Subgroups — C₂×#102×D₂₆₉ #46819369

Order 34432.

1072 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₂₆₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊C₄×D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#22575375	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#22575374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#22575373	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×D₂₆₉	C₂	1	8	yes	maximal, direct product factor
8	C₂×#102×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₂₆₉	—	2	1	no	—
10	C₂²×D₁₀₇₆	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊D₂₆₉	—	2	1	no	—
12	C₂×D₄×D₂₆₉	—	2	2	no	—
13	C₂×D₄×D₂₆₉	—	2	4	no	—
14	C₂×D₄×D₂₆₉	C₄	1	8	yes	complemented
15	C₂×D₄×D₂₆₉	C₂²	1	4	yes	—
16	C₂²×C₄×D₂₆₉	—	2	1	no	—
17	C₂²×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
18	C₂²×#14524130	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₈×D₂₆₉	—	2	4	no	—
20	C₂×C₈×D₂₆₉	C₂²	1	4	yes	complemented
21	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
22	C₂×#16864381	—	2	1	no	—
23	C₄⋊C₄×D₂₆₉	—	2	2	no	—
24	C₄⋊C₄×D₂₆₉	C₂²	1	4	yes	complemented
25	#22575375	C₂²	1	4	yes	complemented
26	#22575374	C₂²	1	4	yes	complemented
27	#22575373	C₂²	1	4	yes	complemented
28	C₂²×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
31	#102×C₂₆₉	C₂²	1	4	yes	complemented
32	C₂³×D₂₆₉	—	2	8	no	—
33	C₂³×D₂₆₉	—	2	4	no	—
34	C₂³×D₂₆₉	—	2	1	no	—
35	C₂³×D₂₆₉	—	4	4	no	—
36	C₂³×D₂₆₉	—	4	4	no	—
37	C₂³×D₂₆₉	D₄	1	1	yes	characteristic
38	C₂×D₁₀₇₆	—	2	1	no	—
39	C₂×D₁₀₇₆	—	2	2	no	—
40	C₂×D₁₀₇₆	C₂×C₄	1	4	yes	complemented
41	C₂×D₁₀₇₆	C₂³	1	2	yes	—
42	C₂×C₂²⋊D₂₆₉	—	2	8	no	—
43	C₂×C₂²⋊D₂₆₉	—	2	4	no	—
44	D₄×D₂₆₉	—	2	16	no	—
45	D₄×D₂₆₉	—	2	8	no	—
46	D₄×D₂₆₉	C₂×C₄	1	16	yes	complemented
47	C₂²×Dic₂₆₉	—	2	1	no	—
48	C₂²×Dic₂₆₉	D₄	1	1	yes	characteristic
49	C₂×C₄×D₂₆₉	—	2	4	no	—
50	C₂×C₄×D₂₆₉	—	4	4	no	—
51	C₂×C₄×D₂₆₉	C₂×C₄	1	4	yes	—
52	C₂×C₄×D₂₆₉	D₄	1	2	yes	—
53	C₂×C₄×D₂₆₉	D₄	1	4	yes	—
54	C₂×C₄×D₂₆₉	C₂³	1	2	yes	—
55	C₂×#14524130	—	2	2	no	—
56	C₂×#14524130	C₂³	1	2	yes	complemented
57	C₈×D₂₆₉	—	2	8	no	—
58	C₄⋊Dic₂₆₉	—	2	1	no	—
59	C₄⋊Dic₂₆₉	C₂³	1	2	yes	complemented
60	#16864381	—	2	4	no	—
61	C₂⁴×C₂₆₉	—	2	1	no	—
62	C₂×D₄×C₂₆₉	—	2	1	no	—
63	C₂×D₄×C₂₆₉	—	2	2	no	—
64	C₂×D₄×C₂₆₉	C₂×C₄	1	4	yes	complemented
65	C₂×D₄×C₂₆₉	C₂³	1	2	yes	—
66	C₂²×C₄×C₂₆₉	—	2	1	no	—
67	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₂₆₉	—	2	2	no	—
69	C₂×C₈×C₂₆₉	C₂³	1	2	yes	complemented
70	C₄⋊C₄×C₂₆₉	—	2	1	no	—
71	C₄⋊C₄×C₂₆₉	C₂³	1	2	yes	complemented
72	C₂²×D₂₆₉	—	2	4	no	—
73	C₂²×D₂₆₉	—	2	2	no	—
74	C₂²×D₂₆₉	—	2	16	no	—
75	C₂²×D₂₆₉	—	4	2	no	—
76	C₂²×D₂₆₉	—	4	2	no	—
