Subgroups — #46813444

Order 34432.

484 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22575444	C₂	1	1	yes	maximal, characteristic
2	#31770503	C₂	1	1	yes	maximal, characteristic
3	#31770490	C₂	1	2	yes	maximal
4	#31770486	C₂	1	2	yes	maximal
5	#31770470	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×Dic₂₆₉	C₂	1	2	yes	maximal
7	C₄⋊C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
9	#31770456	C₂	1	2	yes	maximal
10	C₄²×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
11	#389×C₂₆₉	C₂	1	1	yes	maximal, characteristic
12	#22575444	C₄	1	8	yes	complemented
13	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
16	C₂×#16864381	C₂²	1	2	yes	—
17	C₂×#16864381	C₂²	1	2	yes	—
18	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
23	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
24	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
26	#22575360	C₂²	1	2	yes	—
27	#22575360	C₂²	1	2	yes	—
28	#22575360	C₂²	1	2	yes	—
29	#22575360	C₂²	1	2	yes	—
30	#22575360	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
35	#95×C₂₆₉	C₂²	1	2	yes	—
36	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
37	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
38	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
39	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
40	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₂₆₉	—	2	2	no	—
43	C₄⋊Dic₂₆₉	C₂×C₄	1	8	yes	—
44	C₄⋊Dic₂₆₉	C₂×C₄	1	4	yes	—
45	#16864381	—	2	4	no	—
46	#16864381	C₂×C₄	1	8	yes	—
47	C₄×Dic₂₆₉	—	2	2	no	—
48	C₄×Dic₂₆₉	—	2	2	no	—
49	C₄×Dic₂₆₉	—	2	4	no	—
50	C₄×Dic₂₆₉	—	2	4	no	—
51	C₄×Dic₂₆₉	—	2	2	no	—
52	C₄×Dic₂₆₉	—	2	2	no	—
53	C₄×Dic₂₆₉	C₂×C₄	1	4	yes	—
54	C₄×Dic₂₆₉	Q₈	1	4	yes	—
55	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₆₉	—	2	4	no	—
61	C₄⋊C₄×C₂₆₉	—	2	2	no	—
62	C₄⋊C₄×C₂₆₉	C₂×C₄	1	4	yes	—
63	C₄²×C₂₆₉	—	2	2	no	—
64	C₂×Dic₂₆₉	—	2	1	no	—
65	C₂×Dic₂₆₉	—	2	1	no	—
66	C₂×Dic₂₆₉	—	2	2	no	—
67	C₂×Dic₂₆₉	—	2	1	no	—
68	C₂×Dic₂₆₉	—	2	1	no	—
69	C₂×Dic₂₆₉	—	2	1	no	—
70	C₂×Dic₂₆₉	—	2	1	no	—
71	C₂×Dic₂₆₉	—	2	2	no	—
72	C₂×Dic₂₆₉	—	2	2	no	—
73	C₂×Dic₂₆₉	—	2	2	no	—
74	C₂×Dic₂₆₉	—	2	2	no	—
75	C₂×Dic₂₆₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₂₆₉	C₂²×C₄	1	2	yes	—
77	C₂×Dic₂₆₉	C₂²×C₄	1	4	yes	—
78	C₂×Dic₂₆₉	C₂×Q₈	1	2	yes	—
79	C₂×Dic₂₆₉	C₂×Q₈	1	2	yes	—
80	C₂×Dic₂₆₉	#41	1	2	yes	—
81	C₂×Dic₂₆₉	#41	1	2	yes	—
82	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
83	C₂×C₄×C₂₆₉	—	2	1	no	—
