Subgroups — #3289×C₂₆₉ #46808396

Order 34432.

448 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#578×C₂₆₉	C₂	1	2	yes	maximal, complemented
2	#577×C₂₆₉	C₂	1	2	yes	maximal, complemented
3	#497×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#473×C₂₆₉	C₂	1	2	yes	maximal, complemented
5	#471×C₂₆₉	C₂	1	2	yes	maximal, complemented
6	#456×C₂₆₉	C₂	1	2	yes	maximal, complemented
7	#455×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#454×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#444×C₂₆₉	C₂	1	2	yes	maximal, complemented
10	2₋¹⁺⁴×C₂₆₉	C₂²	1	2	yes	—
11	2₋¹⁺⁴×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
12	H(5,𝔽₂)×C₂₆₉	C₂²	1	1	yes	characteristic
13	#137×C₂₆₉	—	2	2	no	—
14	#137×C₂₆₉	—	2	2	no	—
15	#137×C₂₆₉	—	2	2	no	—
16	#137×C₂₆₉	C₂²	1	2	yes	—
17	#137×C₂₆₉	C₂²	1	2	yes	—
18	#137×C₂₆₉	C₂²	1	2	yes	complemented
19	AGL(1,ℤ/8)×C₂₆₉	—	2	2	no	—
20	AGL(1,ℤ/8)×C₂₆₉	C₂²	1	2	yes	—
21	#135×C₂₆₉	—	2	2	no	—
22	#135×C₂₆₉	—	2	2	no	—
23	#135×C₂₆₉	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₂₆₉	—	2	2	no	—
25	C₂×Q₁₆×C₂₆₉	C₂²	1	2	yes	complemented
26	C₂×SD₁₆×C₂₆₉	—	2	2	no	—
27	C₂×SD₁₆×C₂₆₉	C₂²	1	2	yes	—
28	#131×C₂₆₉	C₂²	1	2	yes	—
29	#131×C₂₆₉	C₂²	1	2	yes	complemented
30	C₄⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
31	#124×C₂₆₉	C₂²	1	1	yes	characteristic
32	#108×C₂₆₉	C₂²	1	2	yes	complemented
33	#104×C₂₆₉	C₂²	1	2	yes	—
34	#104×C₂₆₉	C₂²	1	2	yes	complemented
35	#101×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
36	#101×C₂₆₉	C₂²	1	2	yes	complemented
37	#100×C₂₆₉	C₂²	1	1	yes	characteristic
38	#97×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
39	#41×C₂₆₉	—	2	2	no	—
40	#41×C₂₆₉	—	2	2	no	—
41	#41×C₂₆₉	—	2	2	no	—
42	#41×C₂₆₉	—	2	2	no	—
43	#41×C₂₆₉	—	4	2	no	—
44	#41×C₂₆₉	—	4	1	no	—
45	#41×C₂₆₉	C₂³	1	2	yes	—
46	#41×C₂₆₉	C₂³	1	2	yes	—
47	C₂×Q₈×C₂₆₉	—	2	2	no	—
48	C₂×Q₈×C₂₆₉	—	2	2	no	—
49	C₂×Q₈×C₂₆₉	—	2	2	no	—
50	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₂₆₉	C₂³	1	2	yes	—
52	C₂×D₄×C₂₆₉	—	2	2	no	—
53	C₂×D₄×C₂₆₉	—	4	1	no	—
54	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
55	Q₁₆×C₂₆₉	—	2	2	no	—
56	Q₁₆×C₂₆₉	—	2	2	no	—
57	Q₁₆×C₂₆₉	—	2	2	no	—
58	Q₁₆×C₂₆₉	—	2	2	no	—
59	Q₁₆×C₂₆₉	—	2	2	no	—
60	Q₁₆×C₂₆₉	D₄	1	2	yes	—
61	SD₁₆×C₂₆₉	—	2	2	no	—
62	SD₁₆×C₂₆₉	—	2	2	no	—
