Subgroups — #17952294

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544333	C₂	1	2	yes	maximal
2	#14543987	C₂	1	2	yes	maximal
3	#14543612	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	#11763825	C₂²	1	4	yes	—
6	#11763673	C₄	1	4	yes	complemented
7	#11763671	C₄	1	4	yes	complemented
8	#11763650	—	2	2	no	—
9	C₈×#3624	—	2	2	no	—
10	C₂×C₄×#3622	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	—	2	2	no	—
21	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
23	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
24	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
25	C₂²⋊#3624	—	2	4	no	—
26	C₄⋊#3624	C₂×C₄	1	8	yes	—
27	C₄×#3624	—	2	4	no	—
28	C₄×C₈×C₁₇	—	2	2	no	—
29	C₈×Dic₁₇	—	2	2	no	—
30	C₈×Dic₁₇	C₈	1	4	yes	complemented
31	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
32	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
33	C₈×Dic₁₇	D₄	1	1	yes	characteristic
34	C₈×Dic₁₇	D₄	1	1	yes	characteristic
35	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
36	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
37	C₄×#3622	C₈	1	4	yes	complemented
38	C₂²×C₄×C₁₇	—	2	1	no	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	—	2	1	no	—
41	C₂²×Dic₁₇	#41	1	2	yes	—
42	C₂×#3624	—	2	8	no	—
43	C₂×C₈×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	C₄²	1	2	yes	—
46	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
50	C₄²×C₁₇	—	2	1	no	—
51	C₄²×C₁₇	C₂×C₈	1	2	yes	—
52	C₄×Dic₁₇	—	2	8	no	—
53	C₄×Dic₁₇	—	2	1	no	—
54	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
55	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
56	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×#3622	C₄²	1	1	yes	characteristic
59	C₂×#3622	C₄²	1	1	yes	characteristic
60	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
62	C₂×#3622	C₂×C₈	1	2	yes	—
63	C₂×#3622	C₂×C₈	1	2	yes	—
64	C₂×#3622	C₂×C₈	1	2	yes	—
65	C₂×#3622	M₁₆	1	2	yes	—
66	#8266	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	#117	1	1	yes	characteristic
68	C₂×C₄×C₁₇	—	2	4	no	—
69	C₂×C₄×C₁₇	—	2	2	no	—
70	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂²×C₈	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×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×Dic₁₇	—	2	4	no	—
77	C₂×Dic₁₇	—	2	2	no	—
78	C₂×Dic₁₇	—	2	4	no	—
79	C₂×Dic₁₇	C₄×D₄	1	2	yes	—
80	C₂×Dic₁₇	C₄×Q₈	1	2	yes	—
81	#3624	—	4	8	no	—
82	C₈×C₁₇	—	2	2	no	—
83	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
84	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
85	#3622	C₄×C₈	1	2	yes	—
86	#3622	C₄×C₈	1	2	yes	—
87	#3622	C₄⋊C₈	1	2	yes	—
88	#3622	C₄⋊C₈	1	2	yes	—
89	C₄²×C₈	—	17	1	no	—
90	#1564	—	17	2	no	—
91	#1450	—	17	2	no	—
92	#1276	—	17	2	no	—
93	C₂²×C₁₇	—	2	2	no	—
94	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₁₇	#431	1	1	yes	characteristic
96	C₂²×C₁₇	#431	1	1	yes	characteristic
97	C₄×C₁₇	—	2	4	no	—
98	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
99	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
100	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
102	Dic₁₇	—	2	4	no	—
103	Dic₁₇	—	2	2	no	—
104	Dic₁₇	#443	1	4	yes	—
105	#431	—	17	4	no	—
106	C₂×C₄×C₈	—	17	1	no	—
107	C₂×C₄×C₈	—	17	1	no	—
