Subgroups — C₂×#31771295 #46824397

Order 34432.

1124 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771295	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22575446	C₂	1	2	yes	maximal, complemented
3	C₂×#22575445	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#22575443	C₂	1	2	yes	maximal, complemented
5	C₂×#22575434	C₂	1	2	yes	maximal
6	C₂×#22575432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#22575432	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#22575412	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#22575409	C₂	1	2	yes	maximal, complemented
11	C₂²×C₄⋊Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₅₃₈	C₂²	1	1	yes	characteristic
14	C₂³×Dic₂₆₉	C₂²	1	2	yes	—
15	#22575446	C₂²	1	16	yes	complemented
16	#22575445	C₂²	1	8	yes	complemented
17	#22575443	C₂²	1	16	yes	complemented
18	#22575434	C₂²	1	16	yes	—
19	#22575432	C₂²	1	8	yes	complemented
20	#22575432	C₂²	1	8	yes	complemented
21	C₄×Dic₅₃₈	C₂²	1	8	yes	complemented
22	#22575412	C₂²	1	8	yes	complemented
23	#22575409	C₂²	1	16	yes	complemented
24	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₂₆₉	C₂²	1	8	yes	complemented
29	C₂×#16864381	C₂²	1	2	yes	—
30	C₂×#16864381	C₂²	1	2	yes	—
31	C₂×#16864381	C₂²	1	2	yes	—
32	C₂×#16864381	C₂²	1	2	yes	—
33	C₂×#16864381	C₂²	1	2	yes	—
34	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
35	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
36	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
37	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
38	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
39	C₂²×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
40	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
41	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₂₆₉	C₂²	1	8	yes	—
43	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
47	C₂×Dic₅₃₈	—	2	4	no	—
48	C₂×Dic₅₃₈	C₂³	1	4	yes	—
49	C₂²×Dic₂₆₉	—	2	8	no	—
50	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
51	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
52	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
53	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
54	C₂²×Dic₂₆₉	C₂³	1	8	yes	—
55	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
56	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
57	C₄⋊Dic₂₆₉	C₂³	1	8	yes	—
58	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
59	#16864381	C₂³	1	8	yes	—
60	#16864381	C₂³	1	8	yes	—
61	#16864381	C₂³	1	8	yes	—
62	#16864381	C₂³	1	8	yes	—
63	#16864381	C₂³	1	8	yes	—
64	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
65	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
66	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
67	C₄×Dic₂₆₉	C₂³	1	8	yes	—
68	C₄×Dic₂₆₉	C₂³	1	8	yes	—
69	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
70	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₂₆₉	—	2	4	no	—
72	C₂×D₄×C₂₆₉	C₂³	1	4	yes	—
73	C₂²×C₄×C₂₆₉	—	2	4	no	—
74	C₂²×C₄×C₂₆₉	—	2	4	no	—
75	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
80	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
81	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	—
83	C₂²⋊C₄×C₂₆₉	C₂³	1	4	yes	—
84	C₂²⋊C₄×C₂₆₉	C₂³	1	4	yes	—
85	C₄²×C₂₆₉	C₂³	1	4	yes	—
86	Dic₅₃₈	—	2	8	no	—
87	C₂×Dic₂₆₉	—	2	16	no	—
88	C₂×Dic₂₆₉	—	2	4	no	—
89	C₂×Dic₂₆₉	—	2	4	no	—
90	C₂×Dic₂₆₉	—	2	4	no	—
91	C₂×Dic₂₆₉	—	2	4	no	—
92	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
93	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
