Subgroups — C₂×#31771286 #46824387

Order 34432.

1328 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771286	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22575446	C₂	1	2	yes	maximal, complemented
3	C₂×#22575440	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#22575434	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#22575434	C₂	1	2	yes	maximal, complemented
6	C₂×#22575430	C₂	1	2	yes	maximal, complemented
7	C₂×D₄×Dic₂₆₉	C₂	1	2	yes	maximal, complemented
8	C₂×#22575409	C₂	1	2	yes	maximal, complemented
9	C₂²×C₂²⋊Dic₂₆₉	C₂	1	2	yes	maximal, complemented
10	C₂×#120×C₂₆₉	C₂	1	1	yes	maximal, characteristic
11	C₂²×Dic₅₃₈	C₂²	1	1	yes	characteristic, complemented
12	C₂³×Dic₂₆₉	C₂²	1	2	yes	—
13	C₂³×Dic₂₆₉	C₂²	1	2	yes	—
14	C₂³×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
15	#22575446	C₂²	1	16	yes	complemented
16	#22575440	C₂²	1	8	yes	complemented
17	#22575434	C₂²	1	8	yes	complemented
18	#22575434	C₂²	1	16	yes	complemented
19	#22575430	C₂²	1	16	yes	complemented
20	D₄×Dic₂₆₉	C₂²	1	16	yes	complemented
21	#22575409	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	complemented
23	C₂×#16864381	C₂²	1	2	yes	complemented
24	C₂×#16864381	C₂²	1	2	yes	complemented
25	C₂×#16864381	C₂²	1	2	yes	complemented
26	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
27	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
28	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
29	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
30	C₂×C₂²⋊Dic₂₆₉	C₂²	1	16	yes	complemented
31	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	complemented
32	C₂×C₂²⋊Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	complemented
35	C₂⁵×C₂₆₉	C₂²	1	1	yes	characteristic
36	C₂²×D₄×C₂₆₉	C₂²	1	2	yes	—
37	C₂²×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
38	#120×C₂₆₉	C₂²	1	8	yes	—
39	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₅₃₈	—	4	2	no	—
42	C₂×Dic₅₃₈	C₂³	1	4	yes	complemented
43	C₂²×Dic₂₆₉	—	2	8	no	—
44	C₂²×Dic₂₆₉	—	4	4	no	—
45	C₂²×Dic₂₆₉	—	4	2	no	—
46	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
47	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
48	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
49	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
50	C₂²×Dic₂₆₉	C₂³	1	8	yes	—
51	C₂²×Dic₂₆₉	C₂³	1	8	yes	—
52	C₂²×Dic₂₆₉	C₂³	1	4	yes	complemented
53	C₄⋊Dic₂₆₉	C₂³	1	8	yes	complemented
54	#16864381	C₂³	1	8	yes	complemented
55	#16864381	C₂³	1	8	yes	complemented
56	#16864381	C₂³	1	8	yes	complemented
57	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
58	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
59	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
60	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
61	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	complemented
62	C₂²⋊Dic₂₆₉	C₂³	1	4	yes	complemented
63	C₄×Dic₂₆₉	C₂³	1	4	yes	complemented
64	C₄×Dic₂₆₉	C₂³	1	8	yes	complemented
65	C₂⁴×C₂₆₉	—	2	8	no	—
66	C₂⁴×C₂₆₉	—	4	2	no	—
67	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
