Subgroups — #46819271

Order 34432.

400 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31770504	C₂	1	1	yes	maximal, characteristic
2	#31770503	C₂	1	1	yes	maximal, characteristic
3	#31770501	C₂	1	1	yes	maximal, characteristic
4	#31770491	C₂	1	2	yes	maximal
5	#31770490	C₂	1	2	yes	maximal
6	#31770487	C₂	1	2	yes	maximal
7	#31770486	C₂	1	2	yes	maximal
8	#31770470	C₂	1	1	yes	maximal, characteristic
9	#31770470	C₂	1	1	yes	maximal, characteristic
10	C₄²×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
11	#389×C₂₆₉	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
15	C₂×#16864381	C₂²	1	2	yes	—
16	C₂×#16864381	C₂²	1	2	yes	—
17	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
23	#22575360	C₂²	1	2	yes	—
24	#22575360	C₂²	1	2	yes	—
25	#22575360	C₂²	1	2	yes	—
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	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
33	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
34	#95×C₂₆₉	C₂²	1	2	yes	—
35	C₂²×Dic₂₆₉	C₂³	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	1	yes	characteristic
40	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
41	C₄⋊Dic₂₆₉	—	2	4	no	—
42	C₄⋊Dic₂₆₉	—	2	2	no	—
43	C₄⋊Dic₂₆₉	—	2	2	no	—
44	#16864381	—	2	4	no	—
45	#16864381	—	2	4	no	—
46	C₄×Dic₂₆₉	—	2	2	no	—
47	C₄×Dic₂₆₉	—	2	2	no	—
48	C₄×Dic₂₆₉	—	2	2	no	—
49	C₄×Dic₂₆₉	—	2	2	no	—
50	C₄×Dic₂₆₉	—	2	2	no	—
51	C₄×Dic₂₆₉	Q₈	1	4	yes	—
52	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₂₆₉	—	2	4	no	—
58	C₄⋊C₄×C₂₆₉	—	2	2	no	—
59	C₄⋊C₄×C₂₆₉	—	2	2	no	—
60	C₄²×C₂₆₉	—	2	2	no	—
61	C₂×Dic₂₆₉	—	2	1	no	—
62	C₂×Dic₂₆₉	—	2	1	no	—
63	C₂×Dic₂₆₉	—	2	1	no	—
64	C₂×Dic₂₆₉	—	2	1	no	—
65	C₂×Dic₂₆₉	—	2	1	no	—
66	C₂×Dic₂₆₉	—	2	1	no	—
67	C₂×Dic₂₆₉	—	2	1	no	—
68	C₂×Dic₂₆₉	—	2	1	no	—
69	C₂×Dic₂₆₉	—	2	2	no	—
70	C₂×Dic₂₆₉	—	2	2	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	C₂×Q₈	1	2	yes	—
76	C₂×Dic₂₆₉	C₂×Q₈	1	2	yes	—
77	C₂×Dic₂₆₉	#41	1	2	yes	—
78	C₂×Dic₂₆₉	#41	1	2	yes	—
79	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
80	C₂×C₄×C₂₆₉	—	2	1	no	—
81	C₂×C₄×C₂₆₉	—	2	1	no	—
82	C₂×C₄×C₂₆₉	—	2	1	no	—
83	C₂×C₄×C₂₆₉	—	2	1	no	—
84	C₂×C₄×C₂₆₉	—	2	2	no	—
85	C₂×C₄×C₂₆₉	—	2	2	no	—
86	C₂×C₄×C₂₆₉	—	2	2	no	—
87	C₂×C₄×C₂₆₉	—	2	2	no	—
88	C₂×C₄×C₂₆₉	—	2	2	no	—
89	C₂×C₄×C₂₆₉	—	2	2	no	—
90	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₂₆₉	#41	1	2	yes	—
92	C₂×C₄×C₂₆₉	#41	1	2	yes	—
93	C₂×C₄×C₂₆₉	#41	1	2	yes	—
94	C₂×C₄×C₂₆₉	#41	1	2	yes	—
95	C₂²×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
96	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
97	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
98	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
99	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
101	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
102	C₄×C₂₆₉	—	2	2	no	—
103	C₄×C₂₆₉	—	2	2	no	—
104	C₄×C₂₆₉	—	4	2	no	—
105	C₄×C₂₆₉	—	4	2	no	—
106	C₄×C₂₆₉	#125	1	4	yes	—
107	Dic₂₆₉	—	2	2	no	—
108	Dic₂₆₉	—	2	2	no	—
109	Dic₂₆₉	—	2	2	no	—
110	Dic₂₆₉	—	2	2	no	—
111	Dic₂₆₉	—	4	2	no	—
112	Dic₂₆₉	—	4	2	no	—
113	C₂×C₂₆₉	C₂×#125	1	1	yes	characteristic
114	C₂×C₂₆₉	#529	1	1	yes	characteristic
115	C₂×C₂₆₉	#529	1	1	yes	characteristic
116	C₂×C₂₆₉	#529	1	1	yes	characteristic
117	C₂×C₂₆₉	#529	1	1	yes	characteristic
118	C₂×C₂₆₉	#533	1	1	yes	characteristic
119	C₂×C₂₆₉	#533	1	1	yes	characteristic
120	C₂₆₉	#2864	1	1	yes	characteristic, complemented, Sylow