94	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
95	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
96	C₂³×C₂₆₉	—	2	2	no	—
97	C₂³×C₂₆₉	—	2	2	no	—
98	C₂³×C₂₆₉	—	2	2	no	—
99	C₂³×C₂₆₉	—	2	2	no	—
100	C₂³×C₂₆₉	—	2	2	no	—
101	C₂³×C₂₆₉	#41	1	4	yes	—
102	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
103	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
104	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
105	D₄×C₂₆₉	—	2	8	no	—
106	C₂×C₄×C₂₆₉	—	2	2	no	—
107	C₂×C₄×C₂₆₉	—	2	2	no	—
108	C₂×C₄×C₂₆₉	—	2	2	no	—
109	C₂×C₄×C₂₆₉	—	2	2	no	—
110	C₂×C₄×C₂₆₉	—	2	8	no	—
111	C₂×C₄×C₂₆₉	—	2	8	no	—
112	C₂×C₄×C₂₆₉	#41	1	4	yes	—
113	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
114	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
115	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
116	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
117	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
118	C₂²×C₂₆₉	—	2	8	no	—
119	C₂²×C₂₆₉	—	2	2	no	—
120	C₂²×C₂₆₉	—	2	2	no	—
121	C₂²×C₂₆₉	—	2	2	no	—
122	C₂²×C₂₆₉	—	2	2	no	—
123	C₂²×C₂₆₉	—	2	2	no	—
124	C₂²×C₂₆₉	—	4	4	no	—
125	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
126	C₂²×C₂₆₉	C₂×#41	1	4	yes	—
127	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
128	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
129	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
130	C₄×C₂₆₉	—	2	2	no	—
131	C₄×C₂₆₉	—	2	2	no	—
132	C₄×C₂₆₉	—	2	2	no	—
133	C₄×C₂₆₉	—	2	2	no	—
134	C₄×C₂₆₉	C₂×#41	1	4	yes	—
135	Dic₂₆₉	—	2	4	no	—
136	Dic₂₆₉	—	2	4	no	—
137	Dic₂₆₉	—	2	4	no	—
138	Dic₂₆₉	—	2	4	no	—
139	C₂×C₂₆₉	—	2	4	no	—
140	C₂×C₂₆₉	—	4	2	no	—
141	C₂×C₂₆₉	#552	1	4	yes	complemented
142	C₂×C₂₆₉	C₂²×#41	1	1	yes	characteristic
143	C₂×C₂₆₉	C₂²×#41	1	1	yes	characteristic
144	C₂×C₂₆₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
145	C₂₆₉	C₂×#552	1	1	yes	characteristic, complemented, Sylow
146	C₂×#552	—	269	1	no	maximal, Sylow
147	#552	—	269	16	no	—
148	C₂×#126	—	269	2	no	—
149	C₂×#125	—	269	2	no	—
150	C₂×#125	—	269	1	no	—
151	C₂×#123	—	269	2	no	—
152	C₂×C₂²⋊Q₈	—	269	1	no	—
153	C₂×C₂²⋊Q₈	—	269	1	no	—
154	C₂×C₄×Q₈	—	269	1	no	—
155	C₂×C₄×D₄	D₂₆₉	1	1	yes	characteristic
156	C₂×#117	—	269	1	no	—
157	C₂×#117	—	269	2	no	—
158	C₂²×C₄⋊C₄	—	269	1	no	—
159	C₂²×Q₈	—	269	1	no	—
160	C₂²×D₄	C₂×D₂₆₉	1	1	yes	characteristic
161	C₂³×C₄	—	269	2	no	—
162	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
163	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
164	#126	—	269	16	no	—
165	#125	—	269	16	no	—
166	#125	—	269	8	no	—
167	#123	—	269	16	no	—
168	C₂²⋊Q₈	—	269	8	no	—
169	C₂²⋊Q₈	—	269	8	no	—
170	C₄×Q₈	—	269	8	no	—
171	C₄×D₄	C₂×D₂₆₉	1	8	yes	—
172	#117	—	269	8	no	—
173	#117	—	269	16	no	—
174	C₂×C₄⋊C₄	—	269	1	no	—
175	C₂×C₄⋊C₄	—	269	1	no	—
176	C₂×C₄⋊C₄	—	269	2	no	—
177	C₂×C₄⋊C₄	—	269	8	no	—
178	C₂×C₄⋊C₄	—	269	2	no	—
179	C₂×C₄⋊C₄	—	269	1	no	—
180	C₂×C₄⋊C₄	—	269	2	no	—
181	C₂×C₄⋊C₄	—	269	2	no	—
182	C₂×C₄⋊C₄	—	269	2	no	—
183	C₂×C₄⋊C₄	—	269	2	no	—
184	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	269	2	no	—
186	C₂×C₂²⋊C₄	—	269	2	no	—
187	C₂×C₂²⋊C₄	—	269	2	no	—
188	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
190	C₂×C₄²	—	269	2	no	—
191	C₂×C₄²	—	269	2	no	—
192	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
193	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
194	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
195	C₂×Q₈	—	269	4	no	—
196	C₂×Q₈	—	538	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	C₂²×D₂₆₉	1	4	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	269	8	no	—
