Subgroups — #17953812

Order 4352.

462 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544716	C₂	1	2	yes	maximal
2	C₂×#11763672	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#3624	—	2	4	no	—
5	C₂×C₄⋊#3624	C₂²	1	4	yes	—
6	#11763672	—	2	8	no	—
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₄	1	2	yes	—
9	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
10	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
12	C₂²×#3624	—	2	4	no	—
13	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₇	D₄	1	2	yes	—
17	C₂×C₄×Dic₁₇	Q₈	1	2	yes	—
18	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
19	C₂²×#3622	C₂×C₄	1	2	yes	—
20	C₄⋊#3624	—	2	8	no	—
21	C₄⋊#3624	—	4	8	no	—
22	C₄×C₈×C₁₇	C₈	1	4	yes	complemented
23	C₈×Dic₁₇	—	2	4	no	—
24	C₈×Dic₁₇	D₄	1	4	yes	—
25	C₈×Dic₁₇	Q₈	1	4	yes	—
26	C₄×#3622	—	2	2	no	—
27	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₂²×Dic₁₇	C₄⋊C₄	1	2	yes	—
30	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×#3624	—	4	4	no	—
33	C₂×#3624	—	4	8	no	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
37	C₄²×C₁₇	C₂×C₈	1	2	yes	—
38	C₄²×C₁₇	M₁₆	1	2	yes	—
39	C₄×Dic₁₇	—	2	4	no	—
40	C₄×Dic₁₇	—	2	2	no	—
41	C₄×Dic₁₇	D₈	1	4	yes	—
42	C₄×Dic₁₇	Q₁₆	1	4	yes	—
43	C₄×Dic₁₇	C₂×D₄	1	2	yes	—
44	C₄×Dic₁₇	C₂×Q₈	1	2	yes	—
45	C₂×#3622	—	2	2	no	—
46	C₂×#3622	—	2	2	no	—
47	C₂×#3622	C₄⋊C₄	1	4	yes	—
48	#8806	—	17	1	no	maximal, Sylow
49	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₄⋊C₈	1	4	yes	—
51	C₂×C₄×C₁₇	#107	1	4	yes	—
52	C₂×C₄×C₁₇	#108	1	4	yes	—
53	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
57	C₂×Dic₁₇	—	2	2	no	—
58	C₂×Dic₁₇	—	2	2	no	—
59	C₂×Dic₁₇	—	2	2	no	—
60	C₂×Dic₁₇	#107	1	4	yes	—
61	C₂×Dic₁₇	#127	1	1	yes	characteristic
62	C₂×Dic₁₇	C₄⋊Q₈	1	2	yes	—
63	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
65	C₂×Dic₁₇	C₂×Q₁₆	1	2	yes	—
66	#3624	—	8	4	no	—
67	C₈×C₁₇	C₄⋊C₈	1	8	yes	complemented
68	#3622	—	2	4	no	—
69	#1842	—	17	2	no	—
70	C₄²×C₈	—	17	1	no	—
71	C₂×#335	—	17	4	no	—
72	C₂²×C₁₇	#384	1	1	yes	characteristic
73	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₂²×C₁₇	#423	1	1	yes	characteristic
75	C₂²×C₁₇	C₂×#107	1	2	yes	—
76	C₂²×C₁₇	C₂×#108	1	2	yes	—
77	C₄×C₁₇	#335	1	8	yes	complemented
78	C₄×C₁₇	C₂×C₄⋊C₈	1	2	yes	—
79	C₄×C₁₇	#423	1	2	yes	—
80	Dic₁₇	—	2	4	no	—
81	Dic₁₇	—	2	2	no	—
82	Dic₁₇	#493	1	1	yes	characteristic
83	Dic₁₇	#494	1	1	yes	characteristic
84	Dic₁₇	#500	1	2	yes	—
85	C₂×C₄⋊C₈	—	17	4	no	—
86	C₂×C₄⋊C₈	—	34	4	no	—
87	C₂×C₄×C₈	—	17	2	no	—
88	C₂×C₄×C₈	—	17	2	no	—
89	C₂×C₄×C₈	—	17	1	no	—
90	C₂×C₄×C₈	#595	1	1	yes	characteristic
91	C₄³	—	17	1	no	—
92	#335	—	34	8	no	—
