Subgroups — #16945896

Order 4352.

242 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530538	C₂	1	1	yes	maximal, characteristic
2	D₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#14527066	C₂	1	1	yes	maximal, characteristic
4	#14526854	C₂	1	1	yes	maximal, characteristic, complemented
5	#14526853	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526853	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1370	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587469	—	2	1	no	—
9	C₂×C₁₆×D₁₇	—	2	1	no	—
10	#11762860	C₄	1	1	yes	characteristic
11	#11762860	C₄	1	1	yes	characteristic
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	C₄⋊#63214	C₂²	1	1	yes	characteristic
14	C₄⋊#63214	C₂²	1	1	yes	characteristic
15	#11762627	—	2	1	no	—
16	#11762627	—	2	1	no	—
17	#11762531	C₄	1	1	yes	characteristic
18	#11762531	C₄	1	1	yes	characteristic
19	C₈×#3622	C₂²	1	1	yes	characteristic
20	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₈²×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
24	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
25	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₁₇	—	2	1	no	—
27	C₂×C₁₆×C₁₇	—	2	1	no	—
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
31	#10587491	—	2	1	no	—
32	#10587491	C₈	1	1	yes	characteristic
33	#10587491	C₈	1	1	yes	characteristic
34	#10587485	—	2	1	no	—
35	#10587485	C₈	1	1	yes	characteristic
36	#10587485	C₈	1	1	yes	characteristic
37	C₈×Dic₁₇	D₄	1	1	yes	characteristic
38	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
39	C₂×#63214	—	2	1	no	—
40	C₂×#63214	—	2	1	no	—
41	C₄×#3622	C₂×C₄	1	1	yes	characteristic
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	C₄×#3622	C₂×C₄	1	1	yes	characteristic
44	#10587469	—	4	2	no	—
45	C₁₆×D₁₇	—	4	2	no	—
46	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₇	—	2	1	no	—
50	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
56	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
57	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
58	C₂×#3622	—	2	1	no	—
59	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
60	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#3622	C₂×C₈	1	1	yes	characteristic
62	C₂×#3622	C₂×C₈	1	1	yes	characteristic
63	C₈×D₁₇	—	2	2	no	—
64	C₈×D₁₇	D₈	1	2	yes	—
65	C₈×D₁₇	Q₁₆	1	2	yes	—
66	C₁₆×C₁₇	—	4	1	no	—
67	C₁₆×C₁₇	—	4	1	no	—
68	#63214	—	4	1	no	—
69	#63214	—	4	1	no	—
70	#12005	—	17	1	no	maximal, Sylow
71	C₂²×D₁₇	C₄⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
76	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
77	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
78	C₄×D₁₇	—	2	1	no	—
79	C₄×D₁₇	—	2	1	no	—
80	C₄×D₁₇	—	4	1	no	—
81	C₄×D₁₇	#107	1	2	yes	—
82	C₄×D₁₇	#108	1	2	yes	—
83	C₈×C₁₇	—	2	1	no	—
84	C₈×C₁₇	—	2	1	no	—
85	C₈×C₁₇	—	2	1	no	—
86	C₈×C₁₇	—	4	1	no	—
87	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
88	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
89	#3622	—	2	1	no	—
90	#3622	—	2	1	no	—
91	#3622	—	2	1	no	—
92	#3622	—	4	1	no	—
93	#3622	#107	1	2	yes	—
94	#2150	—	17	1	no	—
95	C₂×C₄⋊C₁₆	—	17	1	no	—
96	#1450	—	17	1	no	—
97	#1370	—	17	1	no	—
98	#1370	—	17	1	no	—
99	#1370	—	17	1	no	—
100	#1370	D₁₇	1	1	yes	characteristic, complemented
101	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂×D₁₇	—	2	1	no	—
103	C₂×D₁₇	—	2	1	no	—
104	C₂×D₁₇	#335	1	2	yes	—
105	C₄×C₁₇	—	2	1	no	—
106	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
107	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
108	C₄×C₁₇	#504	1	1	yes	characteristic
109	C₄×C₁₇	#504	1	1	yes	characteristic
110	Dic₁₇	—	2	1	no	—
111	Dic₁₇	—	2	1	no	—
112	Dic₁₇	#335	1	2	yes	—
113	C₂×M₃₂	—	34	1	no	—
