Subgroups — #17952403

Order 4352.

358 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	#14544331	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#3624	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	#11763650	—	2	2	no	—
12	#11763650	C₄	1	2	yes	—
13	#11763650	C₄	1	2	yes	—
14	C₈×#3624	—	2	2	no	—
15	C₂×#10587490	C₄	1	1	yes	characteristic
16	M₁₆×Dic₁₇	C₂²	1	2	yes	—
17	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
18	#11762573	C₄	1	2	yes	—
19	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
20	C₂²×#3624	—	2	1	no	—
21	C₂²×#3624	Q₈	1	2	yes	—
22	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
25	C₂×#63223	C₂×C₄	1	2	yes	—
26	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
27	C₂²⋊#3624	—	2	2	no	—
28	C₄⋊#3624	C₂×C₄	1	4	yes	—
29	C₄×#3624	—	2	2	no	—
30	C₄×#3624	—	2	1	no	—
31	C₄×#3624	C₂×C₄	1	2	yes	—
32	C₄×#3624	C₂×C₄	1	2	yes	—
33	C₄×#3624	D₄	1	2	yes	—
34	C₁₇×#108	C₈	1	4	yes	complemented
35	#10587490	C₈	1	4	yes	complemented
36	#10587486	—	2	2	no	—
37	C₈×Dic₁₇	—	2	2	no	—
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	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×Dic₁₇	#41	1	1	yes	characteristic
45	C₂×#3624	—	2	2	no	—
46	C₂×#3624	—	2	4	no	—
47	C₂×#3624	—	4	2	no	—
48	C₂×#3624	—	4	2	no	—
49	C₂×#3624	C₄⋊C₄	1	4	yes	—
50	M₁₆×C₁₇	—	2	2	no	—
51	M₁₆×C₁₇	C₂×C₈	1	4	yes	—
52	C₂×C₈×C₁₇	—	2	2	no	—
53	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
57	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
59	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₁₇	#41	1	1	yes	characteristic
61	#63223	—	2	2	no	—
62	#63223	C₂×C₈	1	4	yes	—
63	C₂×#3622	—	2	2	no	—
64	C₂×#3622	—	2	1	no	—
65	C₂×#3622	—	2	1	no	—
66	C₂×#3622	C₄²	1	1	yes	characteristic
67	C₂×#3622	C₄²	1	1	yes	characteristic
68	#8294	—	17	1	no	maximal, Sylow
69	C₂³×C₁₇	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₄×D₄	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	#131	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
76	C₂×Dic₁₇	—	2	2	no	—
77	C₂×Dic₁₇	—	2	2	no	—
78	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×Dic₁₇	#117	1	1	yes	characteristic
80	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
81	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
82	#3624	—	2	4	no	—
83	#3624	—	4	2	no	—
84	#3624	—	4	4	no	—
85	C₈×C₁₇	—	2	4	no	—
86	C₈×C₁₇	C₄×C₈	1	2	yes	—
87	C₈×C₁₇	C₄×C₈	1	2	yes	—
88	#3622	—	2	1	no	—
89	#3622	—	2	1	no	—
90	#3622	—	2	4	no	—
91	C₄×#108	—	17	1	no	—
92	#1564	—	17	1	no	—
93	C₈×M₁₆	—	17	2	no	—
94	C₂×#322	—	17	1	no	—
95	#1278	—	17	2	no	—
96	C₂²×C₁₇	—	2	1	no	—
97	C₂²×C₁₇	—	2	1	no	—
98	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
100	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
101	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
102	C₄×C₁₇	#405	1	1	yes	characteristic
103	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
104	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
105	Dic₁₇	—	2	1	no	—
106	Dic₁₇	—	2	1	no	—
107	Dic₁₇	#430	1	2	yes	—
108	Dic₁₇	#444	1	2	yes	—
109	#431	—	17	2	no	—
110	C₂×#108	—	17	1	no	—
111	C₂×#108	#595	1	1	yes	characteristic
112	C₄×M₁₆	—	17	2	no	—
113	C₂×C₄×C₈	—	17	1	no	—
