Subgroups — #16950760

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762636	C₂	1	4	yes	maximal
2	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
3	#14528944	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	C₂²	1	2	yes	—
6	C₂×#10587489	—	2	1	no	—
7	C₂×#10587489	C₂²	1	2	yes	—
8	#11762679	—	2	1	no	—
9	#11762636	—	2	8	no	—
10	C₄×#63214	C₄	1	4	yes	complemented
11	C₂×C₁₇×#107	—	2	1	no	—
12	C₂×C₁₇×#107	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#384	—	2	1	no	—
15	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₇	—	4	2	no	—
19	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₇	—	2	2	no	—
21	C₂×C₄⋊Dic₁₇	—	4	2	no	—
22	C₁₇×#107	—	2	4	no	—
23	C₁₇×#107	—	4	2	no	—
24	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₇	Q₈	1	2	yes	—
26	#10587489	—	2	4	no	—
27	#10587489	—	4	2	no	—
28	C₂×#63214	C₂×C₄	1	4	yes	—
29	C₂×#63214	D₄	1	4	yes	—
30	C₂×#63214	Q₈	1	4	yes	—
31	C₂²×C₄×C₁₇	—	4	2	no	—
32	C₂²×C₄×C₁₇	D₈	1	2	yes	—
33	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×Dic₁₇	—	4	2	no	—
35	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₁₇	D₈	1	2	yes	—
37	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
38	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
40	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
42	C₄⋊C₄×C₁₇	—	4	2	no	—
43	C₄⋊C₄×C₁₇	—	4	2	no	—
44	C₄⋊C₄×C₁₇	—	8	4	no	—
45	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
46	C₄²×C₁₇	Q₁₆	1	2	yes	—
47	C₄⋊Dic₁₇	—	4	2	no	—
48	C₄⋊Dic₁₇	—	4	2	no	—
49	C₄⋊Dic₁₇	—	8	4	no	—
50	#63214	C₄⋊C₄	1	8	yes	complemented
51	#12906	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	—	8	2	no	—
54	C₂×C₄×C₁₇	—	8	4	no	—
55	C₂×C₄×C₁₇	#107	1	4	yes	—
56	C₂×C₄×C₁₇	SD₃₂	1	4	yes	—
57	C₂×C₄×C₁₇	SD₃₂	1	4	yes	—
58	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×Dic₁₇	—	8	2	no	—
65	C₂×Dic₁₇	—	8	4	no	—
66	C₈×C₁₇	#107	1	4	yes	—
67	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
69	C₂×#367	—	17	4	no	—
70	C₂×C₄×C₁₆	—	17	1	no	—
71	#1845	—	17	1	no	—
72	#1845	D₁₇	1	1	yes	characteristic
73	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
74	C₂²×C₁₇	#493	1	1	yes	characteristic
75	C₂²×C₁₇	#500	1	1	yes	characteristic
76	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
77	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
78	C₄×C₁₇	—	16	2	no	—
79	C₄×C₁₇	#367	1	8	yes	complemented
80	C₄×C₁₇	#384	1	1	yes	characteristic
81	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
82	C₄×C₁₇	#494	1	1	yes	characteristic
83	C₄×C₁₇	#500	1	1	yes	characteristic
84	Dic₁₇	—	16	2	no	—
85	C₂²×C₁₆	—	17	2	no	—
86	C₂×#107	—	2	1	no	—
87	C₂×#107	—	17	2	no	—
88	C₂×#107	—	34	1	no	—
89	C₂×#107	C₂×D₁₇	1	2	yes	—
90	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
91	#384	—	2	1	no	—
92	#384	—	34	1	no	—
93	#367	—	34	8	no	—
94	C₄×C₁₆	—	17	4	no	—
