Subgroups — #16950737

Order 4352.

370 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#63214	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762627	C₂	1	4	yes	maximal
3	#14528470	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	C₂²	1	2	yes	—
6	C₂×#10587474	C₄	1	1	yes	characteristic
7	C₂×#10587486	C₄	1	2	yes	—
8	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
9	#11762655	C₄	1	1	yes	characteristic
10	C₄⋊#63214	—	2	2	no	—
11	#11762627	—	2	8	no	—
12	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×C₄⋊C₁₆	—	2	2	no	—
15	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
19	C₂²×#3622	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₁₇	—	2	2	no	—
21	C₂×C₁₆×C₁₇	—	2	4	no	—
22	C₄×C₈×C₁₇	D₄	1	2	yes	—
23	C₄×C₈×C₁₇	Q₈	1	2	yes	—
24	#10587502	C₈	1	2	yes	—
25	#10587502	C₈	1	2	yes	—
26	#10587486	—	2	4	no	—
27	C₈×Dic₁₇	D₄	1	4	yes	—
28	C₈×Dic₁₇	Q₈	1	4	yes	—
29	C₂×#63214	—	2	4	no	—
30	C₂×#63214	—	2	2	no	—
31	#10587474	—	2	2	no	—
32	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₁₇	—	2	2	no	—
37	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
42	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
43	C₄²×C₁₇	M₁₆	1	2	yes	—
44	C₄×Dic₁₇	C₄⋊C₄	1	4	yes	—
45	C₄×Dic₁₇	M₁₆	1	4	yes	—
46	C₂×#3622	—	2	2	no	—
47	C₂×#3622	—	2	2	no	—
48	C₂×#3622	C₄⋊C₄	1	4	yes	—
49	C₁₆×C₁₇	—	4	2	no	—
50	#63214	—	4	2	no	—
51	#12899	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	—	2	2	no	—
55	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
61	C₂×Dic₁₇	—	2	4	no	—
62	C₂×Dic₁₇	—	2	4	no	—
63	C₂×Dic₁₇	C₄⋊C₈	1	8	yes	—
64	C₈×C₁₇	—	2	2	no	—
65	C₈×C₁₇	#127	1	1	yes	characteristic
66	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
67	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
68	#3622	—	2	4	no	—
69	C₂×C₄⋊C₁₆	—	17	4	no	—
70	C₂×C₄⋊C₁₆	—	17	1	no	—
71	C₂×C₄⋊C₁₆	D₁₇	1	1	yes	characteristic
72	#1726	—	17	1	no	—
73	C₂²×C₁₇	C₂×C₄⋊C₈	1	2	yes	—
74	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₂²×C₁₇	#504	1	2	yes	—
76	C₂²×C₁₇	#504	1	2	yes	—
77	C₄×C₁₇	—	2	2	no	—
78	C₄×C₁₇	—	2	2	no	—
79	C₄×C₁₇	#384	1	1	yes	characteristic
80	C₄×C₁₇	#423	1	2	yes	—
81	C₄×C₁₇	#423	1	1	yes	characteristic
82	Dic₁₇	—	2	8	no	—
83	C₂²×C₁₆	—	17	2	no	—
84	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
85	C₂×#97	—	17	2	no	—
86	C₂×#97	—	17	1	no	—
87	C₂×C₄×C₈	—	17	2	no	—
88	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
89	#376	—	17	1	no	—
90	C₄⋊C₁₆	—	2	2	no	—
91	C₄⋊C₁₆	—	34	8	no	—
92	C₄⋊C₁₆	—	34	2	no	—
93	C₂×C₁₇	#1842	1	1	yes	characteristic
94	C₂×C₁₇	#2151	1	4	yes	—
95	C₂×C₁₇	#2151	1	1	yes	characteristic
96	C₂×C₁₇	#2151	1	1	yes	characteristic
97	C₂²×C₈	—	17	2	no	—
98	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
100	C₂×C₄²	—	17	2	no	—
101	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	34	4	no	—
105	C₂×C₁₆	—	34	2	no	—
106	#97	—	34	4	no	—
107	#97	—	34	2	no	—
108	C₄×C₈	—	17	4	no	—
109	C₄×C₈	—	17	4	no	—
110	C₄×C₈	Dic₃₄	1	2	yes	—
111	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
112	#95	—	17	2	no	—
113	#95	—	17	2	no	—
114	C₁₇	#12899	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₄	—	17	4	no	—
116	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
117	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
118	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	17	4	no	—
121	C₂×C₈	—	34	2	no	—
122	C₂×C₈	—	34	2	no	—
123	C₂×C₈	#63225	1	2	yes	—
124	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
125	C₂×C₈	D₄×D₁₇	1	2	yes	—
126	C₂×C₈	Q₈×D₁₇	1	2	yes	—
127	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
128	C₄²	—	17	4	no	—
129	C₄²	—	17	4	no	—
130	C₄²	#63218	1	2	yes	—
131	C₄²	#63225	1	2	yes	—
132	C₁₆	—	4	2	no	—
133	C₁₆	—	68	2	no	—
134	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	17	8	no	—
138	C₂×C₄	—	34	4	no	—
139	C₂×C₄	—	34	4	no	—
140	C₂×C₄	#10587485	1	2	yes	—
141	C₂×C₄	#10587485	1	2	yes	—
142	C₂×C₄	C₄⋊C₄×D₁₇	1	2	yes	—
143	C₂×C₄	C₂×#63218	1	1	yes	characteristic
144	C₂×C₄	M₁₆×D₁₇	1	2	yes	—
145	C₂×C₄	C₂×#63225	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	34	4	no	—
148	C₈	#10587574	1	2	yes	—
149	C₈	#10587626	1	1	yes	characteristic
150	C₈	#10587630	1	1	yes	characteristic
151	C₂²	C₄⋊C₈×D₁₇	1	2	yes	—
152	C₂²	C₂×#10587485	1	1	yes	characteristic
153	C₂²	#11763225	1	2	yes	—
154	C₂²	#11763227	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	2	2	no	—
157	C₄	—	34	8	no	—
158	C₄	#11762678	1	1	yes	characteristic
159	C₄	#11762831	1	2	yes	—
160	C₄	#11762832	1	1	yes	characteristic
161	C₂	#14528935	1	1	yes	characteristic
162	C₂	#14530541	1	4	yes	—
163	C₂	#14530542	1	1	yes	characteristic
164	C₂	#14530544	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator, socle, prime radical	C₂³×C₁₇	C₂²×C₈
Fitting	C₂×C₁₇×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₇
hypocenter	C₁₇	#12899
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
114. C₁₇	#12899	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#63214	2
C₂×#11762627	2
#14528470	2
C₂×C₁₇×C₄⋊C₁₆	2
#12899	17