Subgroups — #3092×C₂₆₉ #46808199

Order 34432.

458 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#525×C₂₆₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#484×C₂₆₉	C₂	1	2	yes	maximal, complemented
4	#483×C₂₆₉	C₂	1	2	yes	maximal, complemented
5	#481×C₂₆₉	C₂	1	2	yes	maximal, complemented
6	#480×C₂₆₉	C₂	1	2	yes	maximal, complemented
7	#428×C₂₆₉	C₂	1	2	yes	maximal, complemented
8	#417×C₂₆₉	C₂	1	2	yes	maximal, complemented
9	#407×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₆₉	—	2	1	no	—
12	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₆₉	C₂²	1	2	yes	—
14	#123×C₂₆₉	—	2	1	no	—
15	C₂²⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	#121×C₂₆₉	C₂²	1	1	yes	characteristic
18	#121×C₂₆₉	C₂²	1	1	yes	characteristic
19	#120×C₂₆₉	—	2	1	no	—
20	C₄×D₄×C₂₆₉	—	2	1	no	—
21	C₄×D₄×C₂₆₉	—	2	1	no	—
22	#117×C₂₆₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₆₉	—	2	1	no	—
24	C₂×C₄⋊C₄×C₂₆₉	—	2	1	no	—
25	C₂×C₄⋊C₄×C₂₆₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
29	#107×C₂₆₉	C₂²	1	4	yes	complemented
30	#106×C₂₆₉	C₂²	1	4	yes	complemented
31	#103×C₂₆₉	C₂²	1	4	yes	complemented
32	#102×C₂₆₉	C₂²	1	4	yes	complemented
33	C₂²⋊C₈×C₂₆₉	C₂²	1	2	yes	—
34	C₂²⋊C₈×C₂₆₉	C₂²	1	2	yes	—
35	C₂⁴×C₂₆₉	D₄	1	1	yes	characteristic
36	#41×C₂₆₉	—	2	4	no	—
37	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₂₆₉	—	2	1	no	—
39	C₂×D₄×C₂₆₉	—	2	1	no	—
40	C₂×D₄×C₂₆₉	—	2	1	no	—
41	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₆₉	—	2	4	no	—
43	C₂²×C₄×C₂₆₉	—	2	2	no	—
44	C₂²×C₄×C₂₆₉	—	2	2	no	—
45	C₂²×C₄×C₂₆₉	—	2	2	no	—
46	C₂²×C₄×C₂₆₉	—	2	1	no	—
47	C₂²×C₄×C₂₆₉	—	4	4	no	—
48	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
54	M₁₆×C₂₆₉	—	2	4	no	—
55	C₂×C₈×C₂₆₉	C₂³	1	4	yes	—
56	C₄⋊C₄×C₂₆₉	—	2	2	no	—
57	C₄⋊C₄×C₂₆₉	—	2	2	no	—
58	C₄⋊C₄×C₂₆₉	—	2	2	no	—
59	C₄⋊C₄×C₂₆₉	—	2	1	no	—
60	C₄⋊C₄×C₂₆₉	—	2	1	no	—
61	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	—
62	C₄⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
64	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
65	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
66	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
67	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
68	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
69	C₄²×C₂₆₉	—	2	1	no	—
70	C₂³×C₂₆₉	—	2	2	no	—
71	C₂³×C₂₆₉	—	2	2	no	—
72	C₂³×C₂₆₉	—	2	2	no	—
73	C₂³×C₂₆₉	—	2	1	no	—
74	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
76	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₂₆₉	—	2	2	no	—
78	D₄×C₂₆₉	—	2	2	no	—
79	D₄×C₂₆₉	—	4	2	no	—
80	D₄×C₂₆₉	—	4	2	no	—
81	D₄×C₂₆₉	—	4	2	no	—
82	C₂×C₄×C₂₆₉	—	2	2	no	—
83	C₂×C₄×C₂₆₉	—	2	2	no	—
84	C₂×C₄×C₂₆₉	—	2	2	no	—
85	C₂×C₄×C₂₆₉	—	2	2	no	—
86	C₂×C₄×C₂₆₉	—	2	2	no	—
87	C₂×C₄×C₂₆₉	—	2	1	no	—
88	C₂×C₄×C₂₆₉	—	2	1	no	—
