Subgroups — #16949690

Order 4352.

326 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×#11762630	C₂	1	1	yes	maximal, characteristic
2	#14530381	C₂	1	2	yes	maximal
3	C₂×#11762585	C₂	1	1	yes	maximal, characteristic
4	#14526873	C₂	1	2	yes	maximal
5	C₁₇×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂²×#63214	C₂²	1	1	yes	characteristic
8	C₂²×#63214	C₂²	1	1	yes	characteristic
9	#11762630	C₄	1	4	yes	complemented
10	#11762585	—	2	2	no	—
11	#11762585	C₄	1	4	yes	complemented
12	C₄×#63214	—	2	2	no	—
13	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
14	C₁₇×#427	C₄	1	1	yes	characteristic
15	C₁₇×#405	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#339	C₄	1	2	yes	—
18	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₁₇×#117	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₁₇×#108	C₈	1	4	yes	—
24	C₁₇×#107	C₈	1	2	yes	—
25	C₁₇×#106	C₈	1	2	yes	—
26	C₁₇×#97	—	2	2	no	—
27	C₄×C₈×C₁₇	—	2	2	no	—
28	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
30	#10587483	—	2	2	no	—
31	#10587483	C₂×C₄	1	4	yes	—
32	C₂×#63214	—	2	2	no	—
33	C₂×#63214	—	2	1	no	—
34	C₂×#63214	—	2	1	no	—
35	C₂×#63214	C₂×C₄	1	2	yes	—
36	C₂×#63214	C₂×C₄	1	2	yes	—
37	C₂×#63214	D₄	1	2	yes	—
38	C₂×#63214	Q₈	1	2	yes	—
39	C₂²×C₄×C₁₇	—	2	1	no	—
40	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	M₁₆×C₁₇	—	2	2	no	—
42	M₁₆×C₁₇	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	C₄²	1	2	yes	—
45	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
53	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
55	C₂²⋊C₄×C₁₇	—	2	2	no	—
56	C₄²×C₁₇	—	2	2	no	—
57	C₄²×C₁₇	—	2	1	no	—
58	C₄²×C₁₇	C₄²	1	2	yes	—
59	#63214	—	2	2	no	—
60	#63214	—	2	4	no	—
61	#63214	C₄⋊C₄	1	4	yes	—
62	#12665	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	#131	1	1	yes	characteristic
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	4	no	—
66	C₂×C₄×C₁₇	—	4	2	no	—
67	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
73	C₈×C₁₇	—	2	4	no	—
74	C₈×C₁₇	C₄×C₈	1	2	yes	—
75	C₈×C₁₇	C₄×C₈	1	2	yes	—
76	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₈×C₁₇	#117	1	1	yes	characteristic
78	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
79	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
80	C₂×#364	—	17	1	no	—
81	#2108	—	17	2	no	—
82	C₂×#346	—	17	1	no	—
83	#1772	D₁₇	1	1	yes	characteristic
84	#1379	—	17	2	no	—
85	C₂²×C₁₇	—	2	2	no	—
86	C₂²×C₁₇	#405	1	1	yes	characteristic
87	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
88	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
89	C₄×C₁₇	—	4	2	no	—
90	C₄×C₁₇	—	4	4	no	—
91	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
94	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
95	C₂×M₃₂	—	17	2	no	—
96	C₂²×C₁₆	—	17	1	no	—
97	C₂²×C₁₆	—	17	1	no	—
98	C₂×#108	Dic₁₇	1	1	yes	characteristic
99	#427	Dic₁₇	1	1	yes	characteristic
100	#405	C₂×D₁₇	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
102	#364	—	17	4	no	—
103	#346	—	17	4	no	—
104	#346	—	34	2	no	—
105	C₄×C₁₆	—	34	2	no	—
106	#339	Dic₁₇	1	2	yes	—
107	C₂×C₁₇	—	2	1	no	—
108	C₂×C₁₇	—	2	1	no	—
109	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₁₇	#2171	1	2	yes	—
111	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
112	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
114	#117	C₂×Dic₁₇	1	2	yes	—
115	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
116	M₃₂	—	17	4	no	—
117	M₃₂	—	34	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	C₂×C₁₆	—	34	2	no	—
123	C₂×C₁₆	—	34	1	no	—
124	C₂×C₁₆	—	34	1	no	—
125	#108	#3622	1	4	yes	—
126	#107	#3622	1	2	yes	—
127	#106	#3622	1	2	yes	—
128	#97	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	C₄×D₁₇	1	2	yes	—
131	C₄×C₈	C₄×D₁₇	1	2	yes	—
132	C₁₇	#12665	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
135	M₁₆	—	2	2	no	—
136	M₁₆	C₂×#3622	1	4	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	C₂×#3622	1	1	yes	characteristic
139	C₂×C₈	C₂×#3622	1	1	yes	characteristic
140	C₂×C₈	C₂×#3622	1	2	yes	—
141	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
142	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
144	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
145	C₂×C₈	#63254	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
147	C₂×C₈	#63256	1	1	yes	characteristic
148	C₄⋊C₄	C₂×#3622	1	4	yes	—
149	C₂²⋊C₄	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	1	no	—
152	C₄²	C₄×Dic₁₇	1	2	yes	—
153	C₁₆	—	17	4	no	—
154	C₁₆	—	34	2	no	—
155	C₁₆	—	34	4	no	—
156	C₂³	#10587634	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
161	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
163	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
164	C₂×C₄	Q₈×Dic₁₇	1	1	yes	characteristic
165	C₂×C₄	#10587634	1	1	yes	characteristic
166	C₈	—	2	4	no	—
167	C₈	C₄×#3622	1	2	yes	—
168	C₈	C₄×#3622	1	2	yes	—
169	C₈	#10587573	1	1	yes	characteristic
170	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
171	C₈	#10587578	1	1	yes	characteristic
172	C₈	#10587579	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	#11762765	1	1	yes	characteristic
175	C₂²	D₄×#3622	1	1	yes	characteristic
176	C₂²	Q₈×#3622	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	4	4	no	—
179	C₄	C₄⋊C₄×Dic₁₇	1	1	yes	characteristic
180	C₄	C₂×C₄×#3622	1	1	yes	characteristic
181	C₄	D₄×#3622	1	1	yes	characteristic
182	C₄	Q₈×#3622	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	C₄⋊C₄×#3622	1	1	yes	characteristic
186	C₂	#14530626	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1772	C₂
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1772	D₁₇
hypocenter	C₁₇	#12665
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
9. #11762630	C₄	semidirect
11. #11762585	C₄	semidirect
132. C₁₇	#12665	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762630	2
#14530381	2
C₂×#11762585	2
#14526873	2
C₁₇×#1772	2
#12665	17