108	C₂×C₄×C₈	—	17	1	no	—
109	C₂×C₄×C₈	—	17	1	no	—
110	C₂×C₄×C₈	—	17	1	no	—
111	C₂×C₄×C₈	#595	1	1	yes	characteristic
112	C₄³	—	17	1	no	—
113	#335	—	17	4	no	—
114	#334	—	17	4	no	—
115	#322	—	34	2	no	—
116	C₈²	—	34	2	no	—
117	C₂×C₁₇	—	2	1	no	—
118	C₂×C₁₇	—	2	1	no	—
119	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
120	C₂×C₁₇	#1772	1	2	yes	—
121	C₂²×C₈	—	17	1	no	—
122	C₂²×C₈	—	17	1	no	—
123	C₂²×C₈	C₂×#595	1	1	yes	characteristic
124	C₂²×C₈	C₂×#595	1	1	yes	characteristic
125	C₂×C₄²	—	17	1	no	—
126	C₂×C₄²	—	17	1	no	—
127	C₂×C₄²	—	34	2	no	—
128	C₂×C₄²	C₂×#595	1	1	yes	characteristic
129	C₄⋊C₈	—	17	8	no	—
130	C₂²⋊C₈	—	34	4	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	17	4	no	—
133	C₄×C₈	—	17	2	no	—
134	C₄×C₈	—	17	2	no	—
135	C₄×C₈	—	17	1	no	—
136	C₄×C₈	—	17	1	no	—
137	C₄×C₈	—	17	1	no	—
138	C₄×C₈	—	17	1	no	—
139	C₄×C₈	—	17	4	no	—
140	C₄×C₈	—	34	4	no	—
141	C₄×C₈	—	34	2	no	—
142	C₁₇	#8266	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	17	2	no	—
145	C₂²×C₄	—	34	1	no	—
146	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	17	2	no	—
150	C₂×C₈	—	17	2	no	—
151	C₂×C₈	—	17	2	no	—
152	C₂×C₈	—	17	2	no	—
153	C₂×C₈	—	17	1	no	—
154	C₂×C₈	—	17	1	no	—
155	C₂×C₈	—	17	1	no	—
156	C₂×C₈	—	17	1	no	—
157	C₂×C₈	—	34	8	no	—
158	C₂×C₈	#63242	1	2	yes	—
159	C₂×C₈	C₄×#595	1	2	yes	—
160	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
161	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
162	C₂×C₈	#63247	1	2	yes	—
163	C₄²	—	2	1	no	—
164	C₄²	—	17	2	no	—
165	C₄²	—	17	1	no	—
166	C₄²	—	17	1	no	—
167	C₄²	—	17	2	no	—
168	C₄²	—	34	8	no	—
169	C₄²	—	34	1	no	—
170	C₄²	#63242	1	2	yes	—
171	C₂³	#10587677	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	17	2	no	—
175	C₂×C₄	—	17	2	no	—
176	C₂×C₄	—	34	4	no	—
177	C₂×C₄	—	34	4	no	—
178	C₂×C₄	—	34	2	no	—
179	C₂×C₄	C₂×#63242	1	1	yes	characteristic
180	C₂×C₄	C₂×#63242	1	1	yes	characteristic
181	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
184	C₂×C₄	C₂×#63247	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	17	2	no	—
187	C₈	—	17	2	no	—
188	C₈	—	17	2	no	—
189	C₈	—	17	2	no	—
190	C₈	—	68	8	no	—
191	C₈	#10587544	1	2	yes	—
192	C₈	#10587544	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	C₄×C₄⋊#595	1	1	yes	characteristic
195	C₂²	#11763826	1	1	yes	characteristic
196	C₂²	#11763827	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	17	4	no	—
199	C₄	—	34	4	no	—
200	C₄	—	34	2	no	—
201	C₄	C₄×#63242	1	1	yes	characteristic
202	C₄	C₄×#63247	1	1	yes	characteristic
203	C₄	#11763810	1	1	yes	characteristic
204	C₄	C₂×#10587544	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	C₄×#10587544	1	1	yes	characteristic
208	C₂	#14544586	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄⋊#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₇	#117
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#8266
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #11763673	C₄	semidirect
7. #11763671	C₄	semidirect
30. C₈×Dic₁₇	C₈	semidirect
37. C₄×#3622	C₈	semidirect
142. C₁₇	#8266	semidirect

Maximal subgroups

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

subgroup	index
#14544333	2
#14543987	2
#14543612	2
C₄×C₈×Dic₁₇	2
#8266	17