202	C₂²×C₄	—	269	2	no	—
203	C₂²×C₄	—	269	2	no	—
204	C₂²×C₄	—	269	2	no	—
205	C₂²×C₄	—	269	2	no	—
206	C₂²×C₄	—	538	8	no	—
207	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
208	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
209	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
212	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
214	C₄⋊C₄	—	269	4	no	—
215	C₄⋊C₄	—	269	4	no	—
216	C₄⋊C₄	—	269	8	no	—
217	C₄⋊C₄	—	269	8	no	—
218	C₄⋊C₄	—	269	4	no	—
219	C₄⋊C₄	—	269	8	no	—
220	C₄⋊C₄	—	269	8	no	—
221	C₄⋊C₄	—	269	8	no	—
222	C₄⋊C₄	—	269	8	no	—
223	C₄⋊C₄	C₂²×D₂₆₉	1	4	yes	—
224	C₂²⋊C₄	—	269	8	no	—
225	C₂²⋊C₄	—	269	8	no	—
226	C₂²⋊C₄	—	269	8	no	—
227	C₂²⋊C₄	C₂²×D₂₆₉	1	4	yes	—
228	C₂²⋊C₄	C₂²×D₂₆₉	1	4	yes	—
229	C₄²	—	269	8	no	—
230	C₄²	—	269	8	no	—
231	C₄²	C₂²×D₂₆₉	1	4	yes	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	#16864402	1	4	yes	—
238	C₂³	C₂³×D₂₆₉	1	2	yes	—
239	C₂³	C₂³×D₂₆₉	1	2	yes	—
240	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
241	Q₈	—	538	8	no	—
242	D₄	—	2	8	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	269	4	no	—
250	C₂×C₄	—	269	4	no	—
251	C₂×C₄	—	269	4	no	—
252	C₂×C₄	—	269	4	no	—
253	C₂×C₄	—	538	4	no	—
254	C₂×C₄	—	538	16	no	—
255	C₂×C₄	—	538	4	no	—
256	C₂×C₄	—	538	4	no	—
257	C₂×C₄	—	538	4	no	—
258	C₂×C₄	#16864401	1	4	yes	—
259	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
260	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
261	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
262	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
263	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×#16864401	1	2	yes	—
272	C₂²	C₂×#16864402	1	4	yes	—
273	C₂²	C₂×#16864402	1	2	yes	—
274	C₂²	#22575500	1	2	yes	—
275	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	538	4	no	—
281	C₄	—	538	4	no	—
282	C₄	—	538	4	no	—
283	C₄	—	538	4	no	—
284	C₄	C₂×#16864401	1	4	yes	—
285	C₂	—	2	4	no	—
286	C₂	—	4	2	no	—
287	C₂	#31771295	1	4	yes	direct product factor
288	C₂	C₂²×#16864401	1	1	yes	characteristic
289	C₂	C₂²×#16864402	1	1	yes	characteristic
290	C₂	C₂×#22575500	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₆₉
commutator, cosocle	C₂²×C₂₆₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	C₂×C₄×D₄×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂²×D₄×C₂₆₉	C₂²
hypercenter	C₂×C₄×D₄	D₂₆₉
hypocenter	C₂₆₉	C₂×#552
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #31771295	C₂	direct (N × H)
2. C₂×#22575446	C₂	semidirect
3. C₂×#22575445	C₂	semidirect
4. C₂×#22575443	C₂	semidirect
6. C₂×#22575432	C₂	semidirect
7. C₂×#22575432	C₂	semidirect
8. C₂×C₄×Dic₅₃₈	C₂	semidirect
9. C₂×#22575412	C₂	semidirect
10. C₂×#22575409	C₂	semidirect
11. C₂²×C₄⋊Dic₂₆₉	C₂	semidirect
15. #22575446	C₂²	semidirect
16. #22575445	C₂²	semidirect
17. #22575443	C₂²	semidirect
19. #22575432	C₂²	semidirect
20. #22575432	C₂²	semidirect
21. C₄×Dic₅₃₈	C₂²	semidirect
22. #22575412	C₂²	semidirect
23. #22575409	C₂²	semidirect
28. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
141. C₂×C₂₆₉	#552	semidirect
145. C₂₆₉	C₂×#552	semidirect
287. C₂	#31771295	direct (N × H)

Maximal subgroups

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

subgroup	index
#31771295	2
C₂×#22575446	2
C₂×#22575445	2
C₂×#22575443	2
C₂×#22575434	2
C₂×#22575432	2
C₂×C₄×Dic₅₃₈	2
C₂×#22575412	2
C₂×#22575409	2
C₂²×C₄⋊Dic₂₆₉	2
C₂×C₄×D₄×C₂₆₉	2
C₂×#552	269

Subgroups — C2×#31771295 #46824397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771295 #46824397