Subgroups — #46813508

Order 34432.

600 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22575412	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
3	#31770504	C₂	1	4	yes	maximal, complemented
4	#31770480	C₂	1	4	yes	maximal
5	#31770475	C₂	1	2	yes	maximal, complemented
6	#31770471	C₂	1	2	yes	maximal, complemented
7	C₂×#117×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₂₆₉	C₂²	1	2	yes	—
9	#22575412	—	2	4	no	—
10	C₂×C₄⋊Dic₂₆₉	—	2	4	no	—
11	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₂₆₉	C₂²	1	8	yes	—
16	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
17	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
18	#22575360	C₂²	1	8	yes	—
19	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
20	#117×C₂₆₉	C₄	1	8	yes	complemented
21	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₆₉	C₂²	1	2	yes	—
24	C₂²×Dic₂₆₉	—	2	4	no	—
25	C₂²×Dic₂₆₉	—	4	4	no	—
26	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
27	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
28	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
29	C₄⋊Dic₂₆₉	—	2	4	no	—
30	C₄⋊Dic₂₆₉	—	2	4	no	—
31	C₄⋊Dic₂₆₉	—	2	2	no	—
32	C₄⋊Dic₂₆₉	—	2	2	no	—
33	C₄⋊Dic₂₆₉	—	4	8	no	—
34	C₂²⋊Dic₂₆₉	—	2	4	no	—
35	C₄×Dic₂₆₉	—	2	4	no	—
36	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₆₉	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₂₆₉	D₄	1	4	yes	—
39	C₂²×C₄×C₂₆₉	Q₈	1	4	yes	—
40	C₂²×C₄×C₂₆₉	C₂³	1	4	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	complemented
44	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	8	yes	—
45	C₄²×C₂₆₉	C₂×C₄	1	8	yes	complemented
46	C₂×Dic₂₆₉	—	2	2	no	—
47	C₂×Dic₂₆₉	—	2	2	no	—
48	C₂×Dic₂₆₉	—	2	4	no	—
49	C₂×Dic₂₆₉	—	2	4	no	—
50	C₂×Dic₂₆₉	—	2	8	no	—
51	C₂×Dic₂₆₉	—	2	4	no	—
52	C₂×Dic₂₆₉	—	4	8	no	—
53	C₂³×C₂₆₉	—	2	2	no	—
54	C₂³×C₂₆₉	—	2	2	no	—
55	C₂³×C₂₆₉	C₂²×C₄	1	2	yes	—
56	C₂³×C₂₆₉	C₂×D₄	1	2	yes	—
57	C₂³×C₂₆₉	C₂×Q₈	1	2	yes	—
58	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
59	C₂×C₄×C₂₆₉	—	2	4	no	—
60	C₂×C₄×C₂₆₉	—	2	4	no	—
61	C₂×C₄×C₂₆₉	C₄⋊C₄	1	16	yes	complemented
62	C₂×C₄×C₂₆₉	C₂²×C₄	1	8	yes	—
63	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
69	C₂²×C₂₆₉	—	2	4	no	—
70	C₂²×C₂₆₉	—	2	2	no	—
71	C₂²×C₂₆₉	—	2	2	no	—
72	C₂²×C₂₆₉	—	2	4	no	—
73	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
79	C₄×C₂₆₉	—	2	8	no	—
80	C₄×C₂₆₉	C₂×C₄⋊C₄	1	4	yes	—
81	C₄×C₂₆₉	C₂×C₄⋊C₄	1	4	yes	—
82	Dic₂₆₉	—	4	2	no	—
83	Dic₂₆₉	—	4	4	no	—
84	Dic₂₆₉	—	4	2	no	—
85	C₂×C₂₆₉	—	2	4	no	—
86	C₂×C₂₆₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₂₆₉	#520	1	2	yes	—
88	C₂×C₂₆₉	#535	1	1	yes	characteristic
89	C₂×C₂₆₉	#536	1	1	yes	characteristic
90	C₂×C₂₆₉	#544	1	2	yes	—
91	C₂₆₉	#2317	1	1	yes	characteristic, complemented, Sylow
92	#2317	—	269	1	no	maximal, Sylow
93	C₂×#117	—	269	1	no	—
94	C₂×#117	D₂₆₉	1	1	yes	characteristic
95	C₂²×C₄⋊C₄	—	269	1	no	—
96	#389	—	269	4	no	—
97	#385	—	269	4	no	—
98	#384	—	269	2	no	—
99	#382	—	269	2	no	—
100	C₂³×C₄	—	269	2	no	—
101	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
102	#117	—	538	4	no	—
103	#117	Dic₂₆₉	1	8	yes	complemented
104	C₂×C₄⋊C₄	—	269	2	no	—
