Subgroups — #2451×C₂₂₇ #44396407

Order 29056.

484 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
3	#398×C₂₂₇	C₂	1	2	yes	maximal
4	#395×C₂₂₇	C₂	1	2	yes	maximal
5	#391×C₂₂₇	C₂	1	2	yes	maximal
6	#391×C₂₂₇	C₂	1	1	yes	maximal, characteristic
7	#391×C₂₂₇	C₂	1	1	yes	maximal, characteristic
8	#389×C₂₂₇	C₂	1	1	yes	maximal, characteristic
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	2	yes	—
13	C₂²×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₂₂₇	—	2	4	no	—
15	C₄×Q₈×C₂₂₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
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	2	yes	—
23	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
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	4	no	—
39	C₂×Q₈×C₂₂₇	—	2	2	no	—
40	C₂×Q₈×C₂₂₇	—	2	4	no	—
41	C₂×Q₈×C₂₂₇	—	4	2	no	—
42	C₂×Q₈×C₂₂₇	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₂₇	—	2	4	no	—
55	C₄⋊C₄×C₂₂₇	—	2	4	no	—
56	C₄⋊C₄×C₂₂₇	—	2	2	no	—
57	C₄⋊C₄×C₂₂₇	—	2	2	no	—
58	C₄⋊C₄×C₂₂₇	—	2	2	no	—
59	C₄⋊C₄×C₂₂₇	—	4	2	no	—
60	C₄⋊C₄×C₂₂₇	—	4	2	no	—
61	C₄⋊C₄×C₂₂₇	D₄	1	4	yes	—
62	C₄⋊C₄×C₂₂₇	Q₈	1	4	yes	complemented
63	C₄²×C₂₂₇	—	2	4	no	—
64	C₄²×C₂₂₇	—	2	2	no	—
65	C₄²×C₂₂₇	—	2	2	no	—
66	C₄²×C₂₂₇	—	2	2	no	—
67	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
68	Q₈×C₂₂₇	—	2	8	no	—
69	Q₈×C₂₂₇	—	4	8	no	—
70	C₂×C₄×C₂₂₇	—	2	1	no	—
71	C₂×C₄×C₂₂₇	—	2	1	no	—
72	C₂×C₄×C₂₂₇	—	2	2	no	—
73	C₂×C₄×C₂₂₇	—	2	1	no	—
74	C₂×C₄×C₂₂₇	—	2	1	no	—
75	C₂×C₄×C₂₂₇	—	2	1	no	—
76	C₂×C₄×C₂₂₇	—	2	2	no	—
77	C₂×C₄×C₂₂₇	—	2	2	no	—
78	C₂×C₄×C₂₂₇	—	2	2	no	—
79	C₂×C₄×C₂₂₇	—	2	1	no	—
80	C₂×C₄×C₂₂₇	—	2	1	no	—
81	C₂×C₄×C₂₂₇	—	2	1	no	—
82	C₂×C₄×C₂₂₇	—	2	1	no	—
83	C₂×C₄×C₂₂₇	—	2	2	no	—
84	C₂×C₄×C₂₂₇	—	2	1	no	—
85	C₂×C₄×C₂₂₇	—	2	2	no	—
86	C₂×C₄×C₂₂₇	—	2	1	no	—
87	C₂×C₄×C₂₂₇	—	2	2	no	—
88	C₂×C₄×C₂₂₇	—	2	1	no	—
89	C₂×C₄×C₂₂₇	—	2	2	no	—
90	C₂×C₄×C₂₂₇	—	2	1	no	—
91	C₂×C₄×C₂₂₇	—	2	2	no	—
92	C₂×C₄×C₂₂₇	—	2	1	no	—
93	C₂×C₄×C₂₂₇	—	2	2	no	—
94	C₂×C₄×C₂₂₇	C₂×D₄	1	4	yes	—
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	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₂₂₇	C₂×Q₈	1	4	yes	—
100	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₂₂₇	#41	1	2	yes	—
102	C₂×C₄×C₂₂₇	#41	1	2	yes	—
103	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
104	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
107	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
108	C₂²×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
110	C₄×C₂₂₇	—	2	4	no	—
111	C₄×C₂₂₇	—	2	2	no	—
112	C₄×C₂₂₇	—	2	4	no	—
113	C₄×C₂₂₇	—	2	2	no	—
114	C₄×C₂₂₇	—	2	2	no	—
115	C₄×C₂₂₇	—	2	2	no	—
116	C₄×C₂₂₇	—	4	1	no	—
117	C₄×C₂₂₇	—	4	2	no	—
118	C₄×C₂₂₇	—	4	1	no	—
119	C₄×C₂₂₇	—	4	2	no	—
120	C₄×C₂₂₇	C₂²⋊Q₈	1	4	yes	—
121	C₂×C₂₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₂₂₇	#533	1	1	yes	characteristic
123	C₂×C₂₂₇	#542	1	1	yes	characteristic
124	C₂×C₂₂₇	#546	1	1	yes	characteristic
125	C₂×C₂₂₇	D₄×Q₈	1	1	yes	characteristic
126	C₂×C₂₂₇	#557	1	1	yes	characteristic
127	C₂×C₂₂₇	Q₈²	1	1	yes	characteristic
128	C₂₂₇	#2451	1	1	yes	characteristic, direct product factor, Sylow
129	#2451	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₄⋊Q₈	C₂×C₂₂₇	1	1	yes	characteristic
