Subgroups — #16952944

Order 4352.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539298	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587483	C₂	1	1	yes	maximal, characteristic
3	#14526891	C₂	1	4	yes	maximal
4	C₁₇×#1844	C₂	1	1	yes	maximal, characteristic
5	#11763366	—	2	1	no	—
6	C₂×#10587483	C₂²	1	2	yes	—
7	#11763214	C₄	1	2	yes	complemented
8	#11763211	—	2	1	no	—
9	#11763211	C₄	1	2	yes	complemented
10	#11763207	—	2	1	no	—
11	#11762847	—	2	1	no	—
12	#11762847	C₂²	1	2	yes	—
13	#11762585	—	2	1	no	—
14	C₄×#63214	—	2	1	no	—
15	C₂×C₁₇×#106	—	2	1	no	—
16	C₂×C₁₇×#106	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#384	—	2	1	no	—
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	—	4	2	no	—
23	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
24	#10587612	—	2	2	no	—
25	#10587610	—	4	2	no	—
26	#10587556	—	2	2	no	—
27	#10587555	—	2	2	no	—
28	C₄×Dic₃₄	—	4	2	no	—
29	C₁₇×#106	—	2	4	no	—
30	C₁₇×#106	—	4	2	no	—
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
35	#10587489	—	2	2	no	—
36	#10587489	C₂×C₄	1	4	yes	—
37	#10587488	—	2	2	no	—
38	#10587488	—	2	2	no	—
39	#10587483	—	2	2	no	—
40	#10587483	Q₈	1	4	yes	complemented
41	C₂×#63214	—	2	2	no	—
42	C₂²×C₄×C₁₇	—	4	2	no	—
43	C₂²×C₄×C₁₇	D₈	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₃₄	—	4	2	no	—
46	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
49	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
50	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₁₇	#41	1	2	yes	—
54	C₄⋊C₄×C₁₇	—	4	2	no	—
55	C₄⋊C₄×C₁₇	—	4	2	no	—
56	C₄⋊C₄×C₁₇	—	8	4	no	—
57	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₇	SD₁₆	1	2	yes	—
60	C₂²⋊Dic₁₇	—	4	2	no	—
61	C₄⋊Dic₁₇	—	2	4	no	—
62	C₄⋊Dic₁₇	—	4	2	no	—
63	#63225	—	4	4	no	—
64	C₄×Dic₁₇	—	4	2	no	—
65	#63214	—	2	4	no	—
66	#13287	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	—	2	4	no	—
69	C₂×C₄×C₁₇	—	8	4	no	—
70	C₂×C₄×C₁₇	—	8	2	no	—
71	C₂×C₄×C₁₇	#102	1	2	yes	—
72	C₂×C₄×C₁₇	#102	1	2	yes	—
73	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×Dic₁₇	—	4	4	no	—
80	Dic₃₄	—	8	4	no	—
81	C₈×C₁₇	#103	1	2	yes	—
82	C₈×C₁₇	#103	1	2	yes	—
83	C₈×C₁₇	C₄×Q₈	1	2	yes	—
84	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
85	#3160	—	17	1	no	—
86	C₄×M₃₂	—	17	1	no	—
87	#1844	D₁₇	1	1	yes	characteristic
88	#1386	—	17	4	no	—
89	C₂²×C₁₇	—	2	2	no	—
90	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
91	C₂²×C₁₇	#486	1	1	yes	characteristic
92	C₂²×C₁₇	#500	1	1	yes	characteristic
93	C₄×C₁₇	—	2	2	no	—
94	C₄×C₁₇	—	16	2	no	—
95	C₄×C₁₇	#360	1	4	yes	—
96	C₄×C₁₇	#391	1	1	yes	characteristic
97	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
98	C₄×C₁₇	#487	1	1	yes	characteristic
99	C₄×C₁₇	#498	1	1	yes	characteristic
100	Dic₁₇	—	8	4	no	—
101	#533	—	34	1	no	—
102	C₂×M₃₂	—	17	2	no	—
103	#500	—	17	2	no	—
104	#499	—	17	2	no	—
105	#499	—	34	1	no	—
106	#498	—	34	1	no	—
107	#427	—	17	2	no	—
108	#427	—	34	1	no	—
109	C₂×#106	—	2	1	no	—
110	C₂×#106	C₂×D₁₇	1	2	yes	—
111	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
112	#384	—	2	1	no	—
113	#346	—	34	1	no	—
114	C₄×C₁₆	—	34	1	no	—
