Subgroups — #16928132

Order 4352.

322 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	#14528922	C₂	1	2	yes	maximal
2	#14527543	C₂	1	2	yes	maximal
3	#14527540	C₂	1	2	yes	maximal
4	C₂×C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic
5	#11762833	C₂²	1	4	yes	—
6	#11762574	—	2	4	no	—
7	#11762562	—	2	4	no	—
8	#11762560	—	2	4	no	—
9	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₈²×C₁₇	C₄	1	1	yes	characteristic
13	C₈²×C₁₇	C₄	1	1	yes	characteristic
14	C₈²×C₁₇	C₄	1	1	yes	characteristic
15	C₈²×C₁₇	C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	D₄	1	2	yes	—
17	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
18	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
19	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×#63223	—	2	4	no	—
21	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	D₄	1	2	yes	—
28	C₄×C₈×C₁₇	D₄	1	2	yes	—
29	C₄×C₈×C₁₇	Q₈	1	2	yes	—
30	C₄⋊#3622	—	2	4	no	—
31	C₄⋊#3622	—	2	4	no	—
32	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₁₇	—	2	4	no	—
36	C₂×C₈×C₁₇	—	2	4	no	—
37	C₂×C₈×C₁₇	—	2	4	no	—
38	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
48	#63223	—	4	8	no	—
49	C₂×#3622	—	4	4	no	—
50	#8824	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#135	1	4	yes	—
61	C₈×C₁₇	—	2	2	no	—
62	C₈×C₁₇	—	2	2	no	—
63	C₈×C₁₇	—	2	2	no	—
64	C₈×C₁₇	—	2	2	no	—
65	C₈×C₁₇	#108	1	4	yes	—
66	C₈×C₁₇	#108	1	4	yes	—
67	#3622	—	8	4	no	—
68	#1837	—	17	2	no	—
69	#1567	—	17	2	no	—
70	#1566	—	17	2	no	—
71	C₂×C₈²	D₁₇	1	1	yes	characteristic
72	C₂²×C₁₇	—	2	1	no	—
73	C₂²×C₁₇	—	2	1	no	—
74	C₂²×C₁₇	#384	1	1	yes	characteristic
75	C₂²×C₁₇	#499	1	2	yes	—
76	C₄×C₁₇	—	2	1	no	—
77	C₄×C₁₇	—	2	1	no	—
78	C₄×C₁₇	#427	1	2	yes	—
79	C₄×C₁₇	C₂×#108	1	2	yes	—
80	C₄×C₁₇	C₂×#108	1	2	yes	—
81	C₄×C₁₇	#495	1	2	yes	—
82	#423	—	17	4	no	—
83	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
86	#341	—	34	4	no	—
87	#335	—	34	4	no	—
88	#334	—	34	4	no	—
89	C₈²	Dic₁₇	1	1	yes	characteristic
90	C₈²	Dic₁₇	1	1	yes	characteristic
91	C₈²	Dic₁₇	1	1	yes	characteristic
92	C₈²	Dic₁₇	1	1	yes	characteristic
93	C₂×C₁₇	—	2	2	no	—
94	C₂×C₁₇	#1846	1	1	yes	characteristic
95	C₂×C₁₇	#1848	1	1	yes	characteristic
96	C₂×C₁₇	#1848	1	1	yes	characteristic
97	C₂×M₁₆	—	34	4	no	—
98	C₂²×C₈	Dic₃₄	1	2	yes	—
99	C₂²×C₈	Dic₃₄	1	2	yes	—
100	C₂²×C₈	D₆₈	1	2	yes	—
101	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
102	C₄⋊C₈	—	34	4	no	—
103	C₄⋊C₈	—	34	4	no	—
104	C₄×C₈	Dic₃₄	1	2	yes	—
105	C₄×C₈	D₆₈	1	2	yes	—
106	C₄×C₈	D₆₈	1	2	yes	—
107	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
108	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
113	C₁₇	#8824	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
116	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
117	M₁₆	—	68	8	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	68	4	no	—
122	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
123	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
124	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
125	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
126	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
127	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
128	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
129	C₄²	C₂×D₆₈	1	1	yes	characteristic
130	C₄²	C₂×D₆₈	1	1	yes	characteristic
131	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
132	C₂³	#10587555	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	#10587555	1	1	yes	characteristic
136	C₂×C₄	#10587555	1	1	yes	characteristic
137	C₂×C₄	#10587560	1	1	yes	characteristic
138	C₂×C₄	#10587590	1	4	yes	—
139	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	136	4	no	—
147	C₈	#10587490	1	4	yes	—
148	C₈	#10587490	1	4	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#11762679	1	1	yes	characteristic
152	C₂²	#11763211	1	2	yes	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#11762847	1	2	yes	—
156	C₄	C₂×#10587490	1	2	yes	—
157	C₄	C₂×#10587490	1	2	yes	—
158	C₄	#11763193	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#14528948	1	1	yes	characteristic
161	C₂	#14528956	1	1	yes	characteristic
162	C₂	#14528956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587560
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₈²×C₁₇	C₂
socle	C₂²×C₁₇	#384
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×C₈²	D₁₇
hypocenter	C₁₇	#8824
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
113. C₁₇	#8824	semidirect

Maximal subgroups

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

subgroup	index
#14528922	2
#14527543	2
#14527540	2
C₂×C₈²×C₁₇	2
#8824	17