Subgroups — #16952933

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763211	C₂	1	1	yes	maximal, characteristic
2	#14528944	C₂	1	1	yes	maximal, characteristic
3	#14526876	C₂	1	4	yes	maximal
4	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587556	—	2	1	no	—
6	#11763211	—	2	2	no	—
7	#11763211	C₄	1	4	yes	complemented
8	C₂×#10587489	—	2	1	no	—
9	C₂×#10587489	C₂²	1	2	yes	—
10	C₂×#10587489	C₂²	1	2	yes	—
11	C₂×#10587488	—	2	1	no	—
12	#11762679	—	2	1	no	—
13	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#346	—	2	2	no	—
16	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₇	—	2	2	no	—
20	C₂×C₄⋊Dic₁₇	—	2	2	no	—
21	C₂×C₄⋊Dic₁₇	—	4	2	no	—
22	C₂×#63225	—	4	2	no	—
23	#10587556	—	2	4	no	—
24	#10587556	—	4	2	no	—
25	C₂×C₁₆×C₁₇	—	2	2	no	—
26	C₂×C₁₆×C₁₇	—	2	2	no	—
27	C₂×C₁₆×C₁₇	Q₈	1	4	yes	—
28	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₇	D₄	1	2	yes	—
30	#10587489	—	2	2	no	—
31	#10587489	—	2	4	no	—
32	#10587489	—	4	2	no	—
33	#10587489	C₂×C₄	1	4	yes	—
34	#10587488	—	2	4	no	—
35	C₂²×C₄×C₁₇	D₈	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₇	—	4	2	no	—
38	C₂²×Dic₁₇	—	4	2	no	—
39	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
41	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
42	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₁₇	#41	1	2	yes	—
46	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₁₇	SD₁₆	1	2	yes	—
48	C₄⋊Dic₁₇	—	2	4	no	—
49	C₄⋊Dic₁₇	—	4	2	no	—
50	C₄⋊Dic₁₇	—	4	2	no	—
51	C₄⋊Dic₁₇	—	4	2	no	—
52	C₄⋊Dic₁₇	—	8	4	no	—
53	#63225	—	4	8	no	—
54	C₁₆×C₁₇	—	2	4	no	—
55	#13284	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	—	2	2	no	—
58	C₂×C₄×C₁₇	—	2	2	no	—
59	C₂×C₄×C₁₇	#102	1	4	yes	—
60	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×Dic₁₇	—	4	4	no	—
67	C₂×Dic₁₇	—	8	2	no	—
68	C₂×Dic₁₇	—	8	2	no	—
69	C₂×Dic₁₇	—	8	2	no	—
70	C₂×Dic₁₇	—	8	4	no	—
71	C₈×C₁₇	#103	1	4	yes	—
72	C₈×C₁₇	C₄×Q₈	1	2	yes	—
73	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
74	C₂×#499	—	17	1	no	—
75	C₂×#346	D₁₇	1	1	yes	characteristic
76	#1845	—	17	1	no	—
77	#1380	—	17	4	no	—
78	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
79	C₂²×C₁₇	#486	1	1	yes	characteristic
80	C₂²×C₁₇	#500	1	1	yes	characteristic
81	C₂²×C₁₇	#509	1	2	yes	—
82	C₂²×C₁₇	#510	1	2	yes	—
83	C₄×C₁₇	—	2	4	no	—
84	C₄×C₁₇	#391	1	1	yes	characteristic
85	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
86	C₄×C₁₇	#487	1	1	yes	characteristic
87	C₄×C₁₇	#498	1	1	yes	characteristic
88	Dic₁₇	—	8	4	no	—
89	Dic₁₇	—	16	2	no	—
90	C₂×#125	—	34	1	no	—
91	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
92	#499	—	17	4	no	—
93	#499	—	34	2	no	—
94	C₂×#107	—	17	2	no	—
95	C₂×#107	—	17	2	no	—
96	C₂×#107	—	34	1	no	—
97	C₂×#106	—	34	1	no	—
98	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
99	#384	—	34	1	no	—
100	#346	—	2	2	no	—
101	C₂×C₁₇	#1990	1	1	yes	characteristic
