Subgroups — #2453×C₂₂₇ #44396409

Order 29056.

452 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×#125×C₂₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
3	#400×C₂₂₇	C₂	1	2	yes	maximal
4	#398×C₂₂₇	C₂	1	2	yes	maximal
5	#391×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	#391×C₂₂₇	C₂	1	1	yes	maximal, characteristic
7	#389×C₂₂₇	C₂	1	1	yes	maximal, characteristic
8	#389×C₂₂₇	C₂	1	2	yes	maximal
9	#389×C₂₂₇	C₂	1	1	yes	maximal, characteristic
10	#387×C₂₂₇	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
13	#125×C₂₂₇	—	2	4	no	—
14	C₄×Q₈×C₂₂₇	—	2	4	no	—
15	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₂₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
29	#95×C₂₂₇	C₂²	1	1	yes	characteristic
30	#95×C₂₂₇	C₂²	1	1	yes	characteristic
31	#95×C₂₂₇	C₂²	1	2	yes	—
32	#95×C₂₂₇	C₂²	1	2	yes	—
33	#95×C₂₂₇	C₂²	1	2	yes	—
34	#95×C₂₂₇	C₂²	1	2	yes	—
35	#95×C₂₂₇	C₂²	1	2	yes	—
36	#95×C₂₂₇	C₂²	1	2	yes	—
37	C₂×Q₈×C₂₂₇	—	2	2	no	—
38	C₂×Q₈×C₂₂₇	—	2	2	no	—
39	C₂×Q₈×C₂₂₇	Q₈	1	4	yes	—
40	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₂₇	—	2	4	no	—
52	C₄⋊C₄×C₂₂₇	—	2	4	no	—
53	C₄⋊C₄×C₂₂₇	—	2	4	no	—
54	C₄⋊C₄×C₂₂₇	—	2	2	no	—
55	C₄⋊C₄×C₂₂₇	—	2	2	no	—
56	C₄⋊C₄×C₂₂₇	—	2	4	no	—
57	C₄⋊C₄×C₂₂₇	—	2	2	no	—
58	C₄⋊C₄×C₂₂₇	—	2	4	no	—
59	C₄⋊C₄×C₂₂₇	—	2	2	no	—
60	C₄⋊C₄×C₂₂₇	D₄	1	4	yes	—
61	C₄²×C₂₂₇	—	2	4	no	—
62	C₄²×C₂₂₇	—	2	2	no	—
63	C₄²×C₂₂₇	—	2	2	no	—
64	C₄²×C₂₂₇	—	2	2	no	—
65	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
66	Q₈×C₂₂₇	—	2	8	no	—
67	C₂×C₄×C₂₂₇	—	2	1	no	—
68	C₂×C₄×C₂₂₇	—	2	2	no	—
69	C₂×C₄×C₂₂₇	—	2	1	no	—
70	C₂×C₄×C₂₂₇	—	2	1	no	—
71	C₂×C₄×C₂₂₇	—	2	2	no	—
72	C₂×C₄×C₂₂₇	—	2	2	no	—
73	C₂×C₄×C₂₂₇	—	2	2	no	—
74	C₂×C₄×C₂₂₇	—	2	1	no	—
75	C₂×C₄×C₂₂₇	—	2	1	no	—
76	C₂×C₄×C₂₂₇	—	2	1	no	—
77	C₂×C₄×C₂₂₇	—	2	1	no	—
78	C₂×C₄×C₂₂₇	—	2	2	no	—
79	C₂×C₄×C₂₂₇	—	2	1	no	—
80	C₂×C₄×C₂₂₇	—	2	2	no	—
81	C₂×C₄×C₂₂₇	—	2	2	no	—
82	C₂×C₄×C₂₂₇	—	2	2	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	1	no	—
88	C₂×C₄×C₂₂₇	—	2	1	no	—
89	C₂×C₄×C₂₂₇	—	2	1	no	—
90	C₂×C₄×C₂₂₇	—	2	1	no	—
91	C₂×C₄×C₂₂₇	—	2	1	no	—
92	C₂×C₄×C₂₂₇	—	2	2	no	—
93	C₂×C₄×C₂₂₇	C₂×D₄	1	4	yes	—
94	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₂₇	C₂×Q₈	1	4	yes	—
96	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
97	C₂×C₄×C₂₂₇	#41	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	C₂×#41	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	2₋¹⁺⁴	1	1	yes	characteristic
106	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₂₂₇	—	2	4	no	—
108	C₄×C₂₂₇	—	2	2	no	—
109	C₄×C₂₂₇	—	2	4	no	—
110	C₄×C₂₂₇	—	2	2	no	—
111	C₄×C₂₂₇	—	4	2	no	—
112	C₄×C₂₂₇	—	4	1	no	—
113	C₄×C₂₂₇	—	4	1	no	—
114	C₄×C₂₂₇	—	4	1	no	—
115	C₄×C₂₂₇	—	4	1	no	—
116	C₄×C₂₂₇	—	4	2	no	—
117	C₄×C₂₂₇	C₂²⋊Q₈	1	4	yes	—
118	C₂×C₂₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₂₂₇	#529	1	1	yes	characteristic
120	C₂×C₂₂₇	#541	1	1	yes	characteristic
121	C₂×C₂₂₇	#547	1	1	yes	characteristic
122	C₂×C₂₂₇	#548	1	1	yes	characteristic
123	C₂×C₂₂₇	#557	1	1	yes	characteristic
124	C₂×C₂₂₇	#557	1	1	yes	characteristic
125	C₂₂₇	#2453	1	1	yes	characteristic, direct product factor, Sylow
126	#2453	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×#125	C₂×C₂₂₇	1	1	yes	characteristic
