Subgroups — C₂×#31771232 #46824282

Order 34432.

1080 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	#31771232	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22575452	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22575451	C₂	1	2	yes	maximal, complemented
4	C₂×#22575445	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#22575444	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#22575443	C₂	1	2	yes	maximal, complemented
7	C₂×#22575431	C₂	1	2	yes	maximal
8	C₂×#22575429	C₂	1	2	yes	maximal
9	C₂×#22575411	C₂	1	1	yes	maximal, characteristic
10	C₂×#22575410	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄⋊C₄×D₂₆₉	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₅₃₈	C₂²	1	2	yes	—
14	C₂²×C₄×D₂₆₉	C₂²	1	2	yes	—
15	C₂²×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
16	#22575452	C₂²	1	8	yes	complemented
17	#22575451	C₂²	1	16	yes	complemented
18	#22575445	C₂²	1	8	yes	complemented
19	#22575444	C₂²	1	8	yes	complemented
20	#22575443	C₂²	1	16	yes	complemented
21	#22575431	C₂²	1	16	yes	—
22	#22575429	C₂²	1	16	yes	—
23	#22575411	C₂²	1	8	yes	—
24	#22575410	C₂²	1	8	yes	—
25	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
28	C₂×#16864381	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
34	C₂×#16864381	C₂²	1	2	yes	—
35	C₄⋊C₄×D₂₆₉	C₂²	1	8	yes	—
36	C₂×#16864379	C₂²	1	1	yes	characteristic
37	C₂×#16864379	C₂²	1	1	yes	characteristic
38	C₂×#16864379	C₂²	1	2	yes	—
39	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
40	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₂₆₉	C₂²	1	2	yes	—
42	C₄⋊Q₈×C₂₆₉	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
46	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
47	C₂³×D₂₆₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₅₃₈	—	4	4	no	—
49	C₂×Dic₅₃₈	C₂³	1	8	yes	—
50	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
51	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
52	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
53	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
54	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₂₆₉	—	4	4	no	—
56	C₂×C₄×D₂₆₉	Q₈	1	8	yes	complemented
57	C₂×C₄×D₂₆₉	C₂³	1	8	yes	—
58	C₂×C₄×D₂₆₉	C₂³	1	4	yes	—
59	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
60	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
61	C₄⋊Dic₂₆₉	C₂³	1	8	yes	—
62	#16864381	C₂³	1	8	yes	—
63	#16864381	C₂³	1	4	yes	—
64	#16864381	C₂³	1	4	yes	—
65	#16864381	C₂³	1	8	yes	—
66	#16864381	C₂³	1	8	yes	—
67	#16864381	C₂³	1	8	yes	—
68	#16864381	C₂³	1	8	yes	—
69	#16864379	C₂³	1	4	yes	—
70	#16864379	C₂³	1	4	yes	—
71	#16864379	C₂³	1	8	yes	—
72	C₄×Dic₂₆₉	C₂³	1	4	yes	—
73	C₄×Dic₂₆₉	C₂³	1	8	yes	—
74	C₂×Q₈×C₂₆₉	—	4	4	no	—
75	C₂×Q₈×C₂₆₉	C₂³	1	8	yes	—
76	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
77	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
80	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	—
82	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	—
83	C₄⋊C₄×C₂₆₉	C₂³	1	8	yes	—
84	C₄²×C₂₆₉	C₂³	1	4	yes	—
85	C₂²×D₂₆₉	—	2	2	no	—
86	C₂²×D₂₆₉	—	2	2	no	—
87	C₂²×D₂₆₉	C₂×Q₈	1	4	yes	—
88	C₂²×D₂₆₉	C₂⁴	1	2	yes	—
89	Dic₅₃₈	—	4	8	no	—
90	C₂×Dic₂₆₉	—	2	4	no	—
91	C₂×Dic₂₆₉	—	2	2	no	—
92	C₂×Dic₂₆₉	—	2	4	no	—
93	C₂×Dic₂₆₉	—	2	2	no	—
94	C₂×Dic₂₆₉	C₂×Q₈	1	4	yes	—
95	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