77	C₂²×D₂₆₉	—	4	32	no	—
78	C₂²×D₂₆₉	—	4	8	no	—
79	C₂²×D₂₆₉	—	4	8	no	—
80	C₂²×D₂₆₉	C₂²⋊C₄	1	4	yes	—
81	C₂²×D₂₆₉	D₈	1	4	yes	—
82	C₂²×D₂₆₉	SD₁₆	1	4	yes	—
83	C₂²×D₂₆₉	C₂×D₄	1	2	yes	—
84	D₁₀₇₆	—	2	4	no	—
85	D₁₀₇₆	—	2	2	no	—
86	D₁₀₇₆	C₂²×C₄	1	4	yes	complemented
87	C₂²⋊D₂₆₉	—	2	16	no	—
88	C₂×Dic₂₆₉	—	2	2	no	—
89	C₂×Dic₂₆₉	—	4	2	no	—
90	C₂×Dic₂₆₉	C₂²⋊C₄	1	4	yes	—
91	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
92	C₄×D₂₆₉	—	4	8	no	—
93	C₄×D₂₆₉	C₂²⋊C₄	1	8	yes	—
94	C₄×D₂₆₉	C₂²×C₄	1	4	yes	—
95	C₄×D₂₆₉	C₂×D₄	1	4	yes	—
96	#14524130	—	2	2	no	—
97	C₂³×C₂₆₉	—	2	4	no	—
98	C₂³×C₂₆₉	—	2	2	no	—
99	C₂³×C₂₆₉	—	4	2	no	—
100	C₂³×C₂₆₉	—	4	2	no	—
101	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
102	D₄×C₂₆₉	—	2	4	no	—
103	D₄×C₂₆₉	—	2	2	no	—
104	D₄×C₂₆₉	C₂²×C₄	1	4	yes	complemented
105	C₂×C₄×C₂₆₉	—	2	2	no	—
106	C₂×C₄×C₂₆₉	—	4	2	no	—
107	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
111	C₈×C₂₆₉	—	2	2	no	—
112	C₂²×C₂₆₉	—	2	4	no	—
113	C₂²×C₂₆₉	—	4	2	no	—
114	C₂²×C₂₆₉	—	4	2	no	—
115	C₂²×C₂₆₉	—	4	8	no	—
116	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂²×C₂₆₉	C₂×D₈	1	2	yes	—
118	C₂²×C₂₆₉	C₂×SD₁₆	1	2	yes	—
119	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₂₆₉	—	2	4	no	—
121	C₂×D₂₆₉	—	4	2	no	—
122	C₂×D₂₆₉	—	4	2	no	—
123	C₂×D₂₆₉	—	4	8	no	—
124	C₂×D₂₆₉	—	4	32	no	—
125	C₂×D₂₆₉	#102	1	16	yes	direct product factor
126	C₂×D₂₆₉	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂×D₂₆₉	C₂×D₈	1	4	yes	—
128	C₂×D₂₆₉	C₂×SD₁₆	1	4	yes	—
129	C₄×C₂₆₉	—	4	2	no	—
130	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	2	yes	—
131	C₄×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
133	Dic₂₆₉	—	4	2	no	—
134	Dic₂₆₉	C₂×C₂²⋊C₄	1	4	yes	—
135	C₂×C₂₆₉	—	4	4	no	—
136	C₂×C₂₆₉	C₂×#102	1	4	yes	complemented
137	C₂×C₂₆₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₂₆₉	C₂²×D₈	1	1	yes	characteristic
139	C₂×C₂₆₉	C₂²×SD₁₆	1	1	yes	characteristic
140	D₂₆₉	—	4	4	no	—
141	D₂₆₉	C₂×#102	1	8	yes	direct product factor
142	C₂₆₉	C₂²×#102	1	1	yes	characteristic, complemented, Sylow
143	C₂²×#102	—	269	1	no	maximal, Sylow
144	C₂³×D₄	—	269	1	no	—
145	C₂³×C₈	—	269	1	no	—
146	C₂²×C₄⋊C₄	—	269	1	no	—
147	C₂×#102	—	269	1	no	—
148	C₂×#102	—	269	1	no	—
149	C₂×#102	—	269	1	no	—
150	C₂×#102	—	269	8	no	—
151	C₂×#102	D₂₆₉	1	1	yes	characteristic, direct product factor
152	C₂⁵	—	538	1	no	—
153	C₂²×D₄	—	269	8	no	—
154	C₂²×D₄	—	269	4	no	—
155	C₂²×D₄	—	269	1	no	—
156	C₂²×D₄	—	538	2	no	—
157	C₂²×D₄	—	538	4	no	—
158	C₂²×D₄	—	538	1	no	—
159	C₂²×D₄	C₂×D₂₆₉	1	1	yes	characteristic
160	C₂³×C₄	—	269	1	no	—
161	C₂³×C₄	—	538	1	no	—
162	C₂²×C₈	—	269	1	no	—
163	C₂²×C₈	—	269	4	no	—
