Subgroups — #46822222

Order 34432.

354 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771162	C₂	1	1	yes	maximal, characteristic, complemented
2	#31770909	C₂	1	2	yes	maximal, complemented
3	#31770890	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770885	C₂	1	1	yes	maximal, characteristic, complemented
5	#31770844	C₂	1	2	yes	maximal, complemented
6	#31770832	C₂	1	2	yes	maximal
7	#31770824	C₂	1	2	yes	maximal, complemented
8	#31770815	C₂	1	1	yes	maximal, characteristic, complemented
9	#31770810	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊#14524130	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₂₆₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#16864392	C₂²	1	2	yes	—
13	C₂×#16864390	C₂²	1	2	yes	—
14	C₂²×#14524130	C₂²	1	2	yes	—
15	#22575453	C₂²	1	1	yes	characteristic
16	#22575453	C₂²	1	1	yes	characteristic
17	#22575445	—	2	1	no	—
18	#22575432	—	2	1	no	—
19	#22575432	C₂²	1	2	yes	—
20	C₄×Dic₅₃₈	—	2	1	no	—
21	C₄×Dic₅₃₈	C₂²	1	2	yes	—
22	#22575412	—	2	1	no	—
23	#22575386	C₂²	1	2	yes	—
24	#22575384	C₂²	1	2	yes	—
25	C₄⋊#14524130	C₂²	1	1	yes	characteristic
26	C₄⋊#14524130	C₂²	1	1	yes	characteristic
27	C₄⋊#14524130	C₂²	1	2	yes	—
28	#22575377	C₂²	1	2	yes	—
29	#22575376	C₂²	1	2	yes	—
30	#22575376	C₂²	1	2	yes	—
31	#22575374	C₂²	1	2	yes	—
32	#126×C₂₆₉	—	2	1	no	—
33	#125×C₂₆₉	C₂²	1	1	yes	characteristic
34	#124×C₂₆₉	C₂²	1	1	yes	characteristic
35	#123×C₂₆₉	—	2	1	no	—
36	C₂²⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
37	#121×C₂₆₉	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₂₆₉	—	2	1	no	—
40	C₄×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
41	#117×C₂₆₉	—	2	1	no	—
42	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
43	C₂×Dic₅₃₈	—	2	1	no	—
44	C₂×Dic₅₃₈	C₂³	1	2	yes	—
45	#16864392	—	4	2	no	—
46	#16864390	—	4	2	no	—
47	C₂×#14524130	—	4	2	no	—
48	C₂×#14524130	C₂³	1	2	yes	—
49	C₂×#14524130	C₂³	1	2	yes	—
50	C₄⋊Dic₂₆₉	—	2	1	no	—
51	C₄⋊Dic₂₆₉	—	2	1	no	—
52	C₄⋊Dic₂₆₉	—	2	1	no	—
53	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
54	#16864381	—	2	2	no	—
55	#16864381	—	2	2	no	—
56	C₂²⋊Dic₂₆₉	—	2	2	no	—
57	C₄×Dic₂₆₉	—	2	2	no	—
58	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₂₆₉	—	2	1	no	—
60	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₆₉	—	2	1	no	—
62	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₂₆₉	—	2	1	no	—
66	C₄⋊C₄×C₂₆₉	—	2	1	no	—
67	C₄⋊C₄×C₂₆₉	—	2	1	no	—
68	C₄⋊C₄×C₂₆₉	—	2	1	no	—
69	C₄⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
72	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
73	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
74	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
75	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
76	C₄²×C₂₆₉	—	2	1	no	—
77	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
78	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
79	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
80	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
81	Dic₅₃₈	—	2	2	no	—
82	Dic₅₃₈	—	2	2	no	—
83	Dic₅₃₈	—	4	2	no	—
84	C₂×Dic₂₆₉	—	2	2	no	—
85	C₂×Dic₂₆₉	—	2	2	no	—
86	#14524130	—	2	2	no	—
87	#14524130	—	2	2	no	—
88	C₂³×C₂₆₉	—	2	1	no	—
89	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
90	Q₈×C₂₆₉	—	2	1	no	—
91	Q₈×C₂₆₉	—	2	1	no	—
92	D₄×C₂₆₉	—	2	1	no	—
93	D₄×C₂₆₉	—	2	1	no	—
94	D₄×C₂₆₉	—	4	2	no	—
95	C₂×C₄×C₂₆₉	—	2	2	no	—
96	C₂×C₄×C₂₆₉	—	2	2	no	—
97	C₂×C₄×C₂₆₉	—	2	1	no	—
98	C₂×C₄×C₂₆₉	—	2	1	no	—
99	C₂×C₄×C₂₆₉	—	2	1	no	—
100	C₂×C₄×C₂₆₉	—	2	1	no	—
101	C₂×C₄×C₂₆₉	—	2	1	no	—
102	C₂×C₄×C₂₆₉	—	4	2	no	—
103	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
109	C₂²×C₂₆₉	—	2	1	no	—
110	C₂²×C₂₆₉	—	2	1	no	—
111	C₂²×C₂₆₉	—	4	1	no	—
112	C₂²×C₂₆₉	—	4	1	no	—
113	C₂²×C₂₆₉	—	4	1	no	—
114	C₂²×C₂₆₉	#137	1	2	yes	—
115	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₂₆₉	—	2	1	no	—
117	C₄×C₂₆₉	—	2	1	no	—
118	C₄×C₂₆₉	—	2	1	no	—
119	C₄×C₂₆₉	—	2	1	no	—
120	C₄×C₂₆₉	—	4	1	no	—
121	C₄×C₂₆₉	—	4	1	no	—
122	C₄×C₂₆₉	—	4	1	no	—
123	C₄×C₂₆₉	#135	1	2	yes	—
