Subgroups — #16950763

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 188 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	2	yes	maximal
2	C₂×#11762634	C₂	1	2	yes	maximal
3	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
4	#14528948	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63214	C₂²	1	1	yes	characteristic
7	C₂²×#63214	C₂²	1	1	yes	characteristic
8	C₂×#10587489	C₂²	1	2	yes	—
9	C₂×#10587488	—	2	1	no	—
10	#11762675	—	2	1	no	—
11	#11762636	—	2	4	no	—
12	#11762634	—	2	4	no	—
13	C₄×#63214	C₄	1	4	yes	complemented
14	C₂×C₁₇×#107	C₂²	1	2	yes	—
15	C₂×C₁₇×#106	—	2	1	no	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#382	—	2	1	no	—
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₇	—	2	2	no	—
23	C₁₇×#107	—	2	2	no	—
24	C₁₇×#107	—	2	2	no	—
25	C₁₇×#106	—	4	2	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₇	Q₈	1	2	yes	—
28	C₁₇×#95	—	4	2	no	—
29	#10587502	—	4	2	no	—
30	#10587489	—	2	2	no	—
31	#10587489	—	2	2	no	—
32	#10587488	—	4	2	no	—
33	C₂×#63214	C₂×C₄	1	2	yes	—
34	C₂×#63214	C₂×C₄	1	2	yes	—
35	C₂×#63214	D₄	1	2	yes	—
36	C₂×#63214	D₄	1	2	yes	—
37	C₂×#63214	Q₈	1	2	yes	—
38	C₂×#63214	Q₈	1	2	yes	—
39	C₂²×C₄×C₁₇	—	4	2	no	—
40	C₂²×C₄×C₁₇	D₈	1	2	yes	—
41	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	—	4	2	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₇	D₈	1	2	yes	—
45	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
46	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	C₄⋊C₄×C₁₇	—	4	2	no	—
53	C₄⋊C₄×C₁₇	—	4	2	no	—
54	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
55	C₄²×C₁₇	Q₁₆	1	2	yes	—
56	C₄⋊Dic₁₇	—	4	2	no	—
57	C₄⋊Dic₁₇	—	4	2	no	—
58	#63214	C₄⋊C₄	1	4	yes	—
59	#63214	C₄⋊C₄	1	4	yes	—
60	#12907	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	—	2	4	no	—
63	C₂×C₄×C₁₇	—	8	2	no	—
64	C₂×C₄×C₁₇	—	8	4	no	—
65	C₂×C₄×C₁₇	#107	1	4	yes	—
66	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×Dic₁₇	—	8	2	no	—
73	C₂×Dic₁₇	—	8	4	no	—
74	C₈×C₁₇	#107	1	4	yes	—
75	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
78	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×#367	—	17	2	no	—
80	C₂×#366	—	17	2	no	—
81	C₂×C₄×C₁₆	—	17	1	no	—
82	#1846	—	17	1	no	—
83	#1846	D₁₇	1	1	yes	characteristic
84	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
85	C₂²×C₁₇	#493	1	1	yes	characteristic
86	C₂²×C₁₇	#500	1	1	yes	characteristic
87	C₂²×C₁₇	#508	1	4	yes	—
88	C₄×C₁₇	—	2	4	no	—
89	C₄×C₁₇	—	16	2	no	—
90	C₄×C₁₇	#384	1	1	yes	characteristic
91	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
92	C₄×C₁₇	#494	1	1	yes	characteristic
93	C₄×C₁₇	#500	1	1	yes	characteristic
94	Dic₁₇	—	16	2	no	—
95	C₂²×C₁₆	—	17	1	no	—
96	C₂²×C₁₆	—	17	1	no	—
97	C₂×#107	—	17	2	no	—
98	C₂×#107	C₂×D₁₇	1	2	yes	—
99	C₂×#106	—	2	1	no	—
100	C₂×#106	—	34	1	no	—
101	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
102	#382	—	2	1	no	—
103	#382	—	34	1	no	—
104	#367	—	34	4	no	—
105	#366	—	34	4	no	—
106	C₄×C₁₆	—	17	4	no	—
107	C₂×C₁₇	#1845	1	1	yes	characteristic
108	C₂×C₁₇	#2158	1	2	yes	—
109	C₂×C₁₇	#2249	1	2	yes	—
110	C₂×C₁₇	#2253	1	2	yes	—
111	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	34	2	no	—
115	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
116	C₂×C₁₆	—	17	2	no	—
117	C₂×C₁₆	—	17	2	no	—
118	C₂×C₁₆	—	17	2	no	—
119	C₂×C₁₆	—	17	2	no	—
120	C₂×C₁₆	—	17	2	no	—
121	C₂×C₁₆	—	17	2	no	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#107	—	34	2	no	—
125	#107	—	34	2	no	—
126	#106	—	4	2	no	—
127	#106	—	68	2	no	—
128	C₄×C₈	Dic₃₄	1	2	yes	—
129	C₄×C₈	C₄×D₁₇	1	2	yes	—
130	#95	—	4	2	no	—
131	#95	—	68	2	no	—
132	C₁₇	#12907	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	68	2	no	—
135	C₂²×C₄	#63228	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
137	C₂×C₈	#63225	1	2	yes	—
138	C₂×C₈	#63228	1	2	yes	—
139	C₂×C₈	#63231	1	2	yes	—
140	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
142	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
143	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	68	2	no	—
149	C₄⋊C₄	—	68	2	no	—
150	C₄²	#63225	1	2	yes	—
151	C₄²	#63231	1	2	yes	—
152	C₁₆	—	17	4	no	—
153	C₁₆	—	17	4	no	—
154	C₂³	C₂×#63228	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	8	4	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	136	4	no	—
159	C₂×C₄	—	136	2	no	—
160	C₂×C₄	#10587478	1	4	yes	—
161	C₂×C₄	C₂×#63225	1	1	yes	characteristic
162	C₂×C₄	C₂×#63228	1	1	yes	characteristic
163	C₂×C₄	#10587626	1	1	yes	characteristic
164	C₂×C₄	C₂×#63231	1	1	yes	characteristic
165	C₂×C₄	C₂×#63231	1	1	yes	characteristic
166	C₂×C₄	#10587630	1	1	yes	characteristic
167	C₈	#10587478	1	4	yes	—
168	C₈	#10587574	1	1	yes	characteristic
169	C₈	#10587574	1	1	yes	characteristic
170	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
171	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
172	C₂²	C₂×#10587478	1	1	yes	characteristic
173	C₂²	#11763185	1	1	yes	characteristic
174	C₂²	#11763212	1	1	yes	characteristic
175	C₂²	#11763239	1	4	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	16	2	no	—
178	C₄	—	272	2	no	—
179	C₄	#11762678	1	1	yes	characteristic
180	C₄	C₂×#10587478	1	1	yes	characteristic
181	C₄	#11763188	1	1	yes	characteristic
182	C₄	#11763212	1	1	yes	characteristic
183	C₂	#14528943	1	1	yes	characteristic
184	C₂	#14530566	1	2	yes	—
185	C₂	#14531043	1	2	yes	—
186	C₂	#14531060	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1846	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1846	D₁₇
hypocenter	C₁₇	#12907
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
13. C₄×#63214	C₄	semidirect
132. C₁₇	#12907	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×#11762634	2
C₂×C₄×#63214	2
#14528948	2
C₁₇×#1846	2
#12907	17