164	C₂²×C₈	—	538	4	no	—
165	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic, complemented
166	C₂×C₄⋊C₄	—	269	1	no	—
167	C₂×C₄⋊C₄	—	269	4	no	—
168	C₂×C₄⋊C₄	—	538	2	no	—
169	C₂×C₄⋊C₄	—	538	1	no	—
170	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic, complemented
171	#102	—	269	4	no	—
172	#102	—	269	4	no	—
173	#102	—	269	4	no	—
174	#102	C₂×D₂₆₉	1	4	yes	direct product factor
175	C₂⁴	—	2	1	no	—
176	C₂⁴	—	269	1	no	—
177	C₂⁴	—	538	8	no	—
178	C₂⁴	—	538	4	no	—
179	C₂⁴	—	538	1	no	—
180	C₂⁴	—	1076	4	no	—
181	C₂⁴	—	1076	4	no	—
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	269	16	no	—
185	C₂×D₄	—	269	4	no	—
186	C₂×D₄	—	269	2	no	—
187	C₂×D₄	—	538	16	no	—
188	C₂×D₄	—	538	8	no	—
189	C₂×D₄	—	538	8	no	—
190	C₂×D₄	—	538	4	no	—
191	C₂×D₄	—	538	1	no	—
192	C₂×D₄	—	538	2	no	—
193	C₂×D₄	C₄×D₂₆₉	1	4	yes	complemented
194	C₂×D₄	C₂²×D₂₆₉	1	2	yes	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	269	1	no	—
197	C₂²×C₄	—	269	4	no	—
198	C₂²×C₄	—	269	2	no	—
199	C₂²×C₄	—	269	4	no	—
200	C₂²×C₄	—	269	2	no	—
201	C₂²×C₄	—	538	4	no	—
202	C₂²×C₄	—	538	1	no	—
203	C₂²×C₄	—	1076	4	no	—
204	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	269	2	no	—
207	C₂×C₈	—	538	8	no	—
208	C₂×C₈	—	538	2	no	—
209	C₂×C₈	C₂²×D₂₆₉	1	2	yes	complemented
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	269	2	no	—
212	C₄⋊C₄	—	538	4	no	—
213	C₄⋊C₄	—	538	1	no	—
214	C₄⋊C₄	C₂²×D₂₆₉	1	2	yes	complemented
215	C₂³	—	2	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	269	4	no	—
220	C₂³	—	269	4	no	—
221	C₂³	—	269	4	no	—
222	C₂³	—	269	2	no	—
223	C₂³	—	538	16	no	—
224	C₂³	—	538	4	no	—
225	C₂³	—	538	2	no	—
226	C₂³	—	1076	32	no	—
227	C₂³	—	1076	8	no	—
228	C₂³	—	1076	8	no	—
229	C₂³	—	1076	2	no	—
230	C₂³	—	1076	2	no	—
231	C₂³	D₄×D₂₆₉	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	—	269	4	no	—
235	D₄	—	538	16	no	—
236	D₄	—	538	4	no	—
237	D₄	—	538	2	no	—
238	D₄	C₂×C₄×D₂₆₉	1	4	yes	complemented
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	269	4	no	—
242	C₂×C₄	—	269	2	no	—
243	C₂×C₄	—	269	8	no	—
244	C₂×C₄	—	269	4	no	—
245	C₂×C₄	—	269	4	no	—
246	C₂×C₄	—	538	2	no	—
247	C₂×C₄	—	1076	8	no	—
248	C₂×C₄	—	1076	2	no	—
249	C₂×C₄	C₂×C₄×D₂₆₉	1	2	yes	—
250	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
251	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
252	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	538	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	8	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	269	4	no	—
260	C₂²	—	269	16	no	—
261	C₂²	—	269	4	no	—
262	C₂²	—	269	4	no	—
263	C₂²	—	538	4	no	—
264	C₂²	—	1076	32	no	—
265	C₂²	—	1076	8	no	—