124	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
125	C₄×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
126	Dic₂₆₉	—	4	2	no	—
127	Dic₂₆₉	—	4	2	no	—
128	C₂×C₂₆₉	—	2	2	no	—
129	C₂×C₂₆₉	—	8	1	no	—
130	C₂×C₂₆₉	#537	1	1	yes	characteristic
131	C₂×C₂₆₉	C₂×#135	1	1	yes	characteristic
132	C₂×C₂₆₉	C₂×#137	1	1	yes	characteristic
133	C₂₆₉	#3248	1	1	yes	characteristic, complemented, Sylow
134	#3248	—	269	1	no	maximal, Sylow
135	#533	—	269	1	no	—
136	#529	D₂₆₉	1	1	yes	characteristic
137	#479	—	269	2	no	—
138	#475	—	269	1	no	—
139	#474	—	269	1	no	—
140	#467	—	269	2	no	—
141	#465	—	269	2	no	—
142	#464	—	269	2	no	—
143	#462	—	269	1	no	—
144	#461	—	269	1	no	—
145	C₂×C₄⋊C₈	—	269	1	no	—
146	C₂×Q₁₆	—	269	2	no	—
147	C₂×SD₁₆	—	269	2	no	—
148	C₂²×C₈	—	269	2	no	—
149	C₄⋊Q₈	—	269	1	no	—
150	C₄⋊Q₈	—	269	1	no	—
151	#126	—	2	1	no	—
152	#125	—	538	1	no	—
153	#125	C₂×D₂₆₉	1	1	yes	characteristic
154	#124	C₂×D₂₆₉	1	1	yes	characteristic
155	#123	—	2	1	no	—
156	C₂²⋊Q₈	—	269	2	no	—
157	C₂²⋊Q₈	—	538	1	no	—
158	C₂²⋊Q₈	C₂×D₂₆₉	1	1	yes	characteristic
159	#121	C₂×D₂₆₉	1	1	yes	characteristic
160	C₄×Q₈	—	269	2	no	—
161	C₄×Q₈	—	538	1	no	—
162	C₄×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	C₂×D₂₆₉	1	1	yes	characteristic
165	#117	—	2	1	no	—
166	#117	—	538	1	no	—
167	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
168	#107	—	269	2	no	—
169	#106	—	269	2	no	—
170	C₄⋊C₈	—	269	1	no	—
171	C₄⋊C₈	—	269	1	no	—
172	C₄⋊C₈	—	269	2	no	—
173	#103	—	269	2	no	—
174	#103	—	269	2	no	—
175	#103	—	269	2	no	—
176	#102	—	269	2	no	—
177	C₂×Q₈	—	269	2	no	—
178	C₂×Q₈	—	538	1	no	—
179	C₂×Q₈	C₂²×D₂₆₉	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂²×D₂₆₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
186	Q₁₆	—	1076	2	no	—
187	SD₁₆	—	1076	2	no	—
188	C₂×C₈	—	269	2	no	—
189	C₂×C₈	—	269	2	no	—
190	C₂×C₈	—	1076	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	269	2	no	—
196	C₄⋊C₄	—	538	1	no	—
197	C₄⋊C₄	—	538	1	no	—
198	C₄⋊C₄	—	538	2	no	—
199	C₄⋊C₄	—	538	2	no	—
200	C₄⋊C₄	—	538	1	no	—
201	C₄⋊C₄	C₂²×D₂₆₉	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₂₆₉	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	538	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	538	2	no	—
211	C₄²	C₂²⋊D₂₆₉	1	1	yes	characteristic
212	C₄²	C₂²⋊D₂₆₉	1	1	yes	characteristic
213	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
214	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
215	C₂³	—	2	1	no	—
216	C₂³	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
217	Q₈	—	2	1	no	—
218	Q₈	—	2	1	no	—
219	Q₈	—	538	2	no	—
220	Q₈	—	538	2	no	—
221	Q₈	—	1076	2	no	—
222	D₄	—	2	1	no	—
223	D₄	—	2	1	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	538	2	no	—
234	C₂×C₄	—	538	2	no	—
235	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
241	C₈	—	538	2	no	—
242	C₈	—	538	2	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	#22575483	1	2	yes	—
249	C₂²	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	1076	2	no	—
258	C₄	—	1076	2	no	—
259	C₄	#22575473	1	2	yes	—
260	C₄	#22575499	1	1	yes	characteristic
261	C₄	#22575500	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	—	8	1	no	—
264	C₂	#31771183	1	1	yes	characteristic
265	C₂	C₂×#22575473	1	1	yes	characteristic
266	C₂	C₂×#22575483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#529×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
prime radical	#121×C₂₆₉	C₂²
hypercenter	#529	D₂₆₉
hypocenter	C₂₆₉	#3248
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #31771162	C₂	semidirect
2. #31770909	C₂	semidirect
3. #31770890	C₂	semidirect
4. #31770885	C₂	semidirect
5. #31770844	C₂	semidirect
7. #31770824	C₂	semidirect
8. #31770815	C₂	semidirect
9. #31770810	C₂	semidirect
10. C₂×C₄⋊#14524130	C₂	semidirect
133. C₂₆₉	#3248	semidirect

Maximal subgroups

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

subgroup	index
#31771162	2
#31770909	2
#31770890	2
#31770885	2
#31770844	2
#31770832	2
#31770824	2
#31770815	2
#31770810	2
C₂×C₄⋊#14524130	2
#529×C₂₆₉	2
#3248	269