Subgroups — #17960656

Order 4352.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763698	C₂	1	2	yes	maximal
2	#14543823	C₂	1	2	yes	maximal
3	#14543822	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63216	C₂²	1	4	yes	—
6	#11763698	C₄	1	8	yes	complemented
7	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	C₈	1	2	yes	—
18	C₂×C₄×Dic₁₇	C₈	1	2	yes	—
19	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂³	1	1	yes	characteristic
23	C₂×#63216	—	2	4	no	—
24	C₂×#63216	—	2	4	no	—
25	C₂×#63216	C₂×C₄	1	8	yes	—
26	C₄×C₈×C₁₇	—	2	2	no	—
27	C₈×Dic₁₇	—	2	2	no	—
28	C₈×Dic₁₇	—	2	2	no	—
29	C₈×Dic₁₇	C₈	1	4	yes	—
30	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
31	C₄×#3622	—	2	2	no	—
32	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
33	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
36	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₁₇	—	2	1	no	—
38	C₂×C₈×C₁₇	—	2	1	no	—
39	C₂×C₈×C₁₇	C₄²	1	2	yes	—
40	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
42	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
43	C₄²×C₁₇	—	2	1	no	—
44	C₄²×C₁₇	—	2	1	no	—
45	C₄×Dic₁₇	—	2	4	no	—
46	C₄×Dic₁₇	—	2	1	no	—
47	C₄×Dic₁₇	—	2	1	no	—
48	C₄×Dic₁₇	—	2	4	no	—
49	C₄×Dic₁₇	C₄²	1	2	yes	—
50	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
51	C₂×#3622	—	2	1	no	—
52	C₂×#3622	—	2	1	no	—
53	C₂×#3622	C₂×C₈	1	2	yes	—
54	C₂×#3622	C₂²×C₄	1	2	yes	—
55	C₂×#3622	#41	1	2	yes	—
56	C₂×#3622	#41	1	2	yes	—
57	#63216	—	2	8	no	—
58	#12591	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	M₃₂	1	4	yes	—
63	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
69	C₂×Dic₁₇	—	2	2	no	—
70	C₂×Dic₁₇	—	2	2	no	—
71	C₂×Dic₁₇	—	2	2	no	—
72	C₂×Dic₁₇	—	2	2	no	—
73	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
74	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
75	C₂×Dic₁₇	M₃₂	1	4	yes	—
76	C₈×C₁₇	—	2	2	no	—
77	C₈×C₁₇	#97	1	4	yes	—
78	#3622	—	2	2	no	—
79	#3622	#117	1	4	yes	—
80	C₂×#346	—	17	2	no	—
81	C₄²×C₈	—	17	1	no	—
82	#1374	—	17	2	no	—
83	#1374	—	17	2	no	—
84	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
85	C₂²×C₁₇	#431	1	1	yes	characteristic
86	C₂²×C₁₇	#431	1	1	yes	characteristic
87	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
88	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
89	C₄×C₁₇	—	2	4	no	—
90	C₄×C₁₇	#376	1	1	yes	characteristic
91	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
92	C₄×C₁₇	#431	1	1	yes	characteristic
93	C₄×C₁₇	#431	1	1	yes	characteristic
94	Dic₁₇	—	2	2	no	—
95	Dic₁₇	—	2	2	no	—
96	Dic₁₇	—	2	4	no	—
97	C₂²×C₁₆	—	17	4	no	—
98	C₂×C₄×C₈	—	17	1	no	—
99	C₂×C₄×C₈	—	17	1	no	—
100	C₂×C₄×C₈	—	17	1	no	—
101	C₂×C₄×C₈	—	17	1	no	—
102	C₂×C₄×C₈	—	17	1	no	—
103	C₂×C₄×C₈	#595	1	1	yes	characteristic
104	C₄³	—	17	1	no	—
105	#346	—	17	8	no	—
