Subgroups — #3239×C₂₆₉ #46808346

Order 34432.

362 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#529×C₂₆₉	C₂	1	1	yes	maximal, characteristic
3	#501×C₂₆₉	C₂	1	2	yes	maximal, complemented
4	#489×C₂₆₉	C₂	1	2	yes	maximal, complemented
5	#484×C₂₆₉	C₂	1	2	yes	maximal, complemented
6	#481×C₂₆₉	C₂	1	2	yes	maximal
7	#475×C₂₆₉	C₂	1	2	yes	maximal, complemented
8	#474×C₂₆₉	C₂	1	2	yes	maximal, complemented
9	#432×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₂₆₉	—	2	1	no	—
11	C₄⋊Q₈×C₂₆₉	—	2	1	no	—
12	C₄⋊Q₈×C₂₆₉	—	2	1	no	—
13	C₄⋊Q₈×C₂₆₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₂₆₉	C₂²	1	2	yes	—
15	#126×C₂₆₉	—	2	1	no	—
16	#125×C₂₆₉	C₂²	1	1	yes	characteristic
17	#124×C₂₆₉	C₂²	1	1	yes	characteristic
18	#123×C₂₆₉	—	2	1	no	—
19	C₂²⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
20	#121×C₂₆₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₂₆₉	—	2	1	no	—
23	C₄×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
24	#117×C₂₆₉	—	2	1	no	—
25	C₂×C₄⋊C₄×C₂₆₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
28	#107×C₂₆₉	C₂²	1	4	yes	—
29	#106×C₂₆₉	C₂²	1	4	yes	—
30	C₄⋊C₈×C₂₆₉	C₂²	1	2	yes	—
31	#103×C₂₆₉	C₂²	1	4	yes	—
32	#102×C₂₆₉	C₂²	1	4	yes	—
33	C₂²⋊C₈×C₂₆₉	C₂²	1	2	yes	—
34	#97×C₂₆₉	C₂²	1	2	yes	—
35	C₂×Q₈×C₂₆₉	—	2	2	no	—
36	C₂×Q₈×C₂₆₉	—	2	2	no	—
37	C₂×Q₈×C₂₆₉	—	4	4	no	—
38	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
39	C₂×D₄×C₂₆₉	—	2	1	no	—
40	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₆₉	—	2	2	no	—
42	C₂²×C₄×C₂₆₉	—	2	1	no	—
43	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₂₆₉	C₂³	1	4	yes	—
47	C₄⋊C₄×C₂₆₉	—	2	2	no	—
48	C₄⋊C₄×C₂₆₉	—	2	2	no	—
49	C₄⋊C₄×C₂₆₉	—	2	2	no	—
50	C₄⋊C₄×C₂₆₉	—	2	1	no	—
51	C₄⋊C₄×C₂₆₉	—	2	1	no	—
52	C₄⋊C₄×C₂₆₉	—	2	1	no	—
53	C₄⋊C₄×C₂₆₉	—	2	1	no	—
54	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	—
55	C₄⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
57	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
58	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
59	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
60	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
61	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
62	C₄²×C₂₆₉	—	2	1	no	—
63	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
64	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
65	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
66	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
67	C₂³×C₂₆₉	—	2	1	no	—
68	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₂₆₉	—	2	2	no	—
70	Q₈×C₂₆₉	—	4	4	no	—
