Subgroups — #16933044

Order 4352.

322 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528956	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#14527559	C₂	1	2	yes	maximal
4	#14526686	C₂	1	2	yes	maximal
5	C₁₇×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587490	—	2	1	no	—
7	C₂×#10587490	—	2	1	no	—
8	#11762833	C₂²	1	2	yes	—
9	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	—	2	1	no	—
11	C₄×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
12	#11762563	—	2	1	no	—
13	#11762562	—	2	4	no	—
14	C₁₇×#431	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#335	C₄	1	2	yes	complemented
17	C₁₇×#334	C₄	1	2	yes	complemented
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₇	—	2	1	no	—
22	C₂×C₄×Dic₁₇	—	2	1	no	—
23	C₂×#63223	—	2	2	no	—
24	C₂²×#3622	—	2	1	no	—
25	C₁₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
26	C₁₇×C₂²⋊C₈	—	2	2	no	—
27	C₄×C₈×C₁₇	—	2	2	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	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
32	#10587502	Q₈	1	2	yes	—
33	#10587490	—	4	2	no	—
34	#10587490	—	4	2	no	—
35	C₈×Dic₁₇	—	4	2	no	—
36	C₄⋊#3622	—	2	2	no	—
37	C₄⋊#3622	—	2	1	no	—
38	C₄⋊#3622	—	2	2	no	—
39	C₄⋊#3622	D₄	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂²×Dic₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	2	1	no	—
46	C₂×C₈×C₁₇	—	2	4	no	—
47	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	D₈	1	2	yes	—
52	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
53	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₁₇	#41	1	1	yes	characteristic
57	C₄⋊Dic₁₇	—	2	4	no	—
58	C₄×Dic₁₇	—	4	1	no	—
59	C₄×Dic₁₇	—	4	1	no	—
60	#63223	—	4	2	no	—
61	#63223	—	4	2	no	—
62	C₂×#3622	—	2	2	no	—
63	C₂×#3622	—	4	1	no	—
64	C₂×#3622	—	4	1	no	—
65	C₂×#3622	—	4	2	no	—
66	#9627	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	4	1	no	—
70	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#125	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
80	C₂×Dic₁₇	—	2	4	no	—
81	C₂×Dic₁₇	—	4	2	no	—
82	C₂×Dic₁₇	—	4	2	no	—
83	C₈×C₁₇	—	2	1	no	—
84	C₈×C₁₇	—	2	1	no	—
85	C₈×C₁₇	—	4	4	no	—
86	C₈×C₁₇	#107	1	2	yes	—
87	C₈×C₁₇	#107	1	2	yes	—
88	#3622	—	4	2	no	—
89	#3622	—	8	2	no	—
90	#1848	—	17	1	no	—
91	C₈×C₄⋊C₄	—	17	1	no	—
92	#1571	—	17	2	no	—
93	#1564	D₁₇	1	1	yes	characteristic
94	#1303	—	17	2	no	—
95	C₂²×C₁₇	—	2	1	no	—
96	C₂²×C₁₇	—	2	1	no	—
97	C₂²×C₁₇	#389	1	1	yes	characteristic
98	C₂²×C₁₇	#474	1	1	yes	characteristic
99	C₂²×C₁₇	#477	1	1	yes	characteristic
100	C₄×C₁₇	—	2	1	no	—
101	C₄×C₁₇	—	2	1	no	—
102	C₄×C₁₇	—	2	1	no	—
103	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
104	C₄×C₁₇	#428	1	1	yes	characteristic
105	C₄×C₁₇	#443	1	2	yes	—
106	C₄×C₁₇	#459	1	1	yes	characteristic
107	C₄×C₁₇	#462	1	1	yes	characteristic
108	Dic₁₇	—	4	4	no	—
109	#431	C₂×D₁₇	1	2	yes	—
110	C₂×#108	—	34	1	no	—
111	C₂×#108	—	34	1	no	—
112	#423	—	17	2	no	—
113	C₂×C₄⋊C₈	—	17	1	no	—
114	C₂×C₄×C₈	—	34	1	no	—
115	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
116	C₄×C₄⋊C₄	—	17	1	no	—
117	#336	—	34	1	no	—
118	#335	—	34	4	no	—
119	#335	Dic₁₇	1	2	yes	complemented
120	#334	Dic₁₇	1	2	yes	complemented
121	C₂×C₁₇	—	2	1	no	—
122	C₂×C₁₇	—	2	1	no	—
123	C₂×C₁₇	#1945	1	1	yes	characteristic
124	C₂×C₁₇	#1947	1	1	yes	characteristic
125	C₂×C₁₇	#1953	1	1	yes	characteristic
126	C₂×M₁₆	—	34	2	no	—
127	C₂²×C₈	—	34	1	no	—