95	C₂×C₁₇	#1845	1	1	yes	characteristic
96	C₂×C₁₇	C₂×#367	1	2	yes	—
97	C₂×C₁₇	#2248	1	1	yes	characteristic
98	C₂×C₁₇	#2248	1	1	yes	characteristic
99	C₂×C₁₇	#2254	1	2	yes	—
100	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	4	2	no	—
104	C₂×C₄⋊C₄	—	34	2	no	—
105	C₂×C₄⋊C₄	—	68	2	no	—
106	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
107	C₂×C₁₆	—	17	4	no	—
108	C₂×C₁₆	—	17	4	no	—
109	C₂×C₁₆	—	17	4	no	—
110	#107	—	2	4	no	—
111	#107	—	4	2	no	—
112	#107	—	34	4	no	—
113	#107	—	68	2	no	—
114	C₄×C₈	Dic₃₄	1	2	yes	—
115	C₄×C₈	C₄×D₁₇	1	2	yes	—
116	C₁₇	#12906	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	—	68	2	no	—
119	C₂²×C₄	#63228	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
121	C₂×C₈	#63225	1	2	yes	—
122	C₂×C₈	#63228	1	2	yes	—
123	C₂×C₈	#63231	1	2	yes	—
124	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
125	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
126	C₂×C₈	D₄×D₁₇	1	2	yes	—
127	C₂×C₈	Q₈×D₁₇	1	2	yes	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄⋊C₄	—	68	2	no	—
132	C₄⋊C₄	—	68	2	no	—
133	C₄⋊C₄	—	136	4	no	—
134	C₄²	#63225	1	2	yes	—
135	C₄²	#63231	1	2	yes	—
136	C₁₆	—	17	8	no	—
137	C₂³	C₂×#63228	1	1	yes	characteristic
138	C₂×C₄	—	8	4	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	136	4	no	—
141	C₂×C₄	—	136	2	no	—
142	C₂×C₄	#10587478	1	4	yes	—
143	C₂×C₄	#10587498	1	4	yes	—
144	C₂×C₄	#10587499	1	4	yes	—
145	C₂×C₄	C₂×#63225	1	1	yes	characteristic
146	C₂×C₄	C₂×#63228	1	1	yes	characteristic
147	C₂×C₄	#10587626	1	1	yes	characteristic
148	C₂×C₄	C₂×#63231	1	1	yes	characteristic
149	C₂×C₄	C₂×#63231	1	1	yes	characteristic
150	C₂×C₄	#10587630	1	1	yes	characteristic
151	C₈	#10587478	1	4	yes	—
152	C₈	#10587574	1	2	yes	—
153	C₈	C₄⋊C₄×D₁₇	1	2	yes	—
154	C₂²	C₂×#10587478	1	1	yes	characteristic
155	C₂²	#11763185	1	1	yes	characteristic
156	C₂²	#11763212	1	1	yes	characteristic
157	C₂²	C₂×#10587499	1	2	yes	—
158	C₂²	C₂×#10587498	1	2	yes	—
159	C₄	—	16	2	no	—
160	C₄	—	272	2	no	—
161	C₄	#11762636	1	8	yes	complemented
162	C₄	#11762678	1	1	yes	characteristic
163	C₄	C₂×#10587478	1	1	yes	characteristic
164	C₄	#11763188	1	1	yes	characteristic
165	C₄	#11763212	1	1	yes	characteristic
166	C₂	#14528943	1	1	yes	characteristic
167	C₂	C₂×#11762636	1	2	yes	—
168	C₂	#14531039	1	1	yes	characteristic
169	C₂	#14531041	1	1	yes	characteristic
170	C₂	#14531064	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1845	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1845	D₁₇
hypocenter	C₁₇	#12906
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
10. C₄×#63214	C₄	semidirect
50. #63214	C₄⋊C₄	semidirect
79. C₄×C₁₇	#367	semidirect
116. C₁₇	#12906	semidirect
161. C₄	#11762636	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762636	2
C₂×C₄×#63214	2
#14528944	2
C₁₇×#1845	2
#12906	17