71	Q₈×C₂₆₉	—	4	4	no	—
72	D₄×C₂₆₉	—	2	2	no	—
73	D₄×C₂₆₉	—	4	2	no	—
74	C₂×C₄×C₂₆₉	—	2	2	no	—
75	C₂×C₄×C₂₆₉	—	2	2	no	—
76	C₂×C₄×C₂₆₉	—	2	2	no	—
77	C₂×C₄×C₂₆₉	—	2	1	no	—
78	C₂×C₄×C₂₆₉	—	2	1	no	—
79	C₂×C₄×C₂₆₉	—	2	1	no	—
80	C₂×C₄×C₂₆₉	—	2	4	no	—
81	C₂×C₄×C₂₆₉	—	4	4	no	—
82	C₂×C₄×C₂₆₉	—	4	2	no	—
83	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
89	C₈×C₂₆₉	—	2	4	no	—
90	C₂²×C₂₆₉	—	2	1	no	—
91	C₂²×C₂₆₉	—	2	1	no	—
92	C₂²×C₂₆₉	—	4	1	no	—
93	C₂²×C₂₆₉	—	4	1	no	—
94	C₂²×C₂₆₉	—	4	1	no	—
95	C₂²×C₂₆₉	#137	1	2	yes	—
96	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₂₆₉	—	2	1	no	—
98	C₄×C₂₆₉	—	2	1	no	—
99	C₄×C₂₆₉	—	2	1	no	—
100	C₄×C₂₆₉	—	2	1	no	—
101	C₄×C₂₆₉	—	4	1	no	—
102	C₄×C₂₆₉	—	4	1	no	—
103	C₄×C₂₆₉	—	4	4	no	—
104	C₄×C₂₆₉	—	4	1	no	—
105	C₄×C₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
106	C₄×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
107	C₂×C₂₆₉	—	2	2	no	—
108	C₂×C₂₆₉	—	8	1	no	—
109	C₂×C₂₆₉	#536	1	1	yes	characteristic
110	C₂×C₂₆₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₂₆₉	C₂×#137	1	1	yes	characteristic
112	C₂₆₉	#3239	1	1	yes	characteristic, direct product factor, Sylow
113	#3239	C₂₆₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₄⋊Q₈	C₂×C₂₆₉	1	1	yes	characteristic, complemented
115	#529	C₂×C₂₆₉	1	1	yes	characteristic
116	#501	C₂×C₂₆₉	1	2	yes	complemented
117	#489	C₂×C₂₆₉	1	2	yes	complemented
118	#484	C₂×C₂₆₉	1	2	yes	complemented
119	#481	C₂×C₂₆₉	1	2	yes	—
120	#475	C₂×C₂₆₉	1	2	yes	complemented
121	#474	C₂×C₂₆₉	1	2	yes	complemented
122	#432	C₂×C₂₆₉	1	1	yes	characteristic, complemented
123	C₂²×Q₈	—	2	1	no	—
124	C₄⋊Q₈	—	2	1	no	—
125	C₄⋊Q₈	—	2	1	no	—
126	C₄⋊Q₈	C₂²×C₂₆₉	1	2	yes	—
127	C₄⋊Q₈	C₂²×C₂₆₉	1	2	yes	—
128	#126	—	2	1	no	—
129	#125	C₂²×C₂₆₉	1	1	yes	characteristic
130	#124	C₂²×C₂₆₉	1	1	yes	characteristic
131	#123	—	2	1	no	—
132	C₂²⋊Q₈	C₂²×C₂₆₉	1	1	yes	characteristic
133	#121	C₂²×C₂₆₉	1	1	yes	characteristic
134	C₄×Q₈	C₂²×C₂₆₉	1	1	yes	characteristic
135	C₄×D₄	—	2	1	no	—
136	C₄×D₄	C₂²×C₂₆₉	1	1	yes	characteristic
137	#117	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	C₂²×C₂₆₉	1	2	yes	—
140	C₂×C₄²	C₂²×C₂₆₉	1	1	yes	characteristic
141	#107	C₂²×C₂₆₉	1	4	yes	—
142	#106	C₂²×C₂₆₉	1	4	yes	—
143	C₄⋊C₈	C₂²×C₂₆₉	1	2	yes	—
144	#103	C₂²×C₂₆₉	1	4	yes	—
145	#102	C₂²×C₂₆₉	1	4	yes	—
146	C₂²⋊C₈	C₂²×C₂₆₉	1	2	yes	—
147	#97	C₂²×C₂₆₉	1	2	yes	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	C₂³×C₂₆₉	1	1	yes	characteristic
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	C₂³×C₂₆₉	1	1	yes	characteristic