63	SD₁₆×C₂₆₉	—	2	2	no	—
64	SD₁₆×C₂₆₉	—	2	2	no	—
65	SD₁₆×C₂₆₉	—	2	2	no	—
66	SD₁₆×C₂₆₉	—	2	2	no	—
67	SD₁₆×C₂₆₉	D₄	1	2	yes	—
68	SD₁₆×C₂₆₉	D₄	1	2	yes	—
69	D₈×C₂₆₉	—	2	2	no	—
70	D₈×C₂₆₉	D₄	1	2	yes	—
71	M₁₆×C₂₆₉	—	2	2	no	—
72	M₁₆×C₂₆₉	—	2	2	no	—
73	M₁₆×C₂₆₉	C₂³	1	2	yes	—
74	M₁₆×C₂₆₉	C₂³	1	2	yes	—
75	C₂×C₈×C₂₆₉	—	2	2	no	—
76	C₂×C₈×C₂₆₉	—	2	2	no	—
77	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
78	C₄⋊C₄×C₂₆₉	—	4	1	no	—
79	C₂²⋊C₄×C₂₆₉	—	4	1	no	—
80	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
81	C₂³×C₂₆₉	—	2	1	no	—
82	C₂³×C₂₆₉	—	4	1	no	—
83	Q₈×C₂₆₉	—	2	2	no	—
84	Q₈×C₂₆₉	—	2	2	no	—
85	Q₈×C₂₆₉	—	2	2	no	—
86	Q₈×C₂₆₉	—	4	2	no	—
87	Q₈×C₂₆₉	—	4	1	no	—
88	Q₈×C₂₆₉	C₂×D₄	1	2	yes	—
89	Q₈×C₂₆₉	C₂×D₄	1	2	yes	—
90	D₄×C₂₆₉	—	2	2	no	—
91	D₄×C₂₆₉	—	2	2	no	—
92	D₄×C₂₆₉	—	2	2	no	—
93	D₄×C₂₆₉	—	4	2	no	—
94	D₄×C₂₆₉	—	4	1	no	—
95	D₄×C₂₆₉	—	4	2	no	—
96	D₄×C₂₆₉	C₂×D₄	1	2	yes	—
97	D₄×C₂₆₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₂₆₉	—	2	2	no	—
99	C₂×C₄×C₂₆₉	—	2	2	no	—
100	C₂×C₄×C₂₆₉	—	2	1	no	—
101	C₂×C₄×C₂₆₉	—	2	1	no	—
102	C₂×C₄×C₂₆₉	—	2	2	no	—
103	C₂×C₄×C₂₆₉	—	4	1	no	—
104	C₂×C₄×C₂₆₉	—	4	2	no	—
105	C₂×C₄×C₂₆₉	—	4	2	no	—
106	C₂×C₄×C₂₆₉	—	4	2	no	—
107	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
108	C₈×C₂₆₉	—	2	2	no	—
109	C₈×C₂₆₉	—	2	2	no	—
110	C₈×C₂₆₉	C₂×D₄	1	2	yes	—
111	C₈×C₂₆₉	C₂×D₄	1	2	yes	—
112	C₂²×C₂₆₉	—	2	2	no	—
113	C₂²×C₂₆₉	—	2	2	no	—
114	C₂²×C₂₆₉	—	4	1	no	—
115	C₂²×C₂₆₉	—	8	1	no	—
116	C₂²×C₂₆₉	—	16	1	no	—
117	C₂²×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
118	C₄×C₂₆₉	—	2	2	no	—
119	C₄×C₂₆₉	—	2	2	no	—
120	C₄×C₂₆₉	—	4	1	no	—
121	C₄×C₂₆₉	—	4	1	no	—
122	C₄×C₂₆₉	—	4	2	no	—
123	C₄×C₂₆₉	C₂²×D₄	1	2	yes	—
124	C₂×C₂₆₉	—	2	1	no	—
125	C₂×C₂₆₉	—	4	2	no	—
126	C₂×C₂₆₉	—	4	2	no	—
127	C₂×C₂₆₉	—	8	1	no	—
128	C₂×C₂₆₉	D₄²	1	1	yes	characteristic
129	C₂₆₉	#3289	1	1	yes	characteristic, direct product factor, Sylow
130	#3289	C₂₆₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	#578	C₂×C₂₆₉	1	2	yes	complemented
132	#577	C₂×C₂₆₉	1	2	yes	complemented
133	#497	C₂×C₂₆₉	1	1	yes	characteristic, complemented
134	#473	C₂×C₂₆₉	1	2	yes	complemented
135	#471	C₂×C₂₆₉	1	2	yes	complemented
136	#456	C₂×C₂₆₉	1	2	yes	complemented
137	#455	C₂×C₂₆₉	1	1	yes	characteristic, complemented