266	C₂²	—	1076	2	no	—
267	C₂²	—	1076	2	no	—
268	C₂²	C₂²⋊C₄×D₂₆₉	1	2	yes	—
269	C₂²	D₈×D₂₆₉	1	2	yes	—
270	C₂²	SD₁₆×D₂₆₉	1	2	yes	—
271	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
272	C₄	—	4	2	no	—
273	C₄	—	269	4	no	—
274	C₄	—	1076	2	no	—
275	C₄	C₂²⋊C₄×D₂₆₉	1	2	yes	—
276	C₄	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
277	C₄	C₂×D₄×D₂₆₉	1	1	yes	characteristic
278	C₂	—	4	4	no	—
279	C₂	—	269	8	no	—
280	C₂	—	1076	4	no	—
281	C₂	#102×D₂₆₉	1	4	yes	direct product factor
282	C₂	C₂×C₂²⋊C₄×D₂₆₉	1	1	yes	characteristic
283	C₂	C₂×D₈×D₂₆₉	1	1	yes	characteristic
284	C₂	C₂×SD₁₆×D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₂₆₉
commutator	C₄×C₂₆₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	C₂×#102×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂×D₄
prime radical	C₂²×D₄×D₂₆₉	C₂
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	C₂×#102	D₂₆₉
hypocenter	C₂₆₉	C₂²×#102
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#102

Complements

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

N	H ≅ G/N	type
2. C₂²×C₈×D₂₆₉	C₂	semidirect
3. C₂×C₄⋊C₄×D₂₆₉	C₂	semidirect
4. C₂×#22575375	C₂	semidirect
5. C₂×#22575374	C₂	semidirect
6. C₂×#22575373	C₂	semidirect
7. #102×D₂₆₉	C₂	direct (N × H)
8. C₂×#102×C₂₆₉	C₂	semidirect
14. C₂×D₄×D₂₆₉	C₄	semidirect
18. C₂²×#14524130	C₂²	semidirect
20. C₂×C₈×D₂₆₉	C₂²	semidirect
21. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
24. C₄⋊C₄×D₂₆₉	C₂²	semidirect
25. #22575375	C₂²	semidirect
26. #22575374	C₂²	semidirect
27. #22575373	C₂²	semidirect
29. C₂²×C₈×C₂₆₉	C₂²	semidirect
30. C₂×C₄⋊C₄×C₂₆₉	C₂²	semidirect
31. #102×C₂₆₉	C₂²	semidirect
40. C₂×D₁₀₇₆	C₂×C₄	semidirect
46. D₄×D₂₆₉	C₂×C₄	semidirect
56. C₂×#14524130	C₂³	semidirect
59. C₄⋊Dic₂₆₉	C₂³	semidirect
64. C₂×D₄×C₂₆₉	C₂×C₄	semidirect
69. C₂×C₈×C₂₆₉	C₂³	semidirect
71. C₄⋊C₄×C₂₆₉	C₂³	semidirect
86. D₁₀₇₆	C₂²×C₄	semidirect
104. D₄×C₂₆₉	C₂²×C₄	semidirect
125. C₂×D₂₆₉	#102	direct (N × H)
136. C₂×C₂₆₉	C₂×#102	semidirect
141. D₂₆₉	C₂×#102	direct (N × H)
142. C₂₆₉	C₂²×#102	semidirect
151. C₂×#102	D₂₆₉	direct (N × H)
165. C₂²×C₈	C₂×D₂₆₉	semidirect
170. C₂×C₄⋊C₄	C₂×D₂₆₉	semidirect
174. #102	C₂×D₂₆₉	direct (N × H)
193. C₂×D₄	C₄×D₂₆₉	semidirect
209. C₂×C₈	C₂²×D₂₆₉	semidirect
214. C₄⋊C₄	C₂²×D₂₆₉	semidirect
238. D₄	C₂×C₄×D₂₆₉	semidirect
281. C₂	#102×D₂₆₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₄×D₂₆₉	2
C₂²×C₈×D₂₆₉	2
C₂×C₄⋊C₄×D₂₆₉	2
C₂×#22575375	2
C₂×#22575374	2
C₂×#22575373	2
#102×D₂₆₉	2
C₂×#102×C₂₆₉	2
C₂²×#102	269

Subgroups — C2×#102×D269 #46819369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#102×D₂₆₉ #46819369