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	D₄×C₂₆₉	1	1	yes	characteristic
157	C₂²×C₄	D₄×C₂₆₉	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₂₆₉	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₂₆₉	1	4	yes	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	C₂³×C₂₆₉	1	4	yes	—
168	C₄⋊C₄	C₂³×C₂₆₉	1	1	yes	characteristic
169	C₄⋊C₄	C₂³×C₂₆₉	1	1	yes	characteristic
170	C₂²⋊C₄	—	2	1	no	—
171	C₂²⋊C₄	—	2	1	no	—
172	C₂²⋊C₄	—	2	1	no	—
173	C₂²⋊C₄	—	2	1	no	—
174	C₂²⋊C₄	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	D₄×C₂₆₉	1	1	yes	characteristic
177	C₄²	D₄×C₂₆₉	1	1	yes	characteristic
178	C₄²	C₂³×C₂₆₉	1	1	yes	characteristic
179	C₄²	C₂³×C₂₆₉	1	1	yes	characteristic
180	C₂³	—	2	1	no	—
181	C₂³	C₂×D₄×C₂₆₉	1	1	yes	characteristic
182	Q₈	—	2	2	no	—
183	Q₈	—	4	4	no	—
184	Q₈	—	4	4	no	—
185	D₄	—	2	2	no	—
186	D₄	—	4	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂×D₄×C₂₆₉	1	1	yes	characteristic
197	C₂×C₄	C₂×D₄×C₂₆₉	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄×C₂₆₉	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₂₆₉	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₂₆₉	1	1	yes	characteristic
201	C₂×C₄	C₂⁴×C₂₆₉	1	1	yes	characteristic
202	C₈	—	2	4	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	4	1	no	—
206	C₂²	—	4	1	no	—
207	C₂²	—	4	1	no	—
208	C₂²	#137×C₂₆₉	1	2	yes	—
209	C₂²	C₂²×D₄×C₂₆₉	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	1	no	—
215	C₄	—	4	4	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	AGL(1,ℤ/8)×C₂₆₉	1	2	yes	—
219	C₄	2₋¹⁺⁴×C₂₆₉	1	2	yes	—
220	C₂	—	2	2	no	—
221	C₂	—	8	1	no	—
222	C₂	#536×C₂₆₉	1	1	yes	characteristic
223	C₂	C₂×AGL(1,ℤ/8)×C₂₆₉	1	1	yes	characteristic
224	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	#121×C₂₆₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₆₉ ◁ C₂×C₄²×C₂₆₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₂₆₉	C₂	semidirect
3. #501×C₂₆₉	C₂	semidirect
4. #489×C₂₆₉	C₂	semidirect
5. #484×C₂₆₉	C₂	semidirect
7. #475×C₂₆₉	C₂	semidirect
8. #474×C₂₆₉	C₂	semidirect
9. #432×C₂₆₉	C₂	semidirect
112. C₂₆₉	#3239	direct (N × H)
113. #3239	C₂₆₉	direct (N × H)
114. C₂×C₄⋊Q₈	C₂×C₂₆₉	semidirect
116. #501	C₂×C₂₆₉	semidirect
117. #489	C₂×C₂₆₉	semidirect
118. #484	C₂×C₂₆₉	semidirect
120. #475	C₂×C₂₆₉	semidirect
121. #474	C₂×C₂₆₉	semidirect
122. #432	C₂×C₂₆₉	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×C₂₆₉	2
#529×C₂₆₉	2
#501×C₂₆₉	2
#489×C₂₆₉	2
#484×C₂₆₉	2
#481×C₂₆₉	2
#475×C₂₆₉	2
#474×C₂₆₉	2
#432×C₂₆₉	2
#3239	269

Subgroups — #3239×C269 #46808346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3239×C₂₆₉ #46808346