138	#454	C₂×C₂₆₉	1	1	yes	characteristic, complemented
139	#444	C₂×C₂₆₉	1	2	yes	complemented
140	2₋¹⁺⁴	C₂²×C₂₆₉	1	2	yes	—
141	2₋¹⁺⁴	C₂²×C₂₆₉	1	1	yes	characteristic, complemented
142	H(5,𝔽₂)	C₂²×C₂₆₉	1	1	yes	characteristic
143	#137	—	2	2	no	—
144	#137	—	2	2	no	—
145	#137	—	2	2	no	—
146	#137	C₂²×C₂₆₉	1	2	yes	—
147	#137	C₂²×C₂₆₉	1	2	yes	—
148	#137	C₂²×C₂₆₉	1	2	yes	complemented
149	AGL(1,ℤ/8)	—	2	2	no	—
150	AGL(1,ℤ/8)	C₂²×C₂₆₉	1	2	yes	—
151	#135	—	2	2	no	—
152	#135	—	2	2	no	—
153	#135	C₂²×C₂₆₉	1	2	yes	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×Q₁₆	C₂²×C₂₆₉	1	2	yes	complemented
156	C₂×SD₁₆	—	2	2	no	—
157	C₂×SD₁₆	C₂²×C₂₆₉	1	2	yes	—
158	#131	C₂²×C₂₆₉	1	2	yes	—
159	#131	C₂²×C₂₆₉	1	2	yes	complemented
160	C₄⋊Q₈	C₂²×C₂₆₉	1	1	yes	characteristic, complemented
161	#124	C₂²×C₂₆₉	1	1	yes	characteristic
162	#108	C₂²×C₂₆₉	1	2	yes	complemented
163	#104	C₂²×C₂₆₉	1	2	yes	—
164	#104	C₂²×C₂₆₉	1	2	yes	complemented
165	#101	C₂²×C₂₆₉	1	1	yes	characteristic, complemented
166	#101	C₂²×C₂₆₉	1	2	yes	complemented
167	#100	C₂²×C₂₆₉	1	1	yes	characteristic
168	#97	C₂²×C₂₆₉	1	1	yes	characteristic, complemented
169	#41	—	2	2	no	—
170	#41	—	2	2	no	—
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	#41	—	4	2	no	—
174	#41	—	4	1	no	—
175	#41	C₂³×C₂₆₉	1	2	yes	—
176	#41	C₂³×C₂₆₉	1	2	yes	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	C₂³×C₂₆₉	1	1	yes	characteristic
181	C₂×Q₈	C₂³×C₂₆₉	1	2	yes	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	4	1	no	—
184	C₂×D₄	C₂³×C₂₆₉	1	1	yes	characteristic
185	Q₁₆	—	2	2	no	—
186	Q₁₆	—	2	2	no	—
187	Q₁₆	—	2	2	no	—
188	Q₁₆	—	2	2	no	—
189	Q₁₆	—	2	2	no	—
190	Q₁₆	D₄×C₂₆₉	1	2	yes	—
191	SD₁₆	—	2	2	no	—
192	SD₁₆	—	2	2	no	—
193	SD₁₆	—	2	2	no	—
194	SD₁₆	—	2	2	no	—
195	SD₁₆	—	2	2	no	—
196	SD₁₆	—	2	2	no	—
197	SD₁₆	D₄×C₂₆₉	1	2	yes	—
198	SD₁₆	D₄×C₂₆₉	1	2	yes	—
199	D₈	—	2	2	no	—
200	D₈	D₄×C₂₆₉	1	2	yes	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	C₂³×C₂₆₉	1	2	yes	—
204	M₁₆	C₂³×C₂₆₉	1	2	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂³×C₂₆₉	1	2	yes	—
208	C₄⋊C₄	—	4	1	no	—
209	C₂²⋊C₄	—	4	1	no	—
210	C₄²	C₂³×C₂₆₉	1	1	yes	characteristic
211	C₂³	—	2	1	no	—
212	C₂³	—	4	1	no	—
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	4	1	no	—
218	Q₈	C₂×D₄×C₂₆₉	1	2	yes	—
219	Q₈	C₂×D₄×C₂₆₉	1	2	yes	—
220	D₄	—	2	2	no	—