84	C₂×C₄×C₂₆₉	—	2	1	no	—
85	C₂×C₄×C₂₆₉	—	2	2	no	—
86	C₂×C₄×C₂₆₉	—	2	2	no	—
87	C₂×C₄×C₂₆₉	—	2	1	no	—
88	C₂×C₄×C₂₆₉	—	2	1	no	—
89	C₂×C₄×C₂₆₉	—	2	2	no	—
90	C₂×C₄×C₂₆₉	—	2	2	no	—
91	C₂×C₄×C₂₆₉	—	2	2	no	—
92	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₂₆₉	#41	1	2	yes	—
96	C₂×C₄×C₂₆₉	#41	1	2	yes	—
97	C₂×C₄×C₂₆₉	#41	1	2	yes	—
98	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
101	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
102	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
103	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
104	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
105	C₄×C₂₆₉	—	2	4	no	—
106	C₄×C₂₆₉	—	2	2	no	—
107	C₄×C₂₆₉	—	2	2	no	—
108	C₄×C₂₆₉	—	4	2	no	—
109	C₄×C₂₆₉	C₄×Q₈	1	4	yes	—
110	Dic₂₆₉	—	2	2	no	—
111	Dic₂₆₉	—	2	2	no	—
112	Dic₂₆₉	—	2	4	no	—
113	Dic₂₆₉	—	2	2	no	—
114	Dic₂₆₉	—	4	2	no	—
115	Dic₂₆₉	#125	1	4	yes	complemented
116	C₂×C₂₆₉	C₂×C₄×Q₈	1	1	yes	characteristic
117	C₂×C₂₆₉	C₄×#41	1	1	yes	characteristic
118	C₂×C₂₆₉	C₄×#41	1	1	yes	characteristic
119	C₂×C₂₆₉	C₂×#125	1	1	yes	characteristic
120	C₂×C₂₆₉	#529	1	1	yes	characteristic
121	C₂×C₂₆₉	#529	1	1	yes	characteristic
122	C₂×C₂₆₉	#533	1	1	yes	characteristic
123	C₂₆₉	C₄×#125	1	1	yes	characteristic, complemented, Sylow
124	C₄×#125	—	269	1	no	maximal, Sylow
125	C₂×#125	—	269	1	no	—
126	#389	—	269	1	no	—
127	#389	D₂₆₉	1	1	yes	characteristic
128	#387	—	269	2	no	—
129	#387	—	269	2	no	—
130	#382	—	269	1	no	—
131	C₄×C₄⋊C₄	—	269	2	no	—
132	C₄×C₄⋊C₄	—	269	1	no	—
133	C₄×C₄⋊C₄	—	269	1	no	—
134	C₄×C₄⋊C₄	—	269	2	no	—
135	C₄³	—	269	1	no	—
136	#125	—	269	8	no	—
137	C₂×C₄⋊C₄	—	269	2	no	—
138	C₂×C₄⋊C₄	—	269	2	no	—
139	C₂×C₄⋊C₄	—	269	1	no	—
140	C₂×C₄⋊C₄	—	269	2	no	—
141	C₂×C₄⋊C₄	—	269	1	no	—
142	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
144	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
145	C₂×C₄²	—	269	1	no	—
146	C₂×C₄²	—	269	1	no	—
147	C₂×C₄²	—	269	1	no	—
148	C₂×C₄²	—	269	1	no	—
149	C₂×C₄²	—	269	1	no	—
150	C₂×C₄²	—	269	1	no	—
151	C₂×C₄²	—	269	2	no	—
152	C₂×C₄²	—	269	2	no	—
153	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
154	#95	—	269	2	no	—
155	#95	—	269	2	no	—
156	#95	—	269	2	no	—
157	#95	—	269	2	no	—
158	#95	—	269	2	no	—
159	#95	C₂×D₂₆₉	1	2	yes	—
160	C₂²×C₄	—	269	1	no	—
161	C₂²×C₄	—	269	1	no	—
162	C₂²×C₄	—	269	1	no	—
163	C₂²×C₄	—	269	1	no	—
164	C₂²×C₄	—	269	2	no	—
165	C₂²×C₄	—	269	2	no	—
166	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