121	#2864	—	269	1	no	maximal, Sylow
122	#389	—	269	1	no	—
123	#389	—	269	1	no	—
124	#389	—	269	1	no	—
125	#389	D₂₆₉	1	1	yes	characteristic
126	#387	—	269	2	no	—
127	#387	—	269	2	no	—
128	#387	—	269	2	no	—
129	#387	—	269	2	no	—
130	#382	—	269	1	no	—
131	#382	—	269	1	no	—
132	C₄³	—	269	1	no	—
133	C₂×C₄⋊C₄	—	269	2	no	—
134	C₂×C₄⋊C₄	—	269	2	no	—
135	C₂×C₄⋊C₄	—	269	1	no	—
136	C₂×C₄⋊C₄	—	269	1	no	—
137	C₂×C₄⋊C₄	—	269	2	no	—
138	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
141	C₂×C₄²	—	269	1	no	—
142	C₂×C₄²	—	269	1	no	—
143	C₂×C₄²	—	269	1	no	—
144	C₂×C₄²	—	269	1	no	—
145	C₂×C₄²	—	269	1	no	—
146	C₂×C₄²	—	269	1	no	—
147	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
148	#95	—	269	2	no	—
149	#95	—	269	2	no	—
150	#95	—	269	2	no	—
151	#95	—	269	2	no	—
152	#95	—	269	2	no	—
153	#95	—	269	2	no	—
154	#95	—	269	2	no	—
155	#95	C₂×D₂₆₉	1	2	yes	—
156	C₂²×C₄	—	269	1	no	—
157	C₂²×C₄	—	269	1	no	—
158	C₂²×C₄	—	269	1	no	—
159	C₂²×C₄	—	269	1	no	—
160	C₂²×C₄	—	269	2	no	—
161	C₂²×C₄	—	269	2	no	—
162	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
163	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
164	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	538	4	no	—
171	C₄⋊C₄	—	538	4	no	—
172	C₄⋊C₄	—	538	2	no	—
173	C₄⋊C₄	—	538	2	no	—
174	C₄⋊C₄	—	538	4	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	269	4	no	—
177	C₄²	—	538	2	no	—
178	C₄²	—	538	2	no	—
179	C₄²	—	538	2	no	—
180	C₄²	—	538	2	no	—
181	C₄²	—	538	2	no	—
182	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	269	2	no	—
194	C₂×C₄	—	269	2	no	—
195	C₂×C₄	—	269	2	no	—
196	C₂×C₄	—	269	2	no	—
197	C₂×C₄	—	538	1	no	—
198	C₂×C₄	—	538	1	no	—
199	C₂×C₄	—	538	1	no	—
200	C₂×C₄	—	538	1	no	—
201	C₂×C₄	—	538	1	no	—
202	C₂×C₄	—	538	1	no	—
203	C₂×C₄	—	538	1	no	—
204	C₂×C₄	—	538	1	no	—
205	C₂×C₄	—	538	2	no	—
206	C₂×C₄	—	538	2	no	—
207	C₂×C₄	—	538	2	no	—
208	C₂×C₄	—	538	2	no	—
209	C₂×C₄	—	538	2	no	—
210	C₂×C₄	—	538	2	no	—
211	C₂×C₄	Q₈×D₂₆₉	1	2	yes	—
212	C₂×C₄	#16864401	1	2	yes	—
213	C₂×C₄	#16864401	1	2	yes	—
214	C₂×C₄	#16864402	1	2	yes	—
215	C₂×C₄	#16864403	1	2	yes	—
216	C₂²	C₂×Q₈×D₂₆₉	1	1	yes	characteristic
217	C₂²	#41×D₂₆₉	1	1	yes	characteristic
218	C₂²	#41×D₂₆₉	1	1	yes	characteristic
219	C₂²	C₂×#16864401	1	1	yes	characteristic
220	C₂²	C₂×#16864401	1	1	yes	characteristic
221	C₂²	C₂×#16864402	1	1	yes	characteristic
222	C₂²	C₂×#16864403	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	538	2	no	—
228	C₄	—	538	2	no	—
229	C₄	—	538	2	no	—
230	C₄	—	538	2	no	—
231	C₄	—	1076	2	no	—
232	C₄	—	1076	2	no	—
233	C₄	#22575443	1	4	yes	—
234	C₂	C₂×#22575443	1	1	yes	characteristic
235	C₂	#31771136	1	1	yes	characteristic
236	C₂	#31771137	1	1	yes	characteristic
237	C₂	#31771141	1	1	yes	characteristic
238	C₂	#31771142	1	1	yes	characteristic
239	C₂	#31771160	1	1	yes	characteristic
240	C₂	#31771161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₆₉
commutator, socle, prime radical, cosocle	C₂³×C₂₆₉	C₂⁴
Fitting	#389×C₂₆₉	C₂
hypercenter	#389	D₂₆₉
hypocenter	C₂₆₉	#2864
LCS	G ▷ C₂³×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
120. C₂₆₉	#2864	semidirect

Maximal subgroups

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

subgroup	index
#31770504	2
#31770503	2
#31770501	2
#31770491	2
#31770490	2
#31770487	2
#31770486	2
#31770470	2
C₄²×Dic₂₆₉	2
#389×C₂₆₉	2
#2864	269