105	C₂×C₄⋊C₄	—	269	2	no	—
106	C₂×C₄⋊C₄	—	269	1	no	—
107	C₂×C₄⋊C₄	—	269	1	no	—
108	C₂×C₄⋊C₄	—	269	8	no	—
109	C₂×C₄⋊C₄	—	538	4	no	—
110	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
111	C₂×C₂²⋊C₄	—	269	2	no	—
112	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
113	C₂×C₄²	—	269	2	no	—
114	C₂×C₄²	C₂×D₂₆₉	1	2	yes	—
115	#95	—	269	8	no	—
116	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
117	C₂²×C₄	—	269	2	no	—
118	C₂²×C₄	—	269	4	no	—
119	C₂²×C₄	—	269	2	no	—
120	C₂²×C₄	—	538	4	no	—
121	C₂²×C₄	—	1076	4	no	—
122	C₂²×C₄	C₂×Dic₂₆₉	1	4	yes	—
123	C₂²×C₄	D₁₀₇₆	1	4	yes	—
124	C₂²×C₄	Dic₅₃₈	1	4	yes	—
125	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
126	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
128	C₄⋊C₄	—	538	4	no	—
129	C₄⋊C₄	—	538	4	no	—
130	C₄⋊C₄	—	538	2	no	—
131	C₄⋊C₄	—	538	2	no	—
132	C₄⋊C₄	—	1076	8	no	—
133	C₄⋊C₄	C₂×Dic₂₆₉	1	8	yes	complemented
134	C₂²⋊C₄	—	538	4	no	—
135	C₂²⋊C₄	C₂×Dic₂₆₉	1	8	yes	—
136	C₄²	—	538	4	no	—
137	C₄²	C₂×Dic₂₆₉	1	8	yes	complemented
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂²×Dic₂₆₉	1	2	yes	—
141	C₂³	C₂×D₁₀₇₆	1	2	yes	—
142	C₂³	C₂×Dic₅₃₈	1	2	yes	—
143	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	538	2	no	—
147	C₂×C₄	—	538	2	no	—
148	C₂×C₄	—	538	4	no	—
149	C₂×C₄	—	538	4	no	—
150	C₂×C₄	—	538	8	no	—
151	C₂×C₄	—	538	4	no	—
152	C₂×C₄	—	1076	8	no	—
153	C₂×C₄	C₄⋊Dic₂₆₉	1	16	yes	complemented
154	C₂×C₄	C₂²×Dic₂₆₉	1	8	yes	—
155	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
156	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
157	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
158	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
159	C₂×C₄	C₂×Dic₅₃₈	1	2	yes	—
160	C₂×C₄	C₂×Dic₅₃₈	1	2	yes	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	C₂×C₄⋊Dic₂₆₉	1	4	yes	—
166	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
167	C₂²	C₂²×D₁₀₇₆	1	1	yes	characteristic
168	C₂²	C₂²×Dic₅₃₈	1	1	yes	characteristic
169	C₂²	#22575499	1	2	yes	—
170	C₂²	#22575500	1	2	yes	—
171	C₄	—	2	8	no	—
172	C₄	—	1076	4	no	—
173	C₄	—	1076	2	no	—
174	C₄	—	1076	2	no	—
175	C₄	C₂×C₄⋊Dic₂₆₉	1	4	yes	—
176	C₄	C₂×C₄⋊Dic₂₆₉	1	4	yes	—
177	C₂	—	2	4	no	—
178	C₂	C₂²×C₄⋊Dic₂₆₉	1	1	yes	characteristic
179	C₂	#31771088	1	2	yes	—
180	C₂	#31771174	1	1	yes	characteristic
181	C₂	#31771181	1	1	yes	characteristic
182	C₂	#31771233	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₆₉
commutator	C₂²×C₂₆₉	C₂³×C₄
Fitting	C₂×#117×C₂₆₉	C₂
socle, cosocle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂⁴×C₂₆₉	C₂³
hypercenter	C₂×#117	D₂₆₉
hypocenter	C₂₆₉	#2317
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #31770504	C₂	semidirect
5. #31770475	C₂	semidirect
6. #31770471	C₂	semidirect
20. #117×C₂₆₉	C₄	semidirect
43. C₄⋊C₄×C₂₆₉	C₂×C₄	semidirect
45. C₄²×C₂₆₉	C₂×C₄	semidirect
61. C₂×C₄×C₂₆₉	C₄⋊C₄	semidirect
91. C₂₆₉	#2317	semidirect
103. #117	Dic₂₆₉	semidirect
133. C₄⋊C₄	C₂×Dic₂₆₉	semidirect
137. C₄²	C₂×Dic₂₆₉	semidirect
153. C₂×C₄	C₄⋊Dic₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22575412	2
C₂²×C₄⋊Dic₂₆₉	2
#31770504	2
#31770480	2
#31770475	2
#31770471	2
C₂×#117×C₂₆₉	2
#2317	269