68	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
69	C₂⁴×C₂₆₉	C₂³	1	4	yes	—
70	C₂⁴×C₂₆₉	C₂³	1	2	yes	—
71	C₂×D₄×C₂₆₉	—	2	8	no	—
72	C₂×D₄×C₂₆₉	—	4	2	no	—
73	C₂×D₄×C₂₆₉	C₂³	1	8	yes	—
74	C₂×D₄×C₂₆₉	C₂³	1	4	yes	—
75	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
76	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₂₆₉	C₂³	1	8	yes	—
78	C₂²⋊C₄×C₂₆₉	C₂³	1	4	yes	—
79	Dic₅₃₈	—	4	4	no	—
80	C₂×Dic₂₆₉	—	2	16	no	—
81	C₂×Dic₂₆₉	—	2	4	no	—
82	C₂×Dic₂₆₉	—	2	4	no	—
83	C₂×Dic₂₆₉	—	2	4	no	—
84	C₂×Dic₂₆₉	—	2	4	no	—
85	C₂×Dic₂₆₉	—	4	8	no	—
86	C₂×Dic₂₆₉	—	4	4	no	—
87	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
88	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
89	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
90	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
91	C₂³×C₂₆₉	—	2	4	no	—
92	C₂³×C₂₆₉	—	2	4	no	—
93	C₂³×C₂₆₉	—	2	4	no	—
94	C₂³×C₂₆₉	—	2	16	no	—
95	C₂³×C₂₆₉	—	2	4	no	—
96	C₂³×C₂₆₉	—	2	2	no	—
97	C₂³×C₂₆₉	—	2	2	no	—
98	C₂³×C₂₆₉	—	4	4	no	—
99	C₂³×C₂₆₉	—	4	16	no	—
100	C₂³×C₂₆₉	#41	1	8	yes	—
101	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
102	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
103	C₂³×C₂₆₉	C₂⁴	1	4	yes	—
104	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
105	D₄×C₂₆₉	—	2	16	no	—
106	D₄×C₂₆₉	—	4	4	no	—
107	C₂×C₄×C₂₆₉	—	2	4	no	—
108	C₂×C₄×C₂₆₉	—	2	2	no	—
109	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
110	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
111	C₂²×C₂₆₉	—	2	16	no	—
112	C₂²×C₂₆₉	—	2	4	no	—
113	C₂²×C₂₆₉	—	2	4	no	—
114	C₂²×C₂₆₉	—	2	4	no	—
115	C₂²×C₂₆₉	—	2	4	no	—
116	C₂²×C₂₆₉	—	2	2	no	—
117	C₂²×C₂₆₉	—	2	2	no	—
118	C₂²×C₂₆₉	—	4	16	no	—
119	C₂²×C₂₆₉	—	4	4	no	—
120	C₂²×C₂₆₉	—	4	4	no	—
121	C₂²×C₂₆₉	C₂×#41	1	8	yes	—
122	C₂²×C₂₆₉	C₂×#41	1	4	yes	—
123	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
124	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
125	C₄×C₂₆₉	—	2	4	no	—
126	C₄×C₂₆₉	—	2	2	no	—
127	Dic₂₆₉	—	2	4	no	—
128	Dic₂₆₉	—	2	4	no	—
129	Dic₂₆₉	—	2	4	no	—
130	Dic₂₆₉	—	2	4	no	—
131	C₂×C₂₆₉	—	2	8	no	—
132	C₂×C₂₆₉	—	4	2	no	—
133	C₂×C₂₆₉	—	4	2	no	—
134	C₂×C₂₆₉	#551	1	4	yes	complemented
135	C₂×C₂₆₉	C₂²×#41	1	2	yes	—
136	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
137	C₂₆₉	C₂×#551	1	1	yes	characteristic, complemented, Sylow
138	C₂×#551	—	269	1	no	maximal, Sylow
139	#551	—	269	16	no	—
140	C₂×#126	—	269	2	no	—
141	C₂×#124	—	269	1	no	—
142	C₂×#123	—	269	2	no	—
143	C₂×#123	—	269	1	no	—
144	C₂×C₂²⋊Q₈	—	269	2	no	—
145	C₂×#120	D₂₆₉	1	1	yes	characteristic
146	C₂×C₄×D₄	—	269	2	no	—
147	C₂×#117	—	269	2	no	—
148	C₂²×C₂²⋊C₄	—	269	2	no	—
149	C₂⁵	C₂×D₂₆₉	1	1	yes	characteristic
150	C₂²×Q₈	—	269	1	no	—
151	C₂²×D₄	C₂×D₂₆₉	1	2	yes	—
152	C₂²×D₄	C₂×D₂₆₉	1	1	yes	characteristic
153	C₂³×C₄	—	269	2	no	—
154	C₂³×C₄	—	269	2	no	—
155	C₂³×C₄	—	269	1	no	—
156	#126	—	269	16	no	—
157	#124	—	269	8	no	—
158	#123	—	269	16	no	—
159	#123	—	269	8	no	—
160	C₂²⋊Q₈	—	269	16	no	—
161	#120	C₂×D₂₆₉	1	8	yes	—
162	C₄×D₄	—	269	16	no	—
163	#117	—	269	16	no	—
164	C₂×C₄⋊C₄	—	269	2	no	—
165	C₂×C₄⋊C₄	—	269	2	no	—