221	D₄	—	2	2	no	—
222	D₄	—	2	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	1	no	—
225	D₄	—	4	2	no	—
226	D₄	C₂×D₄×C₂₆₉	1	2	yes	—
227	D₄	C₂×D₄×C₂₆₉	1	2	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂⁴×C₂₆₉	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	C₂×D₄×C₂₆₉	1	2	yes	—
241	C₈	C₂×D₄×C₂₆₉	1	2	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	16	1	no	—
247	C₂²	H(5,𝔽₂)×C₂₆₉	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	2	no	—
253	C₄	C₂²×D₄×C₂₆₉	1	2	yes	—
254	C₂	—	2	1	no	—
255	C₂	—	4	2	no	—
256	C₂	—	4	2	no	—
257	C₂	—	8	1	no	—
258	C₂	D₄²×C₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₆₉	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₆₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₆₉ ◁ C₂×C₄×C₂₆₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #578×C₂₆₉	C₂	semidirect
2. #577×C₂₆₉	C₂	semidirect
3. #497×C₂₆₉	C₂	semidirect
4. #473×C₂₆₉	C₂	semidirect
5. #471×C₂₆₉	C₂	semidirect
6. #456×C₂₆₉	C₂	semidirect
7. #455×C₂₆₉	C₂	semidirect
8. #454×C₂₆₉	C₂	semidirect
9. #444×C₂₆₉	C₂	semidirect
11. 2₋¹⁺⁴×C₂₆₉	C₂²	semidirect
18. #137×C₂₆₉	C₂²	semidirect
25. C₂×Q₁₆×C₂₆₉	C₂²	semidirect
29. #131×C₂₆₉	C₂²	semidirect
30. C₄⋊Q₈×C₂₆₉	C₂²	semidirect
32. #108×C₂₆₉	C₂²	semidirect
34. #104×C₂₆₉	C₂²	semidirect
35. #101×C₂₆₉	C₂²	semidirect
36. #101×C₂₆₉	C₂²	semidirect
38. #97×C₂₆₉	C₂²	semidirect
129. C₂₆₉	#3289	direct (N × H)
130. #3289	C₂₆₉	direct (N × H)
131. #578	C₂×C₂₆₉	semidirect
132. #577	C₂×C₂₆₉	semidirect
133. #497	C₂×C₂₆₉	semidirect
134. #473	C₂×C₂₆₉	semidirect
135. #471	C₂×C₂₆₉	semidirect
136. #456	C₂×C₂₆₉	semidirect
137. #455	C₂×C₂₆₉	semidirect
138. #454	C₂×C₂₆₉	semidirect
139. #444	C₂×C₂₆₉	semidirect
141. 2₋¹⁺⁴	C₂²×C₂₆₉	semidirect
148. #137	C₂²×C₂₆₉	semidirect
155. C₂×Q₁₆	C₂²×C₂₆₉	semidirect
159. #131	C₂²×C₂₆₉	semidirect
160. C₄⋊Q₈	C₂²×C₂₆₉	semidirect
162. #108	C₂²×C₂₆₉	semidirect
164. #104	C₂²×C₂₆₉	semidirect
165. #101	C₂²×C₂₆₉	semidirect
166. #101	C₂²×C₂₆₉	semidirect
168. #97	C₂²×C₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#578×C₂₆₉	2
#577×C₂₆₉	2
#497×C₂₆₉	2
#473×C₂₆₉	2
#471×C₂₆₉	2
#456×C₂₆₉	2
#455×C₂₆₉	2
#454×C₂₆₉	2
#444×C₂₆₉	2
#3289	269

Subgroups — #3289×C269 #46808396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3289×C₂₆₉ #46808396