Subgroups — #17950271

Order 4352.

574 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544498	C₂	1	2	yes	maximal
2	C₂×#11763650	C₂	1	2	yes	maximal
3	C₂×C₈×#3624	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#3624	C₂²	1	4	yes	—
6	#11763675	C₄	1	8	yes	—
7	#11763650	C₄	1	8	yes	complemented
8	C₈×#3624	C₄	1	8	yes	complemented
9	C₂×C₄×#3622	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
16	C₂²×#3624	C₂×C₄	1	8	yes	—
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	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	C₂×C₄	1	4	yes	—
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	C₄×#3624	C₂×C₄	1	8	yes	—
26	C₄×#3624	C₂×C₄	1	8	yes	—
27	C₄×C₈×C₁₇	C₈	1	4	yes	complemented
28	C₈×Dic₁₇	C₈	1	4	yes	complemented
29	C₈×Dic₁₇	C₈	1	4	yes	complemented
30	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
31	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
32	C₄×#3622	C₈	1	4	yes	complemented
33	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
34	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	C₄²	1	2	yes	—
36	C₂²×Dic₁₇	C₂²×C₄	1	2	yes	—
37	C₂×#3624	—	2	16	no	—
38	C₂×#3624	C₄²	1	16	yes	—
39	C₂×C₈×C₁₇	C₄²	1	2	yes	—
40	C₂×C₈×C₁₇	C₄²	1	2	yes	—
41	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
45	C₄²×C₁₇	C₂×C₈	1	2	yes	—
46	C₄²×C₁₇	C₂×C₈	1	2	yes	—
47	C₄×Dic₁₇	—	2	8	no	—
48	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
49	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
50	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
51	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
52	C₂×#3622	C₄²	1	2	yes	—
53	C₂×#3622	C₄²	1	2	yes	—
54	C₂×#3622	C₂×C₈	1	2	yes	—
55	C₂×#3622	C₂×C₈	1	2	yes	—
56	C₂×#3622	C₂×C₈	1	2	yes	—
57	C₂×#3622	C₂×C₈	1	2	yes	—
58	#7841	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	2	4	no	—
61	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
62	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×Dic₁₇	—	2	4	no	—
69	C₂×Dic₁₇	C₄×C₈	1	4	yes	—
70	C₂×Dic₁₇	C₂×C₄²	1	4	yes	—
71	C₂×Dic₁₇	#131	1	8	yes	—
72	#3624	—	2	16	no	—
73	C₈×C₁₇	C₄×C₈	1	4	yes	—
74	C₈×C₁₇	C₄×C₈	1	4	yes	—
75	#3622	C₄×C₈	1	4	yes	—
76	#3622	C₄×C₈	1	4	yes	—
77	#1744	—	17	2	no	—
78	C₄²×C₈	—	17	1	no	—
79	C₂×#322	—	17	2	no	—
80	C₂×C₈²	—	17	2	no	—
81	C₂²×C₁₇	C₄³	1	1	yes	characteristic
82	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂²×C₁₇	#405	1	4	yes	—
85	C₄×C₁₇	—	2	4	no	—
86	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
87	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
89	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
90	Dic₁₇	—	2	4	no	—
91	Dic₁₇	#405	1	8	yes	—
92	C₂×C₄×C₈	—	17	4	no	—
93	C₂×C₄×C₈	—	17	1	no	—
94	C₂×C₄×C₈	—	17	1	no	—
95	C₂×C₄×C₈	—	17	1	no	—
96	C₂×C₄×C₈	—	17	1	no	—
97	C₂×C₄×C₈	—	17	1	no	—
98	C₂×C₄×C₈	#595	1	1	yes	characteristic
99	C₄³	—	17	1	no	—
100	#336	—	17	8	no	—
101	#322	—	17	8	no	—
102	C₈²	—	17	8	no	—
103	C₂×C₁₇	#1450	1	2	yes	—