166	C₂×C₄⋊C₄	—	269	2	no	—
167	C₂×C₄⋊C₄	—	269	2	no	—
168	C₂×C₂²⋊C₄	—	269	2	no	—
169	C₂×C₂²⋊C₄	—	269	2	no	—
170	C₂×C₂²⋊C₄	—	269	2	no	—
171	C₂×C₂²⋊C₄	—	269	2	no	—
172	C₂×C₂²⋊C₄	—	269	16	no	—
173	C₂×C₂²⋊C₄	—	269	2	no	—
174	C₂×C₂²⋊C₄	—	269	1	no	—
175	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
177	C₂×C₄²	—	269	1	no	—
178	C₂×C₄²	—	269	2	no	—
179	C₂⁴	—	2	8	no	—
180	C₂⁴	—	4	2	no	—
181	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
182	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
183	C₂⁴	C₂²×D₂₆₉	1	4	yes	—
184	C₂⁴	C₂²×D₂₆₉	1	2	yes	—
185	C₂×Q₈	—	269	4	no	—
186	C₂×Q₈	—	1076	2	no	—
187	C₂×D₄	—	2	8	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	C₂²×D₂₆₉	1	8	yes	—
190	C₂×D₄	C₂²×D₂₆₉	1	4	yes	—
191	C₂²×C₄	—	269	8	no	—
192	C₂²×C₄	—	269	2	no	—
193	C₂²×C₄	—	269	2	no	—
194	C₂²×C₄	—	269	2	no	—
195	C₂²×C₄	—	269	2	no	—
196	C₂²×C₄	—	269	8	no	—
197	C₂²×C₄	—	269	4	no	—
198	C₂²×C₄	—	538	8	no	—
199	C₂²×C₄	—	1076	2	no	—
200	C₂²×C₄	—	1076	4	no	—
201	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
202	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
203	C₄⋊C₄	—	269	8	no	—
204	C₄⋊C₄	—	269	8	no	—
205	C₄⋊C₄	—	269	8	no	—
206	C₄⋊C₄	—	269	8	no	—
207	C₂²⋊C₄	—	269	8	no	—
208	C₂²⋊C₄	—	269	8	no	—
209	C₂²⋊C₄	—	269	8	no	—
210	C₂²⋊C₄	—	269	8	no	—
211	C₂²⋊C₄	—	269	8	no	—
212	C₂²⋊C₄	—	269	4	no	—
213	C₂²⋊C₄	C₂²×D₂₆₉	1	8	yes	—
214	C₂²⋊C₄	C₂²×D₂₆₉	1	4	yes	—
215	C₄²	—	269	4	no	—
216	C₄²	—	269	8	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	16	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	16	no	—
226	C₂³	#16864401	1	8	yes	—
227	C₂³	C₂³×D₂₆₉	1	2	yes	—
228	C₂³	C₂³×D₂₆₉	1	2	yes	—
229	C₂³	C₂³×D₂₆₉	1	4	yes	—
230	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
231	Q₈	—	1076	4	no	—
232	D₄	—	2	16	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	269	4	no	—
237	C₂×C₄	—	269	4	no	—
238	C₂×C₄	—	269	4	no	—
239	C₂×C₄	—	269	4	no	—
240	C₂×C₄	—	538	16	no	—
241	C₂×C₄	—	538	4	no	—
242	C₂×C₄	—	538	4	no	—
243	C₂×C₄	—	538	4	no	—
244	C₂×C₄	—	538	4	no	—
245	C₂×C₄	—	1076	4	no	—
246	C₂×C₄	—	1076	8	no	—
247	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
248	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	16	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	16	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	4	4	no	—
259	C₂²	C₂×#16864401	1	8	yes	—
260	C₂²	C₂×#16864401	1	4	yes	—
261	C₂²	#22575498	1	2	yes	—
262	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	2	no	—
265	C₄	—	538	4	no	—
266	C₄	—	538	4	no	—
267	C₄	—	538	4	no	—
268	C₄	—	538	4	no	—
269	C₂	—	2	8	no	—
270	C₂	—	4	2	no	—
271	C₂	—	4	2	no	—
272	C₂	#31771286	1	4	yes	direct product factor
273	C₂	C₂²×#16864401	1	2	yes	—
274	C₂	C₂×#22575498	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₆₉
commutator, cosocle	C₂²×C₂₆₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting, prime radical	C₂×#120×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
hypercenter	C₂×#120	D₂₆₉
hypocenter	C₂₆₉	C₂×#551
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×#120

