Subgroups — #16925482

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14527532	C₂	1	2	yes	maximal
2	#14527067	C₂	1	2	yes	maximal
3	#14526617	C₂	1	2	yes	maximal
4	C₄²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	#11762862	C₂²	1	4	yes	—
6	#11762562	C₄	1	4	yes	complemented
7	#11762560	C₄	1	4	yes	complemented
8	#11762530	—	2	2	no	—
9	C₈×#3622	—	2	2	no	—
10	C₂×C₄×C₈×C₁₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₇	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₄³×C₁₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₇	—	2	2	no	—
19	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₁₇	—	2	4	no	—
22	C₄×C₈×C₁₇	C₈	1	8	yes	complemented
23	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	Q₈	1	2	yes	—
28	C₂²⋊#3622	—	2	4	no	—
29	C₄⋊#3622	C₂×C₄	1	8	yes	—
30	C₄×#3622	—	2	4	no	—
31	C₂²×C₄×C₁₇	—	2	2	no	—
32	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	#41	1	2	yes	—
34	C₂×C₈×C₁₇	—	2	2	no	—
35	C₂×C₈×C₁₇	C₄²	1	2	yes	—
36	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
43	C₄²×C₁₇	—	2	8	no	—
44	C₄²×C₁₇	—	2	2	no	—
45	C₄²×C₁₇	C₂×C₈	1	4	yes	—
46	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
47	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×#3622	—	2	8	no	—
50	#8266	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	#117	1	1	yes	characteristic
52	C₂×C₄×C₁₇	—	2	8	no	—
53	C₂×C₄×C₁₇	—	2	4	no	—
54	C₂×C₄×C₁₇	—	2	4	no	—
55	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
58	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
59	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
61	C₈×C₁₇	—	2	2	no	—
62	C₈×C₁₇	C₄×C₈	1	2	yes	—
63	C₈×C₁₇	C₄×C₈	1	2	yes	—
64	C₈×C₁₇	C₄⋊C₈	1	4	yes	—
65	C₈×C₁₇	C₄⋊C₈	1	4	yes	—
66	#3622	—	4	8	no	—
67	C₄²×C₈	D₁₇	1	1	yes	characteristic
68	#1564	—	17	2	no	—
69	#1450	—	17	2	no	—
70	#1276	—	17	2	no	—
71	C₂²×C₁₇	—	2	2	no	—
72	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₁₇	#431	1	2	yes	—
74	C₄×C₁₇	—	2	8	no	—
75	C₄×C₁₇	—	2	2	no	—
76	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
77	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
78	C₄×C₁₇	C₂×C₄⋊C₈	1	2	yes	—
79	C₄×C₁₇	#443	1	4	yes	—
80	#431	—	17	4	no	—
81	C₂×C₄×C₈	Dic₁₇	1	2	yes	—
82	C₂×C₄×C₈	Dic₁₇	1	2	yes	—
83	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
85	C₄³	C₂×D₁₇	1	1	yes	characteristic
86	#335	—	17	4	no	—
87	#334	—	17	4	no	—
88	#322	—	34	2	no	—
89	C₈²	—	34	2	no	—
90	C₂×C₁₇	—	2	2	no	—
91	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₁₇	#1772	1	2	yes	—
93	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
95	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
96	C₂×C₄²	—	2	2	no	—
97	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
98	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
99	C₄⋊C₈	—	17	8	no	—
100	C₂²⋊C₈	—	34	4	no	—
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	—	34	4	no	—
103	C₄×C₈	#3622	1	8	yes	complemented
104	C₄×C₈	Dic₃₄	1	2	yes	—
105	C₄×C₈	C₄×D₁₇	1	2	yes	—
106	C₄×C₈	C₄×D₁₇	1	2	yes	—
107	C₄×C₈	D₆₈	1	1	yes	characteristic
108	C₄×C₈	D₆₈	1	1	yes	characteristic
109	C₁₇	#8266	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	#63252	1	2	yes	—
112	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	34	8	no	—
115	C₂×C₈	C₂×#3622	1	4	yes	—
116	C₂×C₈	C₂×#3622	1	4	yes	—
117	C₂×C₈	#63223	1	4	yes	—
118	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
119	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
120	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
122	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
123	C₄²	—	2	8	no	—
124	C₄²	—	2	2	no	—
125	C₄²	C₂×#3622	1	4	yes	—
126	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
127	C₄²	C₂×C₄×D₁₇	1	2	yes	—
128	C₄²	C₂×D₆₈	1	1	yes	characteristic
129	C₂³	#10587612	1	1	yes	characteristic
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₄×Dic₃₄	1	2	yes	—
134	C₂×C₄	C₄×D₆₈	1	2	yes	—
135	C₂×C₄	C₂²×#3622	1	2	yes	—
136	C₂×C₄	C₂×#63223	1	2	yes	—
137	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	68	8	no	—
141	C₈	C₄×#3622	1	2	yes	—
142	C₈	C₄×#3622	1	2	yes	—
143	C₈	C₄⋊#3622	1	4	yes	—
144	C₈	C₄⋊#3622	1	4	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₄×C₄⋊Dic₁₇	1	1	yes	characteristic
147	C₂²	#11762862	1	2	yes	—
148	C₄	—	2	8	no	—
149	C₄	—	2	2	no	—
150	C₄	C₂×C₄×#3622	1	1	yes	characteristic
151	C₄	C₄×#63223	1	1	yes	characteristic
152	C₄	C₂×C₄⋊#3622	1	2	yes	—
153	C₄	#11762927	1	4	yes	—
154	C₂	—	2	2	no	—
155	C₂	C₄×C₄⋊#3622	1	1	yes	characteristic
156	C₂	#14528663	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₆₈
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄²×C₈×C₁₇	C₂
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₇	#117
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄²×C₈	D₁₇
hypocenter	C₁₇	#8266
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. #11762562	C₄	semidirect
7. #11762560	C₄	semidirect
22. C₄×C₈×C₁₇	C₈	semidirect
103. C₄×C₈	#3622	semidirect
109. C₁₇	#8266	semidirect

Maximal subgroups

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

subgroup	index
#14527532	2
#14527067	2
#14526617	2
C₄²×C₈×C₁₇	2
#8266	17