167	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
168	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	269	8	no	—
174	C₄⋊C₄	—	269	8	no	—
175	C₄⋊C₄	—	269	4	no	—
176	C₄⋊C₄	—	538	2	no	—
177	C₄⋊C₄	—	538	4	no	—
178	C₄⋊C₄	C₄×D₂₆₉	1	4	yes	—
179	C₄²	—	2	2	no	—
180	C₄²	—	269	4	no	—
181	C₄²	—	269	4	no	—
182	C₄²	—	538	2	no	—
183	C₄²	—	538	2	no	—
184	C₄²	—	538	4	no	—
185	C₄²	—	538	4	no	—
186	C₄²	—	538	2	no	—
187	C₄²	—	538	2	no	—
188	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	269	2	no	—
199	C₂×C₄	—	269	2	no	—
200	C₂×C₄	—	269	2	no	—
201	C₂×C₄	—	269	2	no	—
202	C₂×C₄	—	269	2	no	—
203	C₂×C₄	—	269	4	no	—
204	C₂×C₄	—	269	2	no	—
205	C₂×C₄	—	538	1	no	—
206	C₂×C₄	—	538	1	no	—
207	C₂×C₄	—	538	2	no	—
208	C₂×C₄	—	538	1	no	—
209	C₂×C₄	—	538	1	no	—
210	C₂×C₄	—	538	1	no	—
211	C₂×C₄	—	538	1	no	—
212	C₂×C₄	—	538	2	no	—
213	C₂×C₄	—	538	2	no	—
214	C₂×C₄	—	538	2	no	—
215	C₂×C₄	—	538	2	no	—
216	C₂×C₄	C₂×C₄×D₂₆₉	1	4	yes	—
217	C₂×C₄	C₂×C₄×D₂₆₉	1	2	yes	—
218	C₂×C₄	Q₈×D₂₆₉	1	2	yes	—
219	C₂×C₄	#16864401	1	2	yes	—
220	C₂×C₄	#16864401	1	2	yes	—
221	C₂×C₄	#16864403	1	2	yes	—
222	C₂²	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
223	C₂²	C₂×Q₈×D₂₆₉	1	1	yes	characteristic
224	C₂²	#41×D₂₆₉	1	1	yes	characteristic
225	C₂²	#41×D₂₆₉	1	1	yes	characteristic
226	C₂²	C₂×#16864401	1	1	yes	characteristic
227	C₂²	C₂×#16864401	1	1	yes	characteristic
228	C₂²	C₂×#16864403	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	4	no	—
232	C₄	—	4	2	no	—
233	C₄	—	269	4	no	—
234	C₄	—	538	2	no	—
235	C₄	—	538	2	no	—
236	C₄	—	538	2	no	—
237	C₄	—	538	4	no	—
238	C₄	—	1076	2	no	—
239	C₄	#22575418	1	4	yes	—
240	C₂	C₂×#22575418	1	1	yes	characteristic
241	C₂	C₄×#16864401	1	1	yes	characteristic
242	C₂	C₄×#16864403	1	1	yes	characteristic
243	C₂	#125×D₂₆₉	1	1	yes	characteristic
244	C₂	#31771136	1	1	yes	characteristic
245	C₂	#31771141	1	1	yes	characteristic
246	C₂	#31771160	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₆₉
commutator	C₂²×C₂₆₉	C₂³×C₄
Fitting	#389×C₂₆₉	C₂
socle, prime radical, cosocle	C₂³×C₂₆₉	C₂⁴
hypercenter	#389	D₂₆₉
hypocenter	C₂₆₉	C₄×#125
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
12. #22575444	C₄	semidirect
115. Dic₂₆₉	#125	semidirect
123. C₂₆₉	C₄×#125	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22575444	2
#31770503	2
#31770490	2
#31770486	2
#31770470	2
C₄⋊C₄×Dic₂₆₉	2
#31770456	2
C₄²×Dic₂₆₉	2
#389×C₂₆₉	2
C₄×#125	269