89	C₂×C₄×C₂₆₉	—	2	1	no	—
90	C₂×C₄×C₂₆₉	—	2	4	no	—
91	C₂×C₄×C₂₆₉	—	4	4	no	—
92	C₂×C₄×C₂₆₉	—	4	4	no	—
93	C₂×C₄×C₂₆₉	—	4	4	no	—
94	C₂×C₄×C₂₆₉	—	4	2	no	—
95	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₆₉	#41	1	4	yes	—
99	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
100	C₈×C₂₆₉	—	2	4	no	—
101	C₂²×C₂₆₉	—	2	8	no	—
102	C₂²×C₂₆₉	—	2	1	no	—
103	C₂²×C₂₆₉	—	2	1	no	—
104	C₂²×C₂₆₉	—	2	1	no	—
105	C₂²×C₂₆₉	—	2	1	no	—
106	C₂²×C₂₆₉	—	2	1	no	—
107	C₂²×C₂₆₉	—	2	1	no	—
108	C₂²×C₂₆₉	—	4	1	no	—
109	C₂²×C₂₆₉	—	4	1	no	—
110	C₂²×C₂₆₉	—	4	1	no	—
111	C₂²×C₂₆₉	#123	1	2	yes	—
112	C₂²×C₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
113	C₂²×C₂₆₉	#137	1	2	yes	—
114	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₂₆₉	—	2	1	no	—
116	C₄×C₂₆₉	—	2	1	no	—
117	C₄×C₂₆₉	—	4	1	no	—
118	C₄×C₂₆₉	—	4	4	no	—
119	C₄×C₂₆₉	—	4	1	no	—
120	C₄×C₂₆₉	—	4	1	no	—
121	C₄×C₂₆₉	#123	1	2	yes	—
122	C₄×C₂₆₉	C₂×#41	1	2	yes	—
123	C₂×C₂₆₉	—	2	2	no	—
124	C₂×C₂₆₉	—	2	2	no	—
125	C₂×C₂₆₉	—	2	2	no	—
126	C₂×C₂₆₉	—	8	1	no	—
127	C₂×C₂₆₉	C₂×#123	1	1	yes	characteristic
128	C₂×C₂₆₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₂₆₉	C₂×#137	1	1	yes	characteristic
130	C₂₆₉	#3092	1	1	yes	characteristic, direct product factor, Sylow
131	#3092	C₂₆₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	#525	C₂×C₂₆₉	1	1	yes	characteristic
133	C₂²×C₄⋊C₄	C₂×C₂₆₉	1	1	yes	characteristic, complemented
134	#484	C₂×C₂₆₉	1	2	yes	complemented
135	#483	C₂×C₂₆₉	1	2	yes	complemented
136	#481	C₂×C₂₆₉	1	2	yes	complemented
137	#480	C₂×C₂₆₉	1	2	yes	complemented
138	#428	C₂×C₂₆₉	1	2	yes	complemented
139	#417	C₂×C₂₆₉	1	2	yes	complemented
140	#407	C₂×C₂₆₉	1	1	yes	characteristic, complemented
141	C₂×#41	C₂²×C₂₆₉	1	1	yes	characteristic
142	C₂³×C₄	—	2	1	no	—
143	C₂³×C₄	C₂²×C₂₆₉	1	1	yes	characteristic
144	C₂×M₁₆	C₂²×C₂₆₉	1	2	yes	—
145	#123	—	2	1	no	—
146	C₂²⋊Q₈	C₂²×C₂₆₉	1	1	yes	characteristic
147	C₂²⋊Q₈	C₂²×C₂₆₉	1	1	yes	characteristic
148	#121	C₂²×C₂₆₉	1	1	yes	characteristic
149	#121	C₂²×C₂₆₉	1	1	yes	characteristic
150	#120	—	2	1	no	—
151	C₄×D₄	—	2	1	no	—
152	C₄×D₄	—	2	1	no	—
153	#117	C₂²×C₂₆₉	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	C₂²×C₂₆₉	1	2	yes	—
158	C₂×C₄⋊C₄	C₂²×C₂₆₉	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₂₆₉	1	2	yes	—
160	#107	C₂²×C₂₆₉	1	4	yes	complemented
161	#106	C₂²×C₂₆₉	1	4	yes	complemented
162	#103	C₂²×C₂₆₉	1	4	yes	complemented
163	#102	C₂²×C₂₆₉	1	4	yes	complemented
164	C₂²⋊C₈	C₂²×C₂₆₉	1	2	yes	—
165	C₂²⋊C₈	C₂²×C₂₆₉	1	2	yes	—
166	C₂⁴	D₄×C₂₆₉	1	1	yes	characteristic
167	#41	—	2	4	no	—
168	C₂×Q₈	C₂³×C₂₆₉	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	C₂³×C₂₆₉	1	1	yes	characteristic
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	D₄×C₂₆₉	1	1	yes	characteristic
180	C₂²×C₄	D₄×C₂₆₉	1	1	yes	characteristic
