Subgroups — #46809115

Order 34432.

656 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₂₆₉	C₂	1	2	yes	maximal
2	C₂×#22575360	C₂	1	4	yes	maximal
3	C₂²×C₄²×C₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₂₆₉	C₂²	1	4	yes	—
5	C₂×C₄⋊Dic₂₆₉	—	2	4	no	—
6	C₂×C₄⋊Dic₂₆₉	—	2	4	no	—
7	C₂×C₄⋊Dic₂₆₉	C₄	1	8	yes	complemented
8	#22575360	—	2	8	no	—
9	C₂³×C₄×C₂₆₉	C₂²	1	2	yes	—
10	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₂₆₉	—	2	4	no	—
12	C₂×C₄²×C₂₆₉	C₄	1	4	yes	complemented
13	C₂²×Dic₂₆₉	—	2	8	no	—
14	C₂²×Dic₂₆₉	—	2	4	no	—
15	C₂²×Dic₂₆₉	—	2	4	no	—
16	C₂²×Dic₂₆₉	—	2	8	no	—
17	C₂²×Dic₂₆₉	C₂×C₄	1	8	yes	—
18	C₄⋊Dic₂₆₉	—	2	16	no	—
19	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₂₆₉	—	2	4	no	—
21	C₂²×C₄×C₂₆₉	—	2	4	no	—
22	C₂²×C₄×C₂₆₉	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₂₆₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₆₉	D₄	1	4	yes	—
26	C₂²×C₄×C₂₆₉	D₄	1	4	yes	—
27	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
28	C₂²×C₄×C₂₆₉	Q₈	1	4	yes	—
29	C₂²×C₄×C₂₆₉	Q₈	1	2	yes	—
30	C₄²×C₂₆₉	—	2	8	no	—
31	C₂×Dic₂₆₉	—	2	8	no	—
32	C₂×Dic₂₆₉	—	2	16	no	—
33	C₂×Dic₂₆₉	—	4	8	no	—
34	C₂×Dic₂₆₉	—	4	4	no	—
35	C₂×Dic₂₆₉	—	4	4	no	—
36	C₂³×C₂₆₉	C₂²×C₄	1	2	yes	—
37	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
38	C₂³×C₂₆₉	C₂×D₄	1	2	yes	—
39	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
40	C₂³×C₂₆₉	C₂×D₄	1	2	yes	—
41	C₂³×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
42	C₂³×C₂₆₉	C₂×Q₈	1	2	yes	—
43	C₂³×C₂₆₉	#41	1	4	yes	—
44	C₂×C₄×C₂₆₉	—	2	4	no	—
45	C₂×C₄×C₂₆₉	—	2	4	no	—
46	C₂×C₄×C₂₆₉	—	2	8	no	—
47	C₂×C₄×C₂₆₉	—	2	8	no	—
48	C₂×C₄×C₂₆₉	C₄²	1	4	yes	—
49	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	8	yes	—
50	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	4	yes	—
51	C₂×C₄×C₂₆₉	C₄⋊C₄	1	8	yes	—
52	C₂×C₄×C₂₆₉	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₂₆₉	C₄⋊C₄	1	8	yes	complemented
54	C₂²×C₂₆₉	C₂×C₄²	1	1	yes	characteristic
55	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	2	yes	—
56	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	2	yes	—
60	C₂²×C₂₆₉	#117	1	2	yes	—
61	C₂²×C₂₆₉	C₄×D₄	1	4	yes	—
62	C₂²×C₂₆₉	C₄×Q₈	1	4	yes	—
63	C₂²×C₂₆₉	#121	1	4	yes	—
64	C₂²×C₂₆₉	C₂²⋊Q₈	1	4	yes	—
65	C₂²×C₂₆₉	#124	1	2	yes	—
66	C₂²×C₂₆₉	#125	1	2	yes	—
67	C₂²×C₂₆₉	#127	1	1	yes	characteristic
68	C₂²×C₂₆₉	C₄⋊Q₈	1	2	yes	—
69	C₂²×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
70	C₄×C₂₆₉	—	2	8	no	—
71	C₄×C₂₆₉	#95	1	8	yes	complemented
72	Dic₂₆₉	—	4	8	no	—
73	C₂×C₂₆₉	C₂×#95	1	1	yes	characteristic
74	C₂×C₂₆₉	C₄×C₄⋊C₄	1	2	yes	—
75	C₂×C₂₆₉	#380	1	2	yes	—
76	C₂×C₂₆₉	#382	1	1	yes	characteristic
77	C₂×C₂₆₉	#384	1	1	yes	characteristic
78	C₂×C₂₆₉	#389	1	4	yes	—
79	C₂×C₂₆₉	#390	1	2	yes	—
80	C₂×C₂₆₉	#391	1	2	yes	—
81	C₂₆₉	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
82	C₄⋊#95	—	269	1	no	maximal, Sylow
83	C₂²×C₄⋊C₄	—	269	2	no	—
84	C₂²×C₄²	D₂₆₉	1	1	yes	characteristic