96	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
97	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
98	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
99	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
100	C₄×D₂₆₉	—	4	8	no	—
101	C₄×D₂₆₉	C₂×Q₈	1	16	yes	complemented
102	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
103	Q₈×C₂₆₉	—	4	8	no	—
104	C₂×C₄×C₂₆₉	—	2	2	no	—
105	C₂×C₄×C₂₆₉	—	2	4	no	—
106	C₂×C₄×C₂₆₉	—	2	4	no	—
107	C₂×C₄×C₂₆₉	—	2	2	no	—
108	C₂×C₄×C₂₆₉	C₂×Q₈	1	4	yes	—
109	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
110	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
111	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
112	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
113	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
114	C₂²×C₂₆₉	C₂²×Q₈	1	2	yes	—
115	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
116	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
117	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
118	C₂×D₂₆₉	—	2	8	no	—
119	C₂×D₂₆₉	—	2	2	no	—
120	C₂×D₂₆₉	—	2	2	no	—
121	C₂×D₂₆₉	C₂²×Q₈	1	4	yes	—
122	C₄×C₂₆₉	—	2	2	no	—
123	C₄×C₂₆₉	—	2	4	no	—
124	C₄×C₂₆₉	—	2	4	no	—
125	C₄×C₂₆₉	—	2	2	no	—
126	C₄×C₂₆₉	C₂²×Q₈	1	4	yes	—
127	Dic₂₆₉	—	2	4	no	—
128	Dic₂₆₉	—	2	2	no	—
129	Dic₂₆₉	—	2	2	no	—
130	Dic₂₆₉	—	2	4	no	—
131	Dic₂₆₉	C₂²×Q₈	1	4	yes	—
132	C₂×C₂₆₉	#544	1	4	yes	complemented
133	C₂×C₂₆₉	C₂³×Q₈	1	1	yes	characteristic
134	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₂₆₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
136	D₂₆₉	—	2	4	no	—
137	C₂₆₉	C₂×#544	1	1	yes	characteristic, complemented, Sylow
138	C₂×#544	—	269	1	no	maximal, Sylow
139	#544	—	269	16	no	—
140	C₂×C₄⋊Q₈	—	269	2	no	—
141	C₂×C₄⋊Q₈	—	269	1	no	—
142	C₂×C₄⋊Q₈	D₂₆₉	1	1	yes	characteristic, complemented
143	C₂×#125	—	269	2	no	—
144	C₂×#125	—	269	1	no	—
145	C₂×#125	—	269	1	no	—
146	C₂×C₂²⋊Q₈	—	269	2	no	—
147	C₂×C₂²⋊Q₈	—	269	2	no	—
148	C₂×#117	—	269	1	no	—
149	C₂×#117	—	269	1	no	—
150	C₂²×C₄⋊C₄	—	269	1	no	—
151	C₂²×Q₈	—	269	2	no	—
152	C₂²×Q₈	C₂×D₂₆₉	1	2	yes	—
153	C₂³×C₄	—	269	1	no	—
154	C₂³×C₄	—	269	2	no	—
155	C₄⋊Q₈	—	269	16	no	—
156	C₄⋊Q₈	—	269	8	no	—
157	C₄⋊Q₈	C₂×D₂₆₉	1	8	yes	complemented
158	#125	—	269	16	no	—
159	#125	—	269	8	no	—
160	#125	—	269	8	no	—
161	C₂²⋊Q₈	—	269	16	no	—
162	C₂²⋊Q₈	—	269	16	no	—
163	#117	—	269	8	no	—
164	#117	—	269	8	no	—
165	C₂×C₄⋊C₄	—	269	2	no	—
166	C₂×C₄⋊C₄	—	269	1	no	—
167	C₂×C₄⋊C₄	—	269	1	no	—
168	C₂×C₄⋊C₄	—	269	1	no	—
169	C₂×C₄⋊C₄	—	269	1	no	—
170	C₂×C₄⋊C₄	—	269	2	no	—
171	C₂×C₄⋊C₄	—	269	2	no	—
172	C₂×C₄⋊C₄	—	269	2	no	—
173	C₂×C₄⋊C₄	—	269	2	no	—
174	C₂×C₄⋊C₄	—	269	2	no	—
175	C₂×C₄⋊C₄	—	269	8	no	—
176	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
179	C₂×C₂²⋊C₄	—	269	2	no	—
180	C₂×C₂²⋊C₄	—	269	1	no	—
181	C₂×C₂²⋊C₄	—	269	1	no	—
182	C₂×C₄²	—	269	2	no	—
183	C₂×C₄²	—	269	1	no	—
184	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
185	C₂⁴	—	269	1	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	269	8	no	—
188	C₂×Q₈	—	1076	4	no	—
189	C₂×Q₈	C₂²×D₂₆₉	1	8	yes	—
190	C₂²×C₄	—	269	1	no	—
191	C₂²×C₄	—	269	8	no	—
192	C₂²×C₄	—	269	4	no	—
193	C₂²×C₄	—	269	1	no	—
194	C₂²×C₄	—	269	2	no	—
195	C₂²×C₄	—	269	1	no	—
196	C₂²×C₄	—	269	2	no	—
197	C₂²×C₄	—	269	8	no	—
198	C₂²×C₄	—	1076	4	no	—
