Subgroups — #46813363

Order 34432.

500 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31770512	C₂	1	2	yes	maximal
2	#31770504	C₂	1	2	yes	maximal
3	#31770491	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊Dic₂₆₉	C₂	1	2	yes	maximal
6	#31770451	C₂	1	2	yes	maximal
7	C₄²×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Dic₂₆₉	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
13	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
14	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
15	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
16	#22575360	C₂²	1	4	yes	—
17	#22575360	C₂²	1	4	yes	—
18	#22575360	C₂²	1	2	yes	—
19	#22575360	C₂²	1	4	yes	—
20	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₂₆₉	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₆₉	C₂²	1	2	yes	—
26	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
27	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
28	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
29	C₄⋊Dic₂₆₉	—	2	8	no	—
30	C₄⋊Dic₂₆₉	—	2	4	no	—
31	C₄×Dic₂₆₉	—	2	4	no	—
32	C₄×Dic₂₆₉	—	2	4	no	—
33	C₄×Dic₂₆₉	—	2	2	no	—
34	C₄×Dic₂₆₉	—	2	4	no	—
35	C₄×Dic₂₆₉	Q₈	1	4	yes	complemented
36	C₂×Q₈×C₂₆₉	—	2	4	no	—
37	C₂×Q₈×C₂₆₉	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₂₆₉	C₂×C₄	1	8	yes	—
44	C₄⋊C₄×C₂₆₉	C₂×C₄	1	4	yes	—
45	C₄²×C₂₆₉	C₂×C₄	1	8	yes	—
46	C₄²×C₂₆₉	C₂×C₄	1	4	yes	—
47	C₂×Dic₂₆₉	—	2	2	no	—
48	C₂×Dic₂₆₉	—	2	2	no	—
49	C₂×Dic₂₆₉	—	2	2	no	—
50	C₂×Dic₂₆₉	—	2	2	no	—
51	C₂×Dic₂₆₉	—	2	4	no	—
52	C₂×Dic₂₆₉	—	2	4	no	—
53	C₂×Dic₂₆₉	—	2	4	no	—
54	C₂×Dic₂₆₉	C₂×Q₈	1	4	yes	—
55	C₂×Dic₂₆₉	#41	1	4	yes	—
56	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
57	Q₈×C₂₆₉	—	2	8	no	—
58	C₂×C₄×C₂₆₉	—	2	4	no	—
59	C₂×C₄×C₂₆₉	—	2	2	no	—
60	C₂×C₄×C₂₆₉	—	2	4	no	—
61	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₂₆₉	#41	1	4	yes	—
68	C₂×C₄×C₂₆₉	#41	1	2	yes	—
69	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
70	C₂²×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
71	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
72	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
73	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
74	C₄×C₂₆₉	—	2	4	no	—
75	C₄×C₂₆₉	—	2	2	no	—
76	C₄×C₂₆₉	—	2	4	no	—
77	C₄×C₂₆₉	#117	1	4	yes	—
78	C₄×C₂₆₉	C₄×Q₈	1	4	yes	—
79	Dic₂₆₉	—	2	4	no	—
80	Dic₂₆₉	—	2	4	no	—
81	Dic₂₆₉	—	4	4	no	—
82	C₂×C₂₆₉	C₂×#117	1	1	yes	characteristic
83	C₂×C₂₆₉	C₂×C₄×Q₈	1	1	yes	characteristic
84	C₂×C₂₆₉	C₄×#41	1	1	yes	characteristic
85	C₂×C₂₆₉	#529	1	2	yes	—
86	C₂×C₂₆₉	#533	1	2	yes	—
87	C₂₆₉	#2295	1	1	yes	characteristic, complemented, Sylow
88	#2295	—	269	1	no	maximal, Sylow
89	C₂×C₄×Q₈	D₂₆₉	1	1	yes	characteristic
90	#391	—	269	2	no	—
91	#389	—	269	2	no	—
92	#387	—	269	4	no	—
93	C₄×C₄⋊C₄	—	269	1	no	—
94	C₄×C₄⋊C₄	—	269	2	no	—
95	#376	—	269	2	no	—
96	C₄³	—	269	1	no	—