114	C₂×C₄×C₈	—	17	1	no	—
115	C₂×C₄×C₈	—	17	1	no	—
116	#341	—	17	2	no	—
117	#335	—	17	2	no	—
118	#334	—	17	2	no	—
119	#322	—	17	2	no	—
120	#322	—	17	2	no	—
121	#322	—	34	2	no	—
122	C₈²	—	34	2	no	—
123	C₂×C₁₇	—	2	1	no	—
124	C₂×C₁₇	—	2	1	no	—
125	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₁₇	#1773	1	1	yes	characteristic
127	C₂×C₁₇	#1778	1	1	yes	characteristic
128	C₂×M₁₆	—	17	2	no	—
129	C₂×M₁₆	C₂×#595	1	2	yes	—
130	C₂²×C₈	—	17	1	no	—
131	C₂²×C₈	—	17	2	no	—
132	C₂²×C₈	—	34	1	no	—
133	C₂²×C₈	C₂×#595	1	1	yes	characteristic
134	C₂×C₄²	—	17	1	no	—
135	#108	—	17	4	no	—
136	#108	#3624	1	4	yes	complemented
137	C₄⋊C₈	—	17	4	no	—
138	C₂²⋊C₈	—	34	2	no	—
139	#97	—	34	2	no	—
140	C₄×C₈	—	17	2	no	—
141	C₄×C₈	—	17	2	no	—
142	C₄×C₈	—	17	1	no	—
143	C₄×C₈	—	17	1	no	—
144	C₄×C₈	—	17	1	no	—
145	C₄×C₈	—	17	1	no	—
146	C₄×C₈	—	17	2	no	—
147	C₄×C₈	—	34	2	no	—
148	C₄×C₈	—	34	2	no	—
149	C₄×C₈	—	34	1	no	—
150	C₁₇	#8294	1	1	yes	characteristic, complemented, Sylow
151	C₂²×C₄	—	17	1	no	—
152	C₂²×C₄	—	17	1	no	—
153	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	17	4	no	—
156	M₁₆	—	34	2	no	—
157	M₁₆	C₂×#3624	1	4	yes	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	17	4	no	—
160	C₂×C₈	—	17	1	no	—
161	C₂×C₈	—	17	1	no	—
162	C₂×C₈	—	34	1	no	—
163	C₂×C₈	—	34	2	no	—
164	C₂×C₈	—	34	1	no	—
165	C₂×C₈	—	34	2	no	—
166	C₂×C₈	—	34	4	no	—
167	C₂×C₈	—	68	2	no	—
168	C₂×C₈	—	68	2	no	—
169	C₂×C₈	#63242	1	2	yes	—
170	C₂×C₈	C₄×#595	1	1	yes	characteristic
171	C₂×C₈	C₄×#595	1	1	yes	characteristic
172	C₂×C₈	C₂×#3624	1	2	yes	—
173	C₄²	—	17	1	no	—
174	C₄²	—	17	1	no	—
175	C₄²	—	17	1	no	—
176	C₄²	—	17	1	no	—
177	C₂³	Q₈×#595	1	1	yes	characteristic
178	C₂×C₄	—	17	1	no	—
179	C₂×C₄	—	17	1	no	—
180	C₂×C₄	—	17	1	no	—
181	C₂×C₄	—	17	1	no	—
182	C₂×C₄	—	34	2	no	—
183	C₂×C₄	—	34	2	no	—
184	C₂×C₄	C₂×#63242	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
186	C₂×C₄	#10587678	1	1	yes	characteristic
187	C₂×C₄	D₄×#595	1	1	yes	characteristic
188	C₂×C₄	#10587680	1	1	yes	characteristic
189	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	34	4	no	—
192	C₈	—	34	1	no	—
193	C₈	—	34	1	no	—
194	C₈	—	34	4	no	—
195	C₈	—	68	4	no	—
196	C₈	—	68	2	no	—
197	C₈	C₄×#3624	1	2	yes	—
198	C₈	C₄×#3624	1	2	yes	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
202	C₂²	Q₈×#3624	1	1	yes	characteristic
203	C₂²	#11763858	1	1	yes	characteristic
204	C₄	—	17	2	no	—
205	C₄	—	17	2	no	—
206	C₄	—	34	1	no	—
207	C₄	—	34	1	no	—
208	C₄	C₂×C₄×#3624	1	1	yes	characteristic
209	C₄	#11763774	1	1	yes	characteristic
210	C₄	D₄×#3624	1	1	yes	characteristic
211	C₄	#11763837	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	C₄⋊C₄×#3624	1	1	yes	characteristic
215	C₂	#14544593	1	1	yes	characteristic
216	C₂	#14544605	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
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	#8294
LCS	G ▷ 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
34. C₁₇×#108	C₈	semidirect
35. #10587490	C₈	semidirect
136. #108	#3624	semidirect
150. C₁₇	#8294	semidirect

Maximal subgroups

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

subgroup	index
#14544331	2
M₁₆×#3624	2
C₂×#11763650	2
#14543617	2
#108×Dic₁₇	2
#8294	17