Subgroups — #16954521

Order 4352.

298 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762643	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#10587465	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#10587465	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₇×#504	C₂	1	1	yes	maximal, characteristic
5	#11762643	—	2	2	no	—
6	C₂×#10587465	—	2	2	no	—
7	C₂×#10587465	C₂²	1	2	yes	—
8	C₂×#10587465	C₂²	1	2	yes	—
9	C₂×C₁₇×#131	C₄	1	1	yes	characteristic
10	C₁₇×#504	—	2	2	no	—
11	C₁₇×#504	—	2	2	no	—
12	C₂×C₁₇×M₃₂	C₄	1	2	yes	—
13	C₂×C₁₇×M₃₂	C₄	1	1	yes	characteristic
14	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×#41×C₁₇	C₈	1	1	yes	characteristic
17	C₁₇×#131	—	2	2	no	—
18	C₁₇×#131	—	2	2	no	—
19	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇	C₈	1	2	yes	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₁₇×M₃₂	—	2	4	no	—
24	C₁₇×M₃₂	—	2	2	no	—
25	C₂×C₁₆×C₁₇	—	2	2	no	—
26	C₂×C₁₆×C₁₇	—	2	2	no	—
27	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
30	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
31	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
32	#10587465	—	2	2	no	—
33	#10587465	—	2	2	no	—
34	#41×C₁₇	—	2	2	no	—
35	#41×C₁₇	—	2	2	no	—
36	C₂×Q₈×C₁₇	C₁₆	1	1	yes	characteristic
37	C₂×D₄×C₁₇	C₁₆	1	2	yes	—
38	C₂×D₄×C₁₇	C₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
41	M₁₆×C₁₇	—	2	2	no	—
42	M₁₆×C₁₇	—	2	4	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	—	2	2	no	—
45	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
49	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₁₆×C₁₇	—	2	2	no	—
51	C₁₆×C₁₇	C₂²⋊C₄	1	2	yes	—
52	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
54	#13554	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	C₂×C₁₆	1	2	yes	—
56	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
57	Q₈×C₁₇	—	2	2	no	—
58	D₄×C₁₇	—	2	4	no	—
59	D₄×C₁₇	—	2	2	no	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
66	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₈×C₁₇	—	2	2	no	—
68	C₈×C₁₇	C₂²⋊C₈	1	2	yes	—
69	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
71	C₂×#504	D₁₇	1	1	yes	characteristic
72	C₂×M₆₄	—	17	1	no	—
73	C₂²×C₃₂	—	17	1	no	—
74	C₂²⋊C₃₂	—	17	4	no	—
75	C₂²×C₁₇	—	2	2	no	—
76	C₂²×C₁₇	—	2	2	no	—
77	C₂²×C₁₇	—	2	4	no	—
78	C₂²×C₁₇	C₂²⋊C₁₆	1	2	yes	—
79	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
80	C₄×C₁₇	—	2	2	no	—
81	C₄×C₁₇	C₂²⋊C₁₆	1	2	yes	—
82	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
84	C₂×#131	Dic₁₇	1	1	yes	characteristic
85	#504	—	2	2	no	—
86	#504	—	2	2	no	—
87	C₂×M₃₂	Dic₁₇	1	1	yes	characteristic
88	C₂×M₃₂	Dic₁₇	1	2	yes	—
89	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
90	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
91	M₆₄	—	34	2	no	—
92	C₂×C₃₂	—	17	2	no	—
93	C₂×C₃₂	—	17	2	no	—
94	C₂×C₃₂	—	34	2	no	—
95	C₂×C₁₇	—	2	2	no	—
96	C₂×C₁₇	—	4	2	no	—
97	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
98	C₂×C₁₇	#2258	1	2	yes	—
99	C₂×#41	#3622	1	1	yes	characteristic
100	#131	—	2	2	no	—
101	#131	—	2	2	no	—
102	C₂×M₁₆	#3622	1	1	yes	characteristic
103	C₂×M₁₆	#3622	1	2	yes	—
104	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
105	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	4	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
111	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
112	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
113	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
114	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
115	C₃₂	—	34	2	no	—
116	C₃₂	—	34	2	no	—
117	C₁₇	#13554	1	1	yes	characteristic, complemented, Sylow
118	#41	—	2	2	no	—
119	#41	—	2	2	no	—
120	C₂×Q₈	#63214	1	1	yes	characteristic
121	C₂×D₄	#63214	1	2	yes	—
122	C₂×D₄	#63214	1	1	yes	characteristic
123	C₂²×C₄	C₂×#3622	1	2	yes	—
124	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
125	M₁₆	—	2	2	no	—
126	M₁₆	—	2	4	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	C₂×#3622	1	2	yes	—
130	C₂×C₈	C₂×#3622	1	1	yes	characteristic
131	C₂×C₈	#63223	1	2	yes	—
132	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
133	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
134	C₁₆	—	2	2	no	—
135	C₁₆	C₂²⋊Dic₁₇	1	2	yes	—
136	C₁₆	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
137	C₁₆	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
138	C₂³	C₂×#63214	1	2	yes	—
139	C₂³	C₂×#63214	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	D₄	—	2	4	no	—
142	D₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	C₂×#63214	1	2	yes	—
146	C₂×C₄	C₂×#63214	1	1	yes	characteristic
147	C₂×C₄	#10587483	1	2	yes	—
148	C₂×C₄	C₂²⋊#3622	1	2	yes	—
149	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	C₂²⋊#3622	1	2	yes	—
152	C₈	C₂×#63223	1	1	yes	characteristic
153	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂²⋊#63214	1	2	yes	—
158	C₂²	C₂²×#63214	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	C₂²⋊#63214	1	2	yes	—
161	C₄	C₂×C₂²⋊#3622	1	1	yes	characteristic
162	C₄	C₂×#10587483	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	4	2	no	—
165	C₂	C₂×C₂²⋊#63214	1	1	yes	characteristic
166	C₂	#14531078	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₇
commutator, socle	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂×C₁₇×#504	C₂
prime radical	C₂×#41×C₁₇	C₈
cosocle	C₂×C₁₆×C₁₇	C₂³
hypercenter	C₂×#504	D₁₇
hypocenter	C₁₇	#13554
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#11762643	C₂	semidirect
2. C₂²×#10587465	C₂	semidirect
3. C₂²⋊#10587465	C₂	semidirect
117. C₁₇	#13554	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762643	2
C₂²×#10587465	2
C₂²⋊#10587465	2
C₂×C₁₇×#504	2
#13554	17