102	C₂×C₁₇	#2139	1	2	yes	—
103	C₂×C₁₇	#2244	1	1	yes	characteristic
104	C₂×C₁₇	#2245	1	1	yes	characteristic
105	C₂×C₁₇	#2255	1	2	yes	—
106	C₂²×C₈	D₆₈	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
108	#125	—	34	4	no	—
109	#125	—	68	2	no	—
110	C₂×C₄⋊C₄	—	34	2	no	—
111	C₂×C₄⋊C₄	—	34	2	no	—
112	C₂×C₄⋊C₄	—	68	2	no	—
113	C₂×C₄⋊C₄	—	68	2	no	—
114	C₂×C₄²	D₆₈	1	1	yes	characteristic
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	Dic₃₄	1	4	yes	—
118	#107	—	17	4	no	—
119	#107	—	34	2	no	—
120	#107	—	34	4	no	—
121	#107	—	68	2	no	—
122	#106	—	34	4	no	—
123	C₄×C₈	C₄×D₁₇	1	2	yes	—
124	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
125	C₁₇	#13284	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₄	—	68	2	no	—
127	C₂²×C₄	—	68	2	no	—
128	C₂²×C₄	D₁₃₆	1	2	yes	—
129	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
130	C₂×C₈	#63219	1	2	yes	—
131	C₂×C₈	Dic₆₈	1	2	yes	—
132	C₂×C₈	#63227	1	2	yes	—
133	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
134	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
135	C₂×C₈	#63252	1	2	yes	—
136	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
137	C₄⋊C₄	—	34	4	no	—
138	C₄⋊C₄	—	68	8	no	—
139	C₄⋊C₄	—	68	2	no	—
140	C₄⋊C₄	—	68	2	no	—
141	C₄⋊C₄	—	68	2	no	—
142	C₄⋊C₄	—	136	4	no	—
143	C₄²	#63219	1	2	yes	—
144	C₄²	#63227	1	2	yes	—
145	C₁₆	—	2	4	no	—
146	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	68	4	no	—
150	C₂×C₄	—	136	4	no	—
151	C₂×C₄	—	136	2	no	—
152	C₂×C₄	—	136	2	no	—
153	C₂×C₄	—	136	2	no	—
154	C₂×C₄	#10587492	1	4	yes	—
155	C₂×C₄	#10587555	1	1	yes	characteristic
156	C₂×C₄	#10587561	1	1	yes	characteristic
157	C₂×C₄	C₂×#63219	1	1	yes	characteristic
158	C₂×C₄	C₂×#63219	1	1	yes	characteristic
159	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
160	C₂×C₄	C₂×#63227	1	1	yes	characteristic
161	C₈	#10587487	1	4	yes	—
162	C₈	C₄×Dic₃₄	1	2	yes	—
163	C₈	#10587610	1	2	yes	—
164	C₂²	C₂×#10587492	1	1	yes	characteristic
165	C₂²	#11763156	1	1	yes	characteristic
166	C₂²	#11763214	1	1	yes	characteristic
167	C₂²	#11763247	1	2	yes	—
168	C₂²	#11763252	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	272	2	no	—
171	C₄	—	136	4	no	—
172	C₄	#11762715	1	1	yes	characteristic
173	C₄	C₂×#10587487	1	1	yes	characteristic
174	C₄	#11763160	1	1	yes	characteristic
175	C₄	#11763207	1	1	yes	characteristic
176	C₂	#14529745	1	1	yes	characteristic
177	C₂	#14530497	1	2	yes	—
178	C₂	#14531026	1	1	yes	characteristic
179	C₂	#14531030	1	1	yes	characteristic
180	C₂	#14531070	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#346	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	#13284
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
7. #11763211	C₄	semidirect
125. C₁₇	#13284	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763211	2
#14528944	2
#14526876	2
C₂×C₁₇×#346	2
#13284	17