85	C₂×#95	—	269	4	no	—
86	C₂³×C₄	—	269	4	no	—
87	C₂³×C₄	C₂×D₂₆₉	1	2	yes	—
88	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	269	8	no	—
90	C₂×C₄⋊C₄	—	538	4	no	—
91	C₂×C₄⋊C₄	—	538	4	no	—
92	C₂×C₄²	—	2	4	no	—
93	C₂×C₄²	Dic₂₆₉	1	4	yes	complemented
94	#95	—	538	8	no	—
95	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
96	C₂²×C₄	—	2	4	no	—
97	C₂²×C₄	—	2	4	no	—
98	C₂²×C₄	—	269	8	no	—
99	C₂²×C₄	—	538	4	no	—
100	C₂²×C₄	—	538	8	no	—
101	C₂²×C₄	—	538	4	no	—
102	C₂²×C₄	—	538	8	no	—
103	C₂²×C₄	C₄×D₂₆₉	1	4	yes	—
104	C₂²×C₄	C₂×Dic₂₆₉	1	4	yes	—
105	C₂²×C₄	C₂×Dic₂₆₉	1	2	yes	—
106	C₂²×C₄	C₂²⋊D₂₆₉	1	4	yes	—
107	C₂²×C₄	D₁₀₇₆	1	4	yes	—
108	C₂²×C₄	D₁₀₇₆	1	2	yes	—
109	C₂²×C₄	Dic₅₃₈	1	4	yes	—
110	C₂²×C₄	Dic₅₃₈	1	2	yes	—
111	C₄⋊C₄	—	538	16	no	—
112	C₄²	—	2	8	no	—
113	C₂³	C₂×C₄×D₂₆₉	1	2	yes	—
114	C₂³	C₂²×Dic₂₆₉	1	1	yes	characteristic
115	C₂³	C₂×C₂²⋊D₂₆₉	1	2	yes	—
116	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
117	C₂³	C₂×D₁₀₇₆	1	2	yes	—
118	C₂³	C₂×Dic₅₃₈	1	1	yes	characteristic
119	C₂³	C₂×Dic₅₃₈	1	2	yes	—
120	C₂³	#16864402	1	4	yes	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	8	no	—
125	C₂×C₄	—	538	8	no	—
126	C₂×C₄	—	538	16	no	—
127	C₂×C₄	—	1076	8	no	—
128	C₂×C₄	—	1076	4	no	—
129	C₂×C₄	—	1076	4	no	—
130	C₂×C₄	C₄×Dic₂₆₉	1	4	yes	—
131	C₂×C₄	#16864379	1	8	yes	—
132	C₂×C₄	C₂²⋊Dic₂₆₉	1	4	yes	—
133	C₂×C₄	#16864381	1	8	yes	—
134	C₂×C₄	C₄⋊Dic₂₆₉	1	4	yes	—
135	C₂×C₄	C₄⋊Dic₂₆₉	1	8	yes	complemented
136	C₂²	C₂×C₄×Dic₂₆₉	1	1	yes	characteristic
137	C₂²	C₂×#16864379	1	2	yes	—
138	C₂²	C₂×C₂²⋊Dic₂₆₉	1	1	yes	characteristic
139	C₂²	C₂×#16864381	1	2	yes	—
140	C₂²	C₂×C₄⋊Dic₂₆₉	1	1	yes	characteristic
141	C₂²	C₂×C₄⋊Dic₂₆₉	1	2	yes	—
142	C₂²	#22575412	1	2	yes	—
143	C₂²	C₄×D₁₀₇₆	1	4	yes	—
144	C₂²	C₄×Dic₅₃₈	1	4	yes	—
145	C₂²	#22575427	1	4	yes	—
146	C₂²	#22575432	1	4	yes	—
147	C₂²	#22575441	1	2	yes	—
148	C₂²	#22575445	1	2	yes	—
149	C₂²	#22575450	1	1	yes	characteristic
150	C₂²	#22575453	1	2	yes	—
151	C₂²	#22575453	1	1	yes	characteristic
152	C₄	—	2	8	no	—
153	C₄	—	1076	8	no	—
154	C₄	#22575360	1	8	yes	complemented
155	C₂	C₂×#22575360	1	1	yes	characteristic
156	C₂	C₄×C₄⋊Dic₂₆₉	1	2	yes	—
157	C₂	#31770465	1	2	yes	—
158	C₂	#31770471	1	1	yes	characteristic
159	C₂	#31770475	1	1	yes	characteristic
160	C₂	#31770503	1	4	yes	—
161	C₂	#31770507	1	2	yes	—
162	C₂	#31770511	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₂₆₉
commutator	C₂²×C₂₆₉	C₂×C₄²
Fitting	C₂²×C₄²×C₂₆₉	C₂
socle, prime radical, cosocle	C₂⁴×C₂₆₉	C₂³
hypercenter	C₂²×C₄²	D₂₆₉
hypocenter	C₂₆₉	C₄⋊#95
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
7. C₂×C₄⋊Dic₂₆₉	C₄	semidirect
12. C₂×C₄²×C₂₆₉	C₄	semidirect
53. C₂×C₄×C₂₆₉	C₄⋊C₄	semidirect
71. C₄×C₂₆₉	#95	semidirect
81. C₂₆₉	C₄⋊#95	semidirect
93. C₂×C₄²	Dic₂₆₉	semidirect
135. C₂×C₄	C₄⋊Dic₂₆₉	semidirect
154. C₄	#22575360	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₂₆₉	2
C₂×#22575360	2
C₂²×C₄²×C₂₆₉	2
C₄⋊#95	269