C₂²⋊D₂₂₇	1	2	yes	—
172	C₂²×C₄	D₉₀₈	1	2	yes	—
173	C₂²×C₄	Dic₄₅₄	1	2	yes	—
174	C₂²×C₄	Dic₄₅₄	1	2	yes	—
175	C₄⋊C₄	—	454	8	no	—
176	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
177	C₂³	D₄×D₂₂₇	1	2	yes	—
178	C₂³	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
179	C₂³	C₂×D₉₀₈	1	1	yes	characteristic
180	C₂³	C₂×Dic₄₅₄	1	1	yes	characteristic
181	C₂³	C₂×Dic₄₅₄	1	1	yes	characteristic
182	C₂³	#16813883	1	2	yes	—
183	C₂³	#16813883	1	2	yes	—
184	C₂³	#16813884	1	1	yes	characteristic
185	C₂³	#16813884	1	1	yes	characteristic
186	C₂³	#16813885	1	2	yes	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	908	2	no	—
202	C₂×C₄	—	908	2	no	—
203	C₂×C₄	—	908	2	no	—
204	C₂×C₄	—	908	2	no	—
205	C₂×C₄	—	908	2	no	—
206	C₂×C₄	—	908	2	no	—
207	C₂×C₄	—	908	2	no	—
208	C₂×C₄	—	908	2	no	—
209	C₂×C₄	—	454	4	no	—
210	C₂×C₄	—	454	4	no	—
211	C₂×C₄	—	454	4	no	—
212	C₂×C₄	—	454	4	no	—
213	C₂×C₄	—	454	4	no	—
214	C₂×C₄	—	454	4	no	—
215	C₂×C₄	Dic₂₂₇⋊C₄	1	4	yes	—
216	C₂²	C₂×Dic₂₂₇⋊C₄	1	1	yes	characteristic
217	C₂²	#20300246	1	2	yes	—
218	C₂²	(C₂²⋊D₂₂₇)⋊C₄	1	2	yes	—
219	C₂²	C₄×D₉₀₈	1	1	yes	characteristic
220	C₂²	C₄×Dic₄₅₄	1	1	yes	characteristic
221	C₂²	#20300256	1	1	yes	characteristic
222	C₂²	#20300262	1	1	yes	characteristic
223	C₂²	#20300263	1	2	yes	—
224	C₂²	#20300265	1	2	yes	—
225	C₂²	#20300265	1	2	yes	—
226	C₂²	#20300267	1	1	yes	characteristic
227	C₂²	#20300269	1	1	yes	characteristic
228	C₂²	#20300272	1	2	yes	—
229	C₂²	#20300273	1	2	yes	—
230	C₂²	#20300274	1	2	yes	—
231	C₂²	#20300275	1	1	yes	characteristic
232	C₂²	#20300277	1	1	yes	characteristic
233	C₂²	#20300279	1	2	yes	—
234	C₂²	#20300279	1	2	yes	—
235	C₂²	#20300280	1	1	yes	characteristic
236	C₂²	#20300281	1	2	yes	—
237	C₂²	#20300283	1	2	yes	—
238	C₂²	#20300288	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	908	4	no	—
243	C₄	—	908	4	no	—
244	C₂	#28408897	1	1	yes	characteristic
245	C₂	#28408904	1	1	yes	characteristic
246	C₂	#28408919	1	2	yes	—
247	C₂	#28408924	1	2	yes	—
248	C₂	#28408930	1	1	yes	characteristic
249	C₂	#28408944	1	1	yes	characteristic
250	C₂	#28408949	1	1	yes	characteristic
251	C₂	#28408957	1	1	yes	characteristic
252	C₂	#28408964	1	2	yes	—
253	C₂	#28408971	1	1	yes	characteristic
254	C₂	#28408974	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₂₂₇
commutator	C₂³×C₂₂₇	C₂²×C₄
Fitting	C₂×#95×C₂₂₇	C₂
socle, prime radical, cosocle	C₂⁴×C₂₂₇	C₂³
hypercenter	C₂×#95	D₂₂₇
hypocenter	C₂₂₇	#1442
LCS	G ▷ C₂³×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
12. C₂₂₇⋊#95	C₄	semidirect
127. C₂₂₇	#1442	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₂₂₇	2
C₂×C₂₂₇⋊#95	2
C₂×#95×C₂₂₇	2
#1442	227