115	C₂×C₁₇	—	2	1	no	—
116	C₂×C₁₇	—	2	1	no	—
117	C₂×C₁₇	#1990	1	1	yes	characteristic
118	C₂×C₁₇	C₂×#360	1	2	yes	—
119	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
121	C₄⋊Q₈	—	34	2	no	—
122	#125	—	34	2	no	—
123	C₂²⋊Q₈	—	68	2	no	—
124	C₄×Q₈	—	68	2	no	—
125	#117	—	34	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	4	2	no	—
128	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
129	M₃₂	—	17	4	no	—
130	M₃₂	—	34	2	no	—
131	C₂×C₁₆	—	34	2	no	—
132	#107	—	17	4	no	—
133	#107	—	34	2	no	—
134	#106	—	2	4	no	—
135	#106	—	4	2	no	—
136	#106	—	34	2	no	—
137	#106	—	34	2	no	—
138	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
139	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
140	C₄×C₈	D₆₈	1	1	yes	characteristic
141	C₄×C₈	D₆₈	1	1	yes	characteristic
142	C₁₇	#13287	1	1	yes	characteristic, complemented, Sylow
143	C₂×Q₈	—	68	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	#63228	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
147	C₂×C₈	#63227	1	1	yes	characteristic
148	C₂×C₈	#63227	1	1	yes	characteristic
149	C₂×C₈	#63229	1	2	yes	—
150	C₂×C₈	#63231	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
152	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
153	C₂×C₈	#63254	1	2	yes	—
154	C₂×C₈	Q₈×D₁₇	1	2	yes	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	8	4	no	—
158	C₄⋊C₄	—	34	4	no	—
159	C₄⋊C₄	—	68	4	no	—
160	C₄⋊C₄	—	68	2	no	—
161	C₂²⋊C₄	—	68	2	no	—
162	C₄²	—	68	2	no	—
163	C₄²	#63227	1	1	yes	characteristic
164	C₄²	#63227	1	1	yes	characteristic
165	C₄²	#63230	1	2	yes	—
166	C₁₆	—	34	4	no	—
167	C₂³	C₂×#63228	1	1	yes	characteristic
168	Q₈	—	136	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	68	4	no	—
173	C₂×C₄	#10587480	1	2	yes	—
174	C₂×C₄	#10587480	1	2	yes	—
175	C₂×C₄	C₂×#63227	1	1	yes	characteristic
176	C₂×C₄	C₂×#63229	1	1	yes	characteristic
177	C₂×C₄	#10587622	1	1	yes	characteristic
178	C₂×C₄	C₂×#63230	1	1	yes	characteristic
179	C₂×C₄	C₂×#63231	1	1	yes	characteristic
180	C₂×C₄	#10587630	1	1	yes	characteristic
181	C₈	#10587481	1	2	yes	—
182	C₈	#10587481	1	2	yes	—
183	C₈	#10587573	1	2	yes	—
184	C₈	#10587583	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#10587480	1	1	yes	characteristic
187	C₂²	#11763155	1	1	yes	characteristic
188	C₂²	#11763212	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	16	2	no	—
191	C₄	—	136	4	no	—
192	C₄	#11762618	1	4	yes	—
193	C₄	#11762714	1	1	yes	characteristic
194	C₄	C₂×#10587481	1	1	yes	characteristic
195	C₄	#11763159	1	1	yes	characteristic
196	C₄	#11763205	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#14529744	1	1	yes	characteristic
200	C₂	C₂×#11762618	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#10587480
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1844	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1844	D₁₇
hypocenter	C₁₇	#13287
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
7. #11763214	C₄	semidirect
9. #11763211	C₄	semidirect
40. #10587483	Q₈	semidirect
142. C₁₇	#13287	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
C₄×#10587483	2
#14526891	2
C₁₇×#1844	2
#13287	17