181	C₂²×C₄	D₄×C₂₆₉	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₂₆₉	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₂₆₉	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₂₆₉	1	1	yes	characteristic
185	M₁₆	—	2	4	no	—
186	C₂×C₈	C₂³×C₂₆₉	1	4	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	C₂³×C₂₆₉	1	4	yes	—
193	C₄⋊C₄	C₂³×C₂₆₉	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₂₆₉	1	1	yes	characteristic
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	1	no	—
205	C₂³	C₂×D₄×C₂₆₉	1	1	yes	characteristic
206	C₂³	C₂×D₄×C₂₆₉	1	1	yes	characteristic
207	C₂³	C₂×D₄×C₂₆₉	1	1	yes	characteristic
208	Q₈	—	2	2	no	—
209	D₄	—	2	2	no	—
210	D₄	—	4	2	no	—
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×D₄×C₂₆₉	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄×C₂₆₉	1	1	yes	characteristic
228	C₂×C₄	C₂×D₄×C₂₆₉	1	1	yes	characteristic
229	C₂×C₄	#41×C₂₆₉	1	4	yes	—
230	C₂×C₄	C₂⁴×C₂₆₉	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₂²	—	2	8	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	#123×C₂₆₉	1	2	yes	—
243	C₂²	AGL(1,ℤ/8)×C₂₆₉	1	2	yes	—
244	C₂²	#137×C₂₆₉	1	2	yes	—
245	C₂²	C₂²×D₄×C₂₆₉	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	4	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	#123×C₂₆₉	1	2	yes	—
253	C₄	C₂×#41×C₂₆₉	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	—	2	2	no	—
257	C₂	—	8	1	no	—
258	C₂	C₂×#123×C₂₆₉	1	1	yes	characteristic
259	C₂	C₂×AGL(1,ℤ/8)×C₂₆₉	1	1	yes	characteristic
260	C₂	C₂×#137×C₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₆₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₆₉
prime radical	#525×C₂₆₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₆₉ ◁ C₂³×C₄×C₂₆₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₄×C₂₆₉	C₂	semidirect
3. #484×C₂₆₉	C₂	semidirect
4. #483×C₂₆₉	C₂	semidirect
5. #481×C₂₆₉	C₂	semidirect
6. #480×C₂₆₉	C₂	semidirect
7. #428×C₂₆₉	C₂	semidirect
8. #417×C₂₆₉	C₂	semidirect
9. #407×C₂₆₉	C₂	semidirect
29. #107×C₂₆₉	C₂²	semidirect
30. #106×C₂₆₉	C₂²	semidirect
31. #103×C₂₆₉	C₂²	semidirect
32. #102×C₂₆₉	C₂²	semidirect
130. C₂₆₉	#3092	direct (N × H)
131. #3092	C₂₆₉	direct (N × H)
133. C₂²×C₄⋊C₄	C₂×C₂₆₉	semidirect
134. #484	C₂×C₂₆₉	semidirect
135. #483	C₂×C₂₆₉	semidirect
136. #481	C₂×C₂₆₉	semidirect
137. #480	C₂×C₂₆₉	semidirect
138. #428	C₂×C₂₆₉	semidirect
139. #417	C₂×C₂₆₉	semidirect
140. #407	C₂×C₂₆₉	semidirect
160. #107	C₂²×C₂₆₉	semidirect
161. #106	C₂²×C₂₆₉	semidirect
162. #103	C₂²×C₂₆₉	semidirect
163. #102	C₂²×C₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#525×C₂₆₉	2
C₂²×C₄⋊C₄×C₂₆₉	2
#484×C₂₆₉	2
#483×C₂₆₉	2
#481×C₂₆₉	2
#480×C₂₆₉	2
#428×C₂₆₉	2
#417×C₂₆₉	2
#407×C₂₆₉	2
#3092	269

Subgroups — #3092×C269 #46808199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3092×C₂₆₉ #46808199