Subgroups — #17952412

Order 4352.

334 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544331	C₂	1	1	yes	maximal, characteristic
2	#14543991	C₂	1	2	yes	maximal
3	C₂×#11763650	C₂	1	1	yes	maximal, characteristic
4	#14543617	C₂	1	2	yes	maximal
5	#108×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	#11763825	C₂²	1	2	yes	—
7	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
8	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
9	#11763672	C₄	1	2	yes	—
10	#11763670	C₄	1	2	yes	—
11	#11763651	—	2	4	no	—
12	#11763650	C₄	1	2	yes	—
13	#11763650	C₄	1	2	yes	—
14	C₂×#10587490	C₄	1	1	yes	characteristic
15	M₁₆×Dic₁₇	C₂²	1	2	yes	—
16	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	#11762573	C₄	1	2	yes	—
18	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
19	C₂²×#3624	—	2	1	no	—
20	C₂²×#3624	Q₈	1	2	yes	—
21	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
24	C₂×#63223	C₂×C₄	1	2	yes	—
25	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
26	C₂²⋊#3624	—	2	2	no	—
27	C₄⋊#3624	C₂×C₄	1	4	yes	—
28	C₄×#3624	—	2	2	no	—
29	C₄×#3624	—	2	1	no	—
30	C₄×#3624	C₂×C₄	1	2	yes	—
31	C₄×#3624	C₂×C₄	1	2	yes	—
32	C₄×#3624	D₄	1	2	yes	—
33	C₁₇×#108	—	2	2	no	—
34	#10587490	—	2	2	no	—
35	#10587486	—	2	2	no	—
36	C₈×Dic₁₇	—	2	2	no	—
37	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
43	C₂²×Dic₁₇	#41	1	1	yes	characteristic
44	C₂×#3624	—	2	2	no	—
45	C₂×#3624	—	2	4	no	—
46	C₂×#3624	—	4	2	no	—
47	C₂×#3624	—	4	2	no	—
48	C₂×#3624	C₄⋊C₄	1	4	yes	—
49	M₁₆×C₁₇	—	2	2	no	—
50	M₁₆×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	—	2	2	no	—
52	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
55	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
56	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₁₇	#41	1	1	yes	characteristic
60	#63223	—	2	2	no	—
61	#63223	—	2	2	no	—
62	C₂×#3622	—	2	1	no	—
63	C₂×#3622	—	2	1	no	—
64	C₂×#3622	—	2	2	no	—
65	C₂×#3622	C₄²	1	1	yes	characteristic
66	C₂×#3622	C₄²	1	1	yes	characteristic
67	#8295	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
75	C₂×Dic₁₇	—	2	2	no	—
76	C₂×Dic₁₇	—	2	2	no	—
77	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×Dic₁₇	#117	1	1	yes	characteristic
79	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
80	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
81	#3624	—	2	4	no	—
82	#3624	—	4	2	no	—
83	#3624	—	4	4	no	—
84	C₈×C₁₇	—	4	2	no	—
85	C₈×C₁₇	#97	1	2	yes	—
86	C₈×C₁₇	#97	1	2	yes	—
87	#3622	—	2	1	no	—
88	#3622	—	2	1	no	—
89	#3622	—	4	2	no	—
90	C₄×#108	—	17	1	no	—
91	#1564	—	17	1	no	—
92	#1451	—	17	2	no	—
93	C₂×#322	—	17	1	no	—
94	#1278	—	17	2	no	—
95	C₂²×C₁₇	—	2	1	no	—
96	C₂²×C₁₇	—	2	1	no	—
97	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₁₇	#446	1	1	yes	characteristic
99	C₂²×C₁₇	#446	1	1	yes	characteristic
100	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
101	C₄×C₁₇	#405	1	1	yes	characteristic
102	C₄×C₁₇	#436	1	1	yes	characteristic
103	C₄×C₁₇	#436	1	1	yes	characteristic
104	Dic₁₇	—	2	1	no	—
105	Dic₁₇	—	2	1	no	—
106	Dic₁₇	#430	1	2	yes	—
107	Dic₁₇	#444	1	2	yes	—
108	#431	—	17	2	no	—
109	C₂×#108	—	17	1	no	—
110	C₂×#108	#595	1	1	yes	characteristic
111	C₄×M₁₆	—	17	2	no	—