199	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
200	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
203	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
204	C₄⋊C₄	—	269	8	no	—
205	C₄⋊C₄	—	269	4	no	—
206	C₄⋊C₄	—	269	4	no	—
207	C₄⋊C₄	—	269	4	no	—
208	C₄⋊C₄	—	269	4	no	—
209	C₄⋊C₄	—	269	8	no	—
210	C₄⋊C₄	—	269	8	no	—
211	C₄⋊C₄	—	269	8	no	—
212	C₄⋊C₄	—	269	8	no	—
213	C₄⋊C₄	—	269	8	no	—
214	C₄⋊C₄	C₂²×D₂₆₉	1	4	yes	—
215	C₄⋊C₄	C₂²×D₂₆₉	1	4	yes	—
216	C₄⋊C₄	C₂²×D₂₆₉	1	8	yes	—
217	C₂²⋊C₄	—	269	8	no	—
218	C₂²⋊C₄	—	269	4	no	—
219	C₂²⋊C₄	—	269	4	no	—
220	C₄²	—	269	8	no	—
221	C₄²	—	269	4	no	—
222	C₄²	C₂²×D₂₆₉	1	4	yes	—
223	C₂³	—	269	4	no	—
224	C₂³	—	269	2	no	—
225	C₂³	—	538	2	no	—
226	C₂³	—	538	2	no	—
227	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
228	Q₈	—	4	8	no	—
229	Q₈	—	1076	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	269	4	no	—
235	C₂×C₄	—	269	2	no	—
236	C₂×C₄	—	269	16	no	—
237	C₂×C₄	—	269	2	no	—
238	C₂×C₄	—	269	4	no	—
239	C₂×C₄	—	269	2	no	—
240	C₂×C₄	—	269	4	no	—
241	C₂×C₄	—	538	4	no	—
242	C₂×C₄	—	538	2	no	—
243	C₂×C₄	—	538	2	no	—
244	C₂×C₄	—	538	4	no	—
245	C₂×C₄	—	1076	8	no	—
246	C₂×C₄	Q₈×D₂₆₉	1	4	yes	—
247	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
248	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
249	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
250	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
251	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
252	C₂²	—	269	4	no	—
253	C₂²	—	538	2	no	—
254	C₂²	—	538	2	no	—
255	C₂²	—	538	8	no	—
256	C₂²	C₂×Q₈×D₂₆₉	1	2	yes	—
257	C₂²	#22575498	1	2	yes	—
258	C₂²	#22575501	1	2	yes	—
259	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	269	4	no	—
265	C₄	—	538	4	no	—
266	C₄	—	538	2	no	—
267	C₄	—	538	2	no	—
268	C₄	—	538	4	no	—
269	C₄	C₂×Q₈×D₂₆₉	1	4	yes	—
270	C₂	—	538	4	no	—
271	C₂	#31771232	1	4	yes	direct product factor
272	C₂	C₂²×Q₈×D₂₆₉	1	1	yes	characteristic
273	C₂	C₂×#22575498	1	1	yes	characteristic
274	C₂	C₂×#22575501	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₆₉
commutator, cosocle	C₂²×C₂₆₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	C₂×C₄⋊Q₈×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂³×D₂₆₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₆₉
hypocenter	C₂₆₉	C₂×#544
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #31771232	C₂	direct (N × H)
2. C₂×#22575452	C₂	semidirect
3. C₂×#22575451	C₂	semidirect
4. C₂×#22575445	C₂	semidirect
5. C₂×#22575444	C₂	semidirect
6. C₂×#22575443	C₂	semidirect
12. C₂×C₄⋊Q₈×C₂₆₉	C₂	semidirect
16. #22575452	C₂²	semidirect
17. #22575451	C₂²	semidirect
18. #22575445	C₂²	semidirect
19. #22575444	C₂²	semidirect
20. #22575443	C₂²	semidirect
42. C₄⋊Q₈×C₂₆₉	C₂²	semidirect
56. C₂×C₄×D₂₆₉	Q₈	semidirect
101. C₄×D₂₆₉	C₂×Q₈	semidirect
132. C₂×C₂₆₉	#544	semidirect
137. C₂₆₉	C₂×#544	semidirect
142. C₂×C₄⋊Q₈	D₂₆₉	semidirect
157. C₄⋊Q₈	C₂×D₂₆₉	semidirect
271. C₂	#31771232	direct (N × H)

Maximal subgroups

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

subgroup	index
#31771232	2
C₂×#22575452	2
C₂×#22575451	2
C₂×#22575445	2
C₂×#22575444	2
C₂×#22575443	2
C₂×#22575431	2
C₂×#22575429	2
C₂×#22575411	2
C₂×#22575410	2
C₂×C₄⋊C₄×D₂₆₉	2
C₂×C₄⋊Q₈×C₂₆₉	2
C₂×#544	269

Subgroups — C2×#31771232 #46824282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771232 #46824282