128	C₂²×C₈	D₆₈	1	1	yes	characteristic
129	C₂²×C₈	D₆₈	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	34	1	no	—
131	C₂×C₄²	—	34	1	no	—
132	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
133	#108	—	68	2	no	—
134	#108	—	68	2	no	—
135	C₄⋊C₈	—	17	2	no	—
136	C₄⋊C₈	—	34	2	no	—
137	C₄⋊C₈	—	34	1	no	—
138	C₄⋊C₈	—	34	2	no	—
139	C₄⋊C₈	C₂×Dic₁₇	1	4	yes	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	—	68	2	no	—
143	C₄×C₈	Dic₃₄	1	1	yes	characteristic
144	C₄×C₈	Dic₃₄	1	1	yes	characteristic
145	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
147	#95	—	17	2	no	—
148	C₁₇	#9627	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₄	—	34	2	no	—
150	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
151	C₂²×C₄	#63254	1	1	yes	characteristic
152	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
153	M₁₆	—	68	2	no	—
154	M₁₆	—	68	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	34	2	no	—
159	C₂×C₈	—	68	1	no	—
160	C₂×C₈	—	68	1	no	—
161	C₂×C₈	—	68	2	no	—
162	C₂×C₈	D₁₃₆	1	2	yes	—
163	C₂×C₈	Dic₆₈	1	2	yes	—
164	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
165	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
166	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
167	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
168	C₄⋊C₄	—	34	4	no	—
169	C₄²	—	68	1	no	—
170	C₄²	—	68	1	no	—
171	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
172	C₄²	D₄×D₁₇	1	1	yes	characteristic
173	C₄²	#63256	1	1	yes	characteristic
174	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
175	C₂³	#10587583	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	34	4	no	—
179	C₂×C₄	—	68	2	no	—
180	C₂×C₄	—	68	2	no	—
181	C₂×C₄	#10587574	1	1	yes	characteristic
182	C₂×C₄	#10587576	1	1	yes	characteristic
183	C₂×C₄	#10587581	1	1	yes	characteristic
184	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
186	C₂×C₄	#10587594	1	1	yes	characteristic
187	C₂×C₄	#10587595	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
189	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
190	C₂×C₄	Q₈×Dic₁₇	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	4	no	—
194	C₈	—	68	2	no	—
195	C₈	—	136	2	no	—
196	C₈	#10587489	1	2	yes	—
197	C₈	#10587489	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#11762708	1	1	yes	characteristic
201	C₂²	#11763090	1	1	yes	characteristic
202	C₂²	#11763105	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	68	4	no	—
207	C₄	C₂×#10587489	1	1	yes	characteristic
208	C₄	#11762851	1	1	yes	characteristic
209	C₄	#11762926	1	2	yes	—
210	C₄	#11763005	1	1	yes	characteristic
211	C₄	#11763020	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#14529533	1	1	yes	characteristic
215	C₂	#14529543	1	1	yes	characteristic
216	C₂	#14529575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587581
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1564	C₂
socle	C₂²×C₁₇	#389
prime radical	C₂³×C₁₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1564	D₁₇
hypocenter	C₁₇	#9627
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
16. C₁₇×#335	C₄	semidirect
17. C₁₇×#334	C₄	semidirect
119. #335	Dic₁₇	semidirect
120. #334	Dic₁₇	semidirect
148. C₁₇	#9627	semidirect

Maximal subgroups

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

subgroup	index
#14528956	2
C₈×C₄⋊Dic₁₇	2
#14527559	2
#14526686	2
C₁₇×#1564	2
#9627	17