112	C₂×C₄×C₈	—	17	1	no	—
113	C₂×C₄×C₈	—	17	1	no	—
114	C₂×C₄×C₈	—	17	1	no	—
115	#341	—	17	2	no	—
116	#335	—	17	2	no	—
117	#334	—	17	2	no	—
118	#322	—	17	2	no	—
119	#322	—	17	2	no	—
120	#322	—	34	4	no	—
121	C₂×C₁₇	—	2	1	no	—
122	C₂×C₁₇	—	2	1	no	—
123	C₂×C₁₇	#1770	1	1	yes	characteristic
124	C₂×C₁₇	#1773	1	1	yes	characteristic
125	C₂×C₁₇	#1778	1	1	yes	characteristic
126	C₂×M₁₆	—	17	2	no	—
127	C₂×M₁₆	C₂×#595	1	2	yes	—
128	C₂²×C₈	—	17	1	no	—
129	C₂²×C₈	—	17	2	no	—
130	C₂²×C₈	—	34	1	no	—
131	C₂²×C₈	C₂×#595	1	1	yes	characteristic
132	C₂×C₄²	—	17	1	no	—
133	#108	—	2	2	no	—
134	#108	—	34	2	no	—
135	C₄⋊C₈	—	17	4	no	—
136	C₂²⋊C₈	—	34	2	no	—
137	#97	—	34	2	no	—
138	C₄×C₈	—	17	2	no	—
139	C₄×C₈	—	17	2	no	—
140	C₄×C₈	—	17	1	no	—
141	C₄×C₈	—	17	1	no	—
142	C₄×C₈	—	17	1	no	—
143	C₄×C₈	—	17	1	no	—
144	C₄×C₈	—	17	2	no	—
145	C₄×C₈	—	34	2	no	—
146	C₄×C₈	—	34	2	no	—
147	C₄×C₈	—	34	1	no	—
148	C₁₇	#8295	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₄	—	17	1	no	—
150	C₂²×C₄	—	17	1	no	—
151	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	2	no	—
154	M₁₆	—	34	2	no	—
155	M₁₆	—	34	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	17	4	no	—
158	C₂×C₈	—	17	1	no	—
159	C₂×C₈	—	17	1	no	—
160	C₂×C₈	—	34	1	no	—
161	C₂×C₈	—	34	1	no	—
162	C₂×C₈	—	34	2	no	—
163	C₂×C₈	—	34	4	no	—
164	C₂×C₈	—	34	2	no	—
165	C₂×C₈	—	68	2	no	—
166	C₂×C₈	—	68	2	no	—
167	C₂×C₈	#63243	1	2	yes	—
168	C₂×C₈	C₄×#595	1	1	yes	characteristic
169	C₂×C₈	C₄×#595	1	1	yes	characteristic
170	C₂×C₈	#63247	1	2	yes	—
171	C₄²	—	17	1	no	—
172	C₄²	—	17	1	no	—
173	C₄²	—	17	1	no	—
174	C₄²	—	17	1	no	—
175	C₂³	Q₈×#595	1	1	yes	characteristic
176	C₂×C₄	—	17	1	no	—
177	C₂×C₄	—	17	1	no	—
178	C₂×C₄	—	17	1	no	—
179	C₂×C₄	—	17	1	no	—
180	C₂×C₄	—	34	2	no	—
181	C₂×C₄	—	34	2	no	—
182	C₂×C₄	C₂×#63243	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
184	C₂×C₄	#10587678	1	1	yes	characteristic
185	C₂×C₄	D₄×#595	1	1	yes	characteristic
186	C₂×C₄	#10587680	1	1	yes	characteristic
187	C₂×C₄	C₂×#63247	1	1	yes	characteristic
188	C₈	—	4	2	no	—
189	C₈	—	34	4	no	—
190	C₈	—	34	1	no	—
191	C₈	—	34	1	no	—
192	C₈	—	68	4	no	—
193	C₈	—	68	2	no	—
194	C₈	—	68	2	no	—
195	C₈	#10587542	1	2	yes	—
196	C₈	#10587542	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
200	C₂²	#11763859	1	1	yes	characteristic
201	C₂²	#11763860	1	1	yes	characteristic
202	C₄	—	17	2	no	—
203	C₄	—	17	2	no	—
204	C₄	—	34	1	no	—
205	C₄	—	34	1	no	—
206	C₄	C₂×#10587542	1	1	yes	characteristic
207	C₄	#11763774	1	1	yes	characteristic
208	C₄	#11763843	1	1	yes	characteristic
209	C₄	#11763844	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#14544580	1	1	yes	characteristic
213	C₂	#14544593	1	1	yes	characteristic
214	C₂	#14544605	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₁₇×#108	C₄
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₇	C₄×Q₈
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#108	#595
hypocenter	C₁₇	#8295
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
148. C₁₇	#8295	semidirect

Maximal subgroups

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

subgroup	index
#14544331	2
#14543991	2
C₂×#11763650	2
#14543617	2
#108×Dic₁₇	2
#8295	17