Subgroups — #16952934

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×#11762585	C₂	1	1	yes	maximal, characteristic
2	#14528944	C₂	1	1	yes	maximal, characteristic
3	#14526875	C₂	1	4	yes	maximal
4	C₂×C₁₇×#499	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	#11762585	—	2	2	no	—
10	C₂×C₁₇×#125	—	2	1	no	—
11	C₁₇×#499	—	2	2	no	—
12	C₁₇×#499	C₄	1	4	yes	complemented
13	C₂×C₁₇×#107	C₂²	1	2	yes	—
14	C₂×C₁₇×#106	—	2	1	no	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₁₇×#125	—	2	4	no	—
19	C₁₇×#125	—	4	2	no	—
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₇	—	4	2	no	—
22	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₇	—	2	2	no	—
24	C₂×C₄⋊Dic₁₇	—	4	2	no	—
25	C₁₇×#107	—	2	2	no	—
26	C₁₇×#107	C₂×C₄	1	4	yes	—
27	C₁₇×#106	—	2	4	no	—
28	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₇	D₄	1	2	yes	—
30	#10587489	—	2	4	no	—
31	#10587489	—	4	2	no	—
32	C₂×#63214	—	2	2	no	—
33	C₂×#63214	—	2	2	no	—
34	C₂×#63214	Q₈	1	4	yes	—
35	C₂²×C₄×C₁₇	—	4	2	no	—
36	C₂²×C₄×C₁₇	D₈	1	2	yes	—
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₁₇	—	2	4	no	—
47	C₄⋊C₄×C₁₇	—	4	8	no	—
48	C₄⋊C₄×C₁₇	—	4	2	no	—
49	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₁₇	SD₁₆	1	2	yes	—
51	C₄⋊Dic₁₇	—	4	2	no	—
52	C₄⋊Dic₁₇	—	4	2	no	—
53	C₄⋊Dic₁₇	—	8	4	no	—
54	#63214	—	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	—	4	4	no	—
60	C₂×C₄×C₁₇	—	8	2	no	—
61	C₂×C₄×C₁₇	—	8	2	no	—
62	C₂×C₄×C₁₇	#102	1	4	yes	—
63	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
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	D₁₇	1	1	yes	characteristic
75	C₂×#346	—	17	1	no	—
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	—	8	4	no	—
85	C₄×C₁₇	#391	1	1	yes	characteristic
86	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
87	C₄×C₁₇	#487	1	1	yes	characteristic
88	C₄×C₁₇	#498	1	1	yes	characteristic
89	Dic₁₇	—	16	2	no	—
90	C₂×#125	—	2	1	no	—
91	C₂²×C₁₆	—	17	2	no	—
92	#499	—	2	2	no	—
93	#499	Dic₁₇	1	4	yes	complemented
94	C₂×#107	—	17	2	no	—
95	C₂×#107	—	34	1	no	—
96	C₂×#107	C₂×D₁₇	1	2	yes	—
97	C₂×#106	—	2	1	no	—
98	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
99	#384	—	34	1	no	—
100	#346	—	34	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	C₂²⋊D₁₇	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
108	#125	—	2	4	no	—
109	#125	—	4	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	4	2	no	—
112	C₂×C₄⋊C₄	—	34	2	no	—
113	C₂×C₄⋊C₄	—	68	2	no	—
114	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
115	C₂×C₁₆	—	17	4	no	—
116	C₂×C₁₆	—	34	2	no	—
117	C₂×C₁₆	—	34	2	no	—
118	#107	—	2	2	no	—
119	#107	—	34	4	no	—
120	#107	—	68	2	no	—
121	#107	C₂×Dic₁₇	1	4	yes	—
122	#106	—	2	4	no	—
123	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
124	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
125	C₁₇	#13284	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	68	2	no	—
128	C₂²×C₄	#63228	1	2	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
130	C₂×C₈	#63230	1	2	yes	—
131	C₂×C₈	#63231	1	2	yes	—
132	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
133	C₂×C₈	Q₈×D₁₇	1	2	yes	—
134	C₂×C₈	#63256	1	2	yes	—
135	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
136	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	4	8	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	68	2	no	—
141	C₄⋊C₄	—	68	2	no	—
142	C₄⋊C₄	—	136	4	no	—
143	C₄²	#63229	1	2	yes	—
144	C₄²	C₂²⋊Dic₁₇	1	2	yes	—
145	C₁₆	—	34	4	no	—
146	C₂³	C₂×#63228	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	136	4	no	—
153	C₂×C₄	—	136	2	no	—
154	C₂×C₄	#10587503	1	4	yes	—
155	C₂×C₄	C₂×#63229	1	1	yes	characteristic
156	C₂×C₄	C₂×#63230	1	1	yes	characteristic
157	C₂×C₄	C₂×#63231	1	1	yes	characteristic
158	C₂×C₄	#10587630	1	1	yes	characteristic
159	C₂×C₄	#10587633	1	1	yes	characteristic
160	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
161	C₈	#10587506	1	4	yes	—
162	C₈	Q₈×Dic₁₇	1	2	yes	—
163	C₈	#10587632	1	2	yes	—
164	C₂²	C₂×#10587503	1	1	yes	characteristic
165	C₂²	#11763157	1	1	yes	characteristic
166	C₂²	#11763212	1	1	yes	characteristic
167	C₂²	#11763244	1	2	yes	—
168	C₂²	#11763249	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	8	4	no	—
171	C₄	—	272	2	no	—
172	C₄	#11762716	1	1	yes	characteristic
173	C₄	C₂×#10587506	1	1	yes	characteristic
174	C₄	#11763161	1	1	yes	characteristic
175	C₄	#11763205	1	1	yes	characteristic
176	C₂	#14529746	1	1	yes	characteristic
177	C₂	#14530496	1	2	yes	—
178	C₂	#14531027	1	1	yes	characteristic
179	C₂	#14531031	1	1	yes	characteristic
180	C₂	#14531071	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#499	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#499	D₁₇
hypocenter	C₁₇	#13284
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
12. C₁₇×#499	C₄	semidirect
93. #499	Dic₁₇	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×#11762585	2
#14528944	2
#14526875	2
C₂×C₁₇×#499	2
#13284	17