131	C₂×C₄×Q₈	C₂×C₂₂₇	1	1	yes	characteristic
132	#398	C₂×C₂₂₇	1	2	yes	—
133	#395	C₂×C₂₂₇	1	2	yes	—
134	#391	C₂×C₂₂₇	1	2	yes	—
135	#391	C₂×C₂₂₇	1	1	yes	characteristic
136	#391	C₂×C₂₂₇	1	1	yes	characteristic
137	#389	C₂×C₂₂₇	1	1	yes	characteristic
138	#389	C₂×C₂₂₇	1	1	yes	characteristic
139	#387	C₂×C₂₂₇	1	2	yes	—
140	C₄×C₄⋊C₄	C₂×C₂₂₇	1	1	yes	characteristic
141	C₂²×Q₈	C₂²×C₂₂₇	1	2	yes	—
142	C₂²×Q₈	C₂²×C₂₂₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	4	no	—
144	C₄×Q₈	—	2	4	no	—
145	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
147	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
155	C₂×C₄²	C₂²×C₂₂₇	1	2	yes	—
156	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
157	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
158	#95	C₂²×C₂₂₇	1	1	yes	characteristic
159	#95	C₂²×C₂₂₇	1	1	yes	characteristic
160	#95	C₂²×C₂₂₇	1	2	yes	—
161	#95	C₂²×C₂₂₇	1	2	yes	—
162	#95	C₂²×C₂₂₇	1	2	yes	—
163	#95	C₂²×C₂₂₇	1	2	yes	—
164	#95	C₂²×C₂₂₇	1	2	yes	—
165	#95	C₂²×C₂₂₇	1	2	yes	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	Q₈×C₂₂₇	1	4	yes	complemented
172	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
174	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
175	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
177	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
179	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	D₄×C₂₂₇	1	4	yes	—
191	C₄⋊C₄	Q₈×C₂₂₇	1	4	yes	complemented
192	C₄²	—	2	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₂³	C₂⁴×C₂₂₇	1	1	yes	characteristic
197	Q₈	—	2	8	no	—
198	Q₈	—	4	8	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	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	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	C₂×D₄×C₂₂₇	1	4	yes	—
224	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
225	C₂×C₄	C₂×Q₈×C₂₂₇	1	2	yes	—
226	C₂×C₄	C₂×Q₈×C₂₂₇	1	2	yes	—
227	C₂×C₄	C₂×Q₈×C₂₂₇	1	2	yes	—
228	C₂×C₄	C₂×Q₈×C₂₂₇	1	4	yes	—
229	C₂×C₄	C₂×Q₈×C₂₂₇	1	2	yes	—
230	C₂×C₄	#41×C₂₂₇	1	2	yes	—
231	C₂×C₄	#41×C₂₂₇	1	2	yes	—
232	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
233	C₂²	C₂²×Q₈×C₂₂₇	1	1	yes	characteristic
234	C₂²	C₂²×Q₈×C₂₂₇	1	1	yes	characteristic
235	C₂²	C₂×#41×C₂₂₇	1	1	yes	characteristic
236	C₂²	C₂×#41×C₂₂₇	1	1	yes	characteristic
237	C₂²	H(5,𝔽₂)×C₂₂₇	1	1	yes	characteristic
238	C₂²	2₋¹⁺⁴×C₂₂₇	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	2	no	—
249	C₄	C₂²⋊Q₈×C₂₂₇	1	4	yes	—
250	C₂	C₂×C₂²⋊Q₈×C₂₂₇	1	1	yes	characteristic
251	C₂	#533×C₂₂₇	1	1	yes	characteristic
252	C₂	#542×C₂₂₇	1	1	yes	characteristic
253	C₂	#546×C₂₂₇	1	1	yes	characteristic
254	C₂	D₄×Q₈×C₂₂₇	1	1	yes	characteristic
255	C₂	#557×C₂₂₇	1	1	yes	characteristic
256	C₂	Q₈²×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
42. C₂×Q₈×C₂₂₇	Q₈	semidirect
62. C₄⋊C₄×C₂₂₇	Q₈	semidirect
128. C₂₂₇	#2451	direct (N × H)
129. #2451	C₂₂₇	direct (N × H)
171. C₂×Q₈	Q₈×C₂₂₇	semidirect
191. C₄⋊C₄	Q₈×C₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₂₂₇	2
C₂×C₄×Q₈×C₂₂₇	2
#398×C₂₂₇	2
#395×C₂₂₇	2
#391×C₂₂₇	2
#389×C₂₂₇	2
#387×C₂₂₇	2
C₄×C₄⋊C₄×C₂₂₇	2
#2451	227

Subgroups — #2451×C227 #44396407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2451×C₂₂₇ #44396407