114	C₂²×C₁₆	—	34	1	no	—
115	#431	—	17	1	no	—
116	#431	—	17	1	no	—
117	C₂×C₄×C₈	—	17	1	no	—
118	C₄⋊C₁₆	—	17	1	no	—
119	C₄⋊C₁₆	—	17	1	no	—
120	C₄⋊C₁₆	—	34	1	no	—
121	C₄⋊C₁₆	—	34	1	no	—
122	C₄⋊C₁₆	C₂×D₁₇	1	1	yes	characteristic
123	C₄⋊C₁₆	C₂×D₁₇	1	1	yes	characteristic
124	#322	—	17	1	no	—
125	#322	—	17	1	no	—
126	C₈²	—	17	1	no	—
127	C₈²	C₂×D₁₇	1	1	yes	characteristic
128	C₂×C₁₇	C₂×#335	1	1	yes	characteristic
129	C₂×C₁₇	#2151	1	1	yes	characteristic
130	C₂×C₁₇	#2153	1	1	yes	characteristic
131	D₁₇	—	2	2	no	—
132	C₂²×C₈	—	17	1	no	—
133	C₂²×C₈	—	17	1	no	—
134	C₂×C₄²	—	17	1	no	—
135	M₃₂	—	68	2	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	2	1	no	—
138	C₂×C₁₆	—	34	1	no	—
139	C₂×C₁₆	—	34	1	no	—
140	C₂×C₁₆	—	68	2	no	—
141	C₄⋊C₈	—	17	1	no	—
142	C₄⋊C₈	—	17	1	no	—
143	C₄⋊C₈	—	34	1	no	—
144	C₂²⋊C₈	—	17	1	no	—
145	C₂²⋊C₈	—	17	1	no	—
146	C₂²⋊C₈	—	34	1	no	—
147	C₄×C₈	—	17	1	no	—
148	C₄×C₈	—	17	1	no	—
149	C₄×C₈	—	17	1	no	—
150	C₄×C₈	—	17	1	no	—
151	C₄×C₈	—	17	1	no	—
152	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
153	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
154	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
155	C₁₇	#12005	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄	—	17	1	no	—
157	C₂²×C₄	—	17	1	no	—
158	C₂²×C₄	—	17	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	17	1	no	—
161	C₂×C₈	—	17	2	no	—
162	C₂×C₈	—	17	2	no	—
163	C₂×C₈	—	17	1	no	—
164	C₂×C₈	—	17	1	no	—
165	C₂×C₈	—	17	1	no	—
166	C₂×C₈	—	34	2	no	—
167	C₂×C₈	—	34	1	no	—
168	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
169	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
170	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
171	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
172	C₄²	—	17	1	no	—
173	C₄²	—	17	1	no	—
174	C₄²	—	17	1	no	—
175	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
176	C₁₆	—	4	1	no	—
177	C₁₆	—	4	1	no	—
178	C₁₆	—	68	1	no	—
179	C₁₆	—	68	1	no	—
180	C₂³	—	17	1	no	—
181	C₂×C₄	—	17	1	no	—
182	C₂×C₄	—	17	2	no	—
183	C₂×C₄	—	17	2	no	—
184	C₂×C₄	—	17	1	no	—
185	C₂×C₄	—	17	1	no	—
186	C₂×C₄	—	34	1	no	—
187	C₂×C₄	—	34	1	no	—
188	C₂×C₄	—	68	1	no	—
189	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
190	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
191	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	1	no	—
196	C₈	—	17	2	no	—
197	C₈	—	34	1	no	—
198	C₈	—	34	1	no	—
199	C₈	—	34	1	no	—
200	C₈	—	68	1	no	—
201	C₈	D₈×D₁₇	1	1	yes	characteristic
202	C₈	Q₁₆×D₁₇	1	1	yes	characteristic
203	C₂²	—	17	2	no	—
204	C₂²	—	34	1	no	—
205	C₂²	—	34	1	no	—
206	C₂²	C₄⋊C₈×D₁₇	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	17	2	no	—
209	C₄	—	34	1	no	—
210	C₄	—	34	1	no	—
211	C₄	#107×D₁₇	1	1	yes	characteristic
212	C₄	#108×D₁₇	1	1	yes	characteristic
213	C₄	#11763225	1	1	yes	characteristic
214	C₄	#11763227	1	1	yes	characteristic
215	C₂	—	34	2	no	—
216	C₂	D₁₇×#335	1	1	yes	characteristic
217	C₂	#14530541	1	1	yes	characteristic
218	C₂	#14530548	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1370	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂²×D₁₇	C₄⋊C₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#1370	D₁₇
hypocenter	C₁₇	#12005
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

Complements

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

N	H ≅ G/N	type
4. #14526854	C₂	semidirect
5. #14526853	C₂	semidirect
6. #14526853	C₂	semidirect
7. C₁₇×#1370	C₂	semidirect
100. #1370	D₁₇	semidirect
155. C₁₇	#12005	semidirect

Maximal subgroups

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

subgroup	index
#14530538	2
D₁₇×C₄⋊C₁₆	2
#14527066	2
#14526854	2
#14526853	2
C₁₇×#1370	2
#12005	17