128	C₂×C₄×Q₈	C₂×C₂₂₇	1	1	yes	characteristic
129	#400	C₂×C₂₂₇	1	2	yes	—
130	#398	C₂×C₂₂₇	1	2	yes	—
131	#391	C₂×C₂₂₇	1	1	yes	characteristic
132	#391	C₂×C₂₂₇	1	1	yes	characteristic
133	#389	C₂×C₂₂₇	1	1	yes	characteristic
134	#389	C₂×C₂₂₇	1	2	yes	—
135	#389	C₂×C₂₂₇	1	1	yes	characteristic
136	#387	C₂×C₂₂₇	1	2	yes	—
137	C₄×C₄⋊C₄	C₂×C₂₂₇	1	1	yes	characteristic
138	C₂²×Q₈	C₂²×C₂₂₇	1	1	yes	characteristic
139	#125	—	2	4	no	—
140	C₄×Q₈	—	2	4	no	—
141	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
142	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
143	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
148	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
151	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
152	C₂×C₄²	C₂²×C₂₂₇	1	2	yes	—
153	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
155	#95	C₂²×C₂₂₇	1	1	yes	characteristic
156	#95	C₂²×C₂₂₇	1	1	yes	characteristic
157	#95	C₂²×C₂₂₇	1	2	yes	—
158	#95	C₂²×C₂₂₇	1	2	yes	—
159	#95	C₂²×C₂₂₇	1	2	yes	—
160	#95	C₂²×C₂₂₇	1	2	yes	—
161	#95	C₂²×C₂₂₇	1	2	yes	—
162	#95	C₂²×C₂₂₇	1	2	yes	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	Q₈×C₂₂₇	1	4	yes	—
166	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
168	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
172	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
173	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
174	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	D₄×C₂₂₇	1	4	yes	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₂³	C₂⁴×C₂₂₇	1	1	yes	characteristic
192	Q₈	—	2	8	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	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	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	C₂×D₄×C₂₂₇	1	4	yes	—
220	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
221	C₂×C₄	C₂×Q₈×C₂₂₇	1	4	yes	—
222	C₂×C₄	C₂×Q₈×C₂₂₇	1	2	yes	—
223	C₂×C₄	#41×C₂₂₇	1	2	yes	—
224	C₂×C₄	#41×C₂₂₇	1	2	yes	—
225	C₂×C₄	#41×C₂₂₇	1	2	yes	—
226	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
227	C₂²	C₂²×Q₈×C₂₂₇	1	1	yes	characteristic
228	C₂²	C₂×#41×C₂₂₇	1	1	yes	characteristic
229	C₂²	C₂×#41×C₂₂₇	1	1	yes	characteristic
230	C₂²	C₂×#41×C₂₂₇	1	1	yes	characteristic
231	C₂²	2₋¹⁺⁴×C₂₂₇	1	1	yes	characteristic
232	C₂²	2₋¹⁺⁴×C₂₂₇	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	2	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	2	no	—
243	C₄	C₂²⋊Q₈×C₂₂₇	1	4	yes	—
244	C₂	C₂×C₂²⋊Q₈×C₂₂₇	1	1	yes	characteristic
245	C₂	#529×C₂₂₇	1	1	yes	characteristic
246	C₂	#541×C₂₂₇	1	1	yes	characteristic
247	C₂	#547×C₂₂₇	1	1	yes	characteristic
248	C₂	#548×C₂₂₇	1	1	yes	characteristic
249	C₂	#557×C₂₂₇	1	1	yes	characteristic
250	C₂	#557×C₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₂₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₂₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
125. C₂₂₇	#2453	direct (N × H)
126. #2453	C₂₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#125×C₂₂₇	2
C₂×C₄×Q₈×C₂₂₇	2
#400×C₂₂₇	2
#398×C₂₂₇	2
#391×C₂₂₇	2
#389×C₂₂₇	2
#387×C₂₂₇	2
C₄×C₄⋊C₄×C₂₂₇	2
#2453	227

Subgroups — #2453×C227 #44396409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2453×C₂₂₇ #44396409