106	C₂×C₁₇	#1744	1	1	yes	characteristic
107	C₂×C₁₇	C₄×M₃₂	1	1	yes	characteristic
108	C₂×C₁₇	C₄×M₃₂	1	1	yes	characteristic
109	C₂×C₁₇	#2163	1	2	yes	—
110	C₂×C₁₇	#2163	1	2	yes	—
111	C₂²×C₈	—	17	1	no	—
112	C₂²×C₈	—	17	1	no	—
113	C₂²×C₈	C₂×#595	1	1	yes	characteristic
114	C₂²×C₈	C₂×#595	1	1	yes	characteristic
115	C₂×C₄²	—	17	1	no	—
116	C₂×C₄²	—	17	2	no	—
117	C₂×C₄²	—	17	2	no	—
118	C₂×C₄²	—	17	1	no	—
119	C₂×C₄²	C₂×#595	1	1	yes	characteristic
120	C₂×C₁₆	—	17	8	no	—
121	C₂×C₁₆	—	34	4	no	—
122	C₂×C₁₆	—	34	4	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	17	4	no	—
125	C₄×C₈	—	17	4	no	—
126	C₄×C₈	—	34	2	no	—
127	C₄×C₈	—	34	2	no	—
128	C₄×C₈	—	34	2	no	—
129	C₁₇	#12591	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₄	—	17	1	no	—
131	C₂²×C₄	—	17	1	no	—
132	C₂²×C₄	—	17	2	no	—
133	C₂²×C₄	#63242	1	2	yes	—
134	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	17	2	no	—
138	C₂×C₈	—	17	2	no	—
139	C₂×C₈	—	17	2	no	—
140	C₂×C₈	—	17	2	no	—
141	C₂×C₈	—	34	1	no	—
142	C₂×C₈	—	34	1	no	—
143	C₂×C₈	#63242	1	2	yes	—
144	C₂×C₈	#63243	1	2	yes	—
145	C₂×C₈	C₄×#595	1	2	yes	—
146	C₂×C₈	#63247	1	2	yes	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	17	2	no	—
150	C₄²	—	17	2	no	—
151	C₄²	—	34	4	no	—
152	C₄²	—	34	1	no	—
153	C₄²	—	34	1	no	—
154	C₄²	—	34	4	no	—
155	C₁₆	—	34	8	no	—
156	C₂³	C₂×#63242	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	17	2	no	—
160	C₂×C₄	—	17	2	no	—
161	C₂×C₄	—	17	4	no	—
162	C₂×C₄	—	34	2	no	—
163	C₂×C₄	—	34	2	no	—
164	C₂×C₄	—	34	2	no	—
165	C₂×C₄	—	34	2	no	—
166	C₂×C₄	#10587530	1	4	yes	—
167	C₂×C₄	C₂×#63242	1	1	yes	characteristic
168	C₂×C₄	C₂×#63243	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
170	C₂×C₄	#10587677	1	1	yes	characteristic
171	C₂×C₄	#10587677	1	1	yes	characteristic
172	C₂×C₄	C₂×#63247	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	17	4	no	—
175	C₈	—	34	2	no	—
176	C₈	#10587542	1	4	yes	—
177	C₂²	C₄×#63242	1	1	yes	characteristic
178	C₂²	#11763826	1	1	yes	characteristic
179	C₂²	#11763828	1	1	yes	characteristic
180	C₂²	#11763865	1	2	yes	—
181	C₂²	C₂×#10587530	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	34	2	no	—
184	C₄	—	34	2	no	—
185	C₄	—	34	4	no	—
186	C₄	#11763745	1	1	yes	characteristic
187	C₄	C₂×#10587542	1	1	yes	characteristic
188	C₄	#11763826	1	1	yes	characteristic
189	C₄	#11763828	1	1	yes	characteristic
190	C₂	#14544499	1	1	yes	characteristic
191	C₂	C₄×#10587530	1	1	yes	characteristic
192	C₂	#14544991	1	1	yes	characteristic
193	C₂	#14545146	1	2	yes	—
194	C₂	#14545148	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#12591
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #11763698	C₄	semidirect
129. C₁₇	#12591	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763698	2
#14543823	2
#14543822	2
C₄×C₈×Dic₁₇	2
#12591	17