104	C₂×C₁₇	#1450	1	2	yes	—
105	C₂×C₁₇	C₄²×C₈	1	1	yes	characteristic
106	C₂×C₁₇	#1726	1	2	yes	—
107	C₂²×C₈	—	17	1	no	—
108	C₂²×C₈	—	17	1	no	—
109	C₂²×C₈	—	17	8	no	—
110	C₂²×C₈	C₂×#595	1	1	yes	characteristic
111	C₂²×C₈	C₂×#595	1	1	yes	characteristic
112	C₂×C₄²	—	17	1	no	—
113	C₂×C₄²	—	17	1	no	—
114	C₂×C₄²	—	17	4	no	—
115	C₂×C₄²	C₂×#595	1	1	yes	characteristic
116	C₄×C₈	—	17	8	no	—
117	C₄×C₈	—	17	8	no	—
118	C₄×C₈	—	17	4	no	—
119	C₄×C₈	—	17	4	no	—
120	C₄×C₈	—	17	4	no	—
121	C₄×C₈	—	17	4	no	—
122	C₄×C₈	—	17	4	no	—
123	C₄×C₈	#3624	1	4	yes	complemented
124	C₁₇	#7841	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₄	—	17	2	no	—
126	C₂²×C₄	—	17	2	no	—
127	C₂²×C₄	C₄×#595	1	2	yes	—
128	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
129	C₂×C₈	—	17	2	no	—
130	C₂×C₈	—	17	2	no	—
131	C₂×C₈	—	17	2	no	—
132	C₂×C₈	—	17	2	no	—
133	C₂×C₈	—	17	2	no	—
134	C₂×C₈	—	17	2	no	—
135	C₂×C₈	—	17	16	no	—
136	C₂×C₈	—	34	16	no	—
137	C₂×C₈	#63242	1	2	yes	—
138	C₂×C₈	#63242	1	2	yes	—
139	C₂×C₈	C₄×#595	1	2	yes	—
140	C₂×C₈	C₄×#595	1	2	yes	—
141	C₂×C₈	C₂×#3624	1	2	yes	—
142	C₂×C₈	C₂×#3624	1	2	yes	—
143	C₄²	—	17	2	no	—
144	C₄²	—	17	2	no	—
145	C₄²	—	17	2	no	—
146	C₄²	—	17	2	no	—
147	C₄²	—	34	8	no	—
148	C₄²	#63242	1	2	yes	—
149	C₄²	C₂×#3624	1	2	yes	—
150	C₂³	C₂×C₄×#595	1	1	yes	characteristic
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	17	4	no	—
153	C₂×C₄	—	17	8	no	—
154	C₂×C₄	—	17	4	no	—
155	C₂×C₄	—	34	4	no	—
156	C₂×C₄	C₄×#3624	1	4	yes	—
157	C₂×C₄	C₂×#63242	1	1	yes	characteristic
158	C₂×C₄	C₂×#63242	1	1	yes	characteristic
159	C₂×C₄	C₂×#63242	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
162	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
163	C₈	—	17	4	no	—
164	C₈	—	17	4	no	—
165	C₈	—	34	16	no	—
166	C₈	C₄×#3624	1	4	yes	—
167	C₈	C₄×#3624	1	4	yes	—
168	C₂²	C₄²×#595	1	1	yes	characteristic
169	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
170	C₂²	C₄×#63242	1	1	yes	characteristic
171	C₂²	#11763772	1	4	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	17	8	no	—
174	C₄	—	34	4	no	—
175	C₄	C₂×C₄×#3624	1	1	yes	characteristic
176	C₄	C₂×C₄×#3624	1	1	yes	characteristic
177	C₄	C₄×#63242	1	1	yes	characteristic
178	C₄	C₄×#63242	1	1	yes	characteristic
179	C₂	#14543987	1	2	yes	—
180	C₂	#14543988	1	2	yes	—
181	C₂	C₄²×#3624	1	1	yes	characteristic
182	C₂	#14544440	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#595
commutator	C₂×C₁₇	C₄²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#7841
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #11763650	C₄	semidirect
8. C₈×#3624	C₄	semidirect
27. C₄×C₈×C₁₇	C₈	semidirect
28. C₈×Dic₁₇	C₈	semidirect
29. C₈×Dic₁₇	C₈	semidirect
32. C₄×#3622	C₈	semidirect
123. C₄×C₈	#3624	semidirect
124. C₁₇	#7841	semidirect

Maximal subgroups

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

subgroup	index
#14544498	2
C₂×#11763650	2
C₂×C₈×#3624	2
C₄×C₈×Dic₁₇	2
#7841	17