97	C₂²×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
98	C₄×Q₈	Dic₂₆₉	1	8	yes	complemented
99	C₂×C₄⋊C₄	—	269	4	no	—
100	C₂×C₄⋊C₄	—	269	2	no	—
101	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
102	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
103	C₂×C₄²	—	269	1	no	—
104	C₂×C₄²	—	269	2	no	—
105	C₂×C₄²	—	269	2	no	—
106	C₂×C₄²	—	269	1	no	—
107	C₂×C₄²	—	269	2	no	—
108	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
109	C₂×C₄²	C₂×D₂₆₉	1	2	yes	—
110	#95	—	269	4	no	—
111	#95	—	269	4	no	—
112	#95	—	269	2	no	—
113	#95	—	269	4	no	—
114	C₂×Q₈	—	2	4	no	—
115	C₂×Q₈	C₂×Dic₂₆₉	1	4	yes	—
116	C₂²×C₄	—	269	2	no	—
117	C₂²×C₄	—	269	2	no	—
118	C₂²×C₄	—	269	4	no	—
119	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
120	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
122	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
124	C₄⋊C₄	—	538	8	no	—
125	C₄⋊C₄	—	538	4	no	—
126	C₄⋊C₄	C₂×Dic₂₆₉	1	8	yes	—
127	C₄⋊C₄	C₂×Dic₂₆₉	1	4	yes	—
128	C₄²	—	269	4	no	—
129	C₄²	—	538	4	no	—
130	C₄²	—	538	4	no	—
131	C₄²	—	538	2	no	—
132	C₄²	—	538	4	no	—
133	C₄²	C₂×Dic₂₆₉	1	8	yes	—
134	C₄²	C₂×Dic₂₆₉	1	4	yes	—
135	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
136	Q₈	—	2	8	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	269	4	no	—
141	C₂×C₄	—	269	4	no	—
142	C₂×C₄	—	538	2	no	—
143	C₂×C₄	—	538	2	no	—
144	C₂×C₄	—	538	2	no	—
145	C₂×C₄	—	538	2	no	—
146	C₂×C₄	—	538	4	no	—
147	C₂×C₄	—	538	4	no	—
148	C₂×C₄	—	538	4	no	—
149	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
150	C₂×C₄	C₂²×Dic₂₆₉	1	4	yes	—
151	C₂×C₄	C₂²×Dic₂₆₉	1	4	yes	—
152	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
153	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
154	C₂×C₄	Q₈×D₂₆₉	1	2	yes	—
155	C₂×C₄	#16864402	1	4	yes	—
156	C₂×C₄	#16864403	1	2	yes	—
157	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
158	C₂²	C₂×Q₈×D₂₆₉	1	1	yes	characteristic
159	C₂²	#41×D₂₆₉	1	2	yes	—
160	C₂²	C₂×#16864402	1	2	yes	—
161	C₂²	C₂×#16864403	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	538	4	no	—
166	C₄	—	538	4	no	—
167	C₄	—	1076	4	no	—
168	C₄	#22575412	1	4	yes	—
169	C₄	Q₈×Dic₂₆₉	1	4	yes	—
170	C₂	C₂×#22575412	1	1	yes	characteristic
171	C₂	C₂×Q₈×Dic₂₆₉	1	1	yes	characteristic
172	C₂	#41×Dic₂₆₉	1	1	yes	characteristic
173	C₂	#31771142	1	2	yes	—
174	C₂	#31771159	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₆₉
commutator	C₂²×C₂₆₉	C₂³×C₄
Frattini	C₂³	C₂³×D₂₆₉
Fitting	C₂×C₄×Q₈×C₂₆₉	C₂
socle, prime radical, cosocle	C₂³×C₂₆₉	C₂⁴
hypercenter	C₂×C₄×Q₈	D₂₆₉
hypocenter	C₂₆₉	#2295
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×Q₈×C₂₆₉	C₄	semidirect
35. C₄×Dic₂₆₉	Q₈	semidirect
87. C₂₆₉	#2295	semidirect
98. C₄×Q₈	Dic₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31770512	2
#31770504	2
#31770491	2
C₄×C₄⋊Dic₂₆₉	2
#31770451	2
C₄²×Dic₂₆₉	2
C₂×C₄×Q₈×C₂₆₉	2
#2295	269