Complements

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

N	H ≅ G/N	type
1. #31771286	C₂	direct (N × H)
2. C₂×#22575446	C₂	semidirect
3. C₂×#22575440	C₂	semidirect
4. C₂×#22575434	C₂	semidirect
5. C₂×#22575434	C₂	semidirect
6. C₂×#22575430	C₂	semidirect
7. C₂×D₄×Dic₂₆₉	C₂	semidirect
8. C₂×#22575409	C₂	semidirect
9. C₂²×C₂²⋊Dic₂₆₉	C₂	semidirect
11. C₂²×Dic₅₃₈	C₂²	semidirect
14. C₂³×Dic₂₆₉	C₂²	semidirect
15. #22575446	C₂²	semidirect
16. #22575440	C₂²	semidirect
17. #22575434	C₂²	semidirect
18. #22575434	C₂²	semidirect
19. #22575430	C₂²	semidirect
20. D₄×Dic₂₆₉	C₂²	semidirect
21. #22575409	C₂²	semidirect
22. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
23. C₂×#16864381	C₂²	semidirect
24. C₂×#16864381	C₂²	semidirect
25. C₂×#16864381	C₂²	semidirect
30. C₂×C₂²⋊Dic₂₆₉	C₂²	semidirect
31. C₂×C₂²⋊Dic₂₆₉	C₂²	semidirect
32. C₂×C₂²⋊Dic₂₆₉	C₂²	semidirect
33. C₂×C₄×Dic₂₆₉	C₂²	semidirect
34. C₂×C₄×Dic₂₆₉	C₂²	semidirect
42. C₂×Dic₅₃₈	C₂³	semidirect
52. C₂²×Dic₂₆₉	C₂³	semidirect
53. C₄⋊Dic₂₆₉	C₂³	semidirect
54. #16864381	C₂³	semidirect
55. #16864381	C₂³	semidirect
56. #16864381	C₂³	semidirect
61. C₂²⋊Dic₂₆₉	C₂³	semidirect
62. C₂²⋊Dic₂₆₉	C₂³	semidirect
63. C₄×Dic₂₆₉	C₂³	semidirect
64. C₄×Dic₂₆₉	C₂³	semidirect
134. C₂×C₂₆₉	#551	semidirect
137. C₂₆₉	C₂×#551	semidirect
272. C₂	#31771286	direct (N × H)

Maximal subgroups

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

subgroup	index
#31771286	2
C₂×#22575446	2
C₂×#22575440	2
C₂×#22575434	2
C₂×#22575430	2
C₂×D₄×Dic₂₆₉	2
C₂×#22575409	2
C₂²×C₂²⋊Dic₂₆₉	2
C₂×#120×C₂₆₉	2
C₂×#551	269

Subgroups — C2×#31771286 #46824387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771286 #46824387