93	C₂×C₁₇	C₂×#335	1	2	yes	—
94	C₂×C₁₇	#1567	1	2	yes	—
95	C₂×C₁₇	#1842	1	1	yes	characteristic
96	C₂×C₁₇	#1845	1	1	yes	characteristic
97	C₂×C₁₇	#1848	1	1	yes	characteristic
98	C₂²×C₈	—	17	2	no	—
99	C₂²×C₈	—	34	4	no	—
100	C₂²×C₈	C₂×#595	1	2	yes	—
101	C₂×C₄²	—	17	2	no	—
102	C₂×C₄²	—	17	1	no	—
103	C₂×C₄²	—	17	2	no	—
104	C₂×C₄²	—	17	1	no	—
105	C₂×C₄²	C₂×#595	1	1	yes	characteristic
106	C₄⋊C₈	—	34	8	no	—
107	C₄⋊C₈	—	68	8	no	—
108	C₄×C₈	—	17	4	no	—
109	C₄×C₈	—	17	4	no	—
110	C₄×C₈	—	34	4	no	—
111	C₄×C₈	—	34	2	no	—
112	C₄×C₈	#3624	1	4	yes	complemented
113	C₁₇	#8806	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₄	—	17	1	no	—
115	C₂²×C₄	—	17	2	no	—
116	C₂²×C₄	—	17	1	no	—
117	C₂²×C₄	C₄⋊#595	1	2	yes	—
118	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
119	C₂×C₈	—	17	4	no	—
120	C₂×C₈	—	34	2	no	—
121	C₂×C₈	—	34	2	no	—
122	C₂×C₈	—	68	4	no	—
123	C₂×C₈	—	68	8	no	—
124	C₂×C₈	#63243	1	4	yes	—
125	C₂×C₈	C₄⋊#595	1	4	yes	—
126	C₂×C₈	C₂×#3624	1	4	yes	—
127	C₄²	—	17	2	no	—
128	C₄²	—	17	2	no	—
129	C₄²	—	17	4	no	—
130	C₄²	—	17	4	no	—
131	C₄²	—	34	2	no	—
132	C₄²	—	34	4	no	—
133	C₄²	#63243	1	2	yes	—
134	C₄²	C₂×#3624	1	2	yes	—
135	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
136	C₂×C₄	—	17	1	no	—
137	C₂×C₄	—	17	1	no	—
138	C₂×C₄	—	17	2	no	—
139	C₂×C₄	—	17	2	no	—
140	C₂×C₄	—	17	2	no	—
141	C₂×C₄	—	17	4	no	—
142	C₂×C₄	—	34	2	no	—
143	C₂×C₄	—	34	2	no	—
144	C₂×C₄	—	34	2	no	—
145	C₂×C₄	#10587534	1	4	yes	—
146	C₂×C₄	#10587536	1	4	yes	—
147	C₂×C₄	C₄⋊#3624	1	4	yes	—
148	C₂×C₄	C₂×#63243	1	2	yes	—
149	C₂×C₄	C₂×#63243	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄⋊#595	1	2	yes	—
151	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
152	C₈	—	34	4	no	—
153	C₈	—	136	4	no	—
154	C₈	C₄⋊#3624	1	8	yes	complemented
155	C₂²	#11763755	1	1	yes	characteristic
156	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
157	C₂²	#11763813	1	1	yes	characteristic
158	C₂²	C₂×#10587534	1	2	yes	—
159	C₂²	C₂×#10587536	1	2	yes	—
160	C₄	—	17	1	no	—
161	C₄	—	17	1	no	—
162	C₄	—	17	2	no	—
163	C₄	—	34	2	no	—
164	C₄	—	34	4	no	—
165	C₄	#11763672	1	8	yes	complemented
166	C₄	C₂×C₄⋊#3624	1	2	yes	—
167	C₄	#11763813	1	2	yes	—
168	C₂	C₂×#11763672	1	2	yes	—
169	C₂	#14544342	1	2	yes	—
170	C₂	#14544716	1	1	yes	characteristic
171	C₂	#14544720	1	1	yes	characteristic
172	C₂	#14544725	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#8806
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
22. C₄×C₈×C₁₇	C₈	semidirect
67. C₈×C₁₇	C₄⋊C₈	semidirect
77. C₄×C₁₇	#335	semidirect
112. C₄×C₈	#3624	semidirect
113. C₁₇	#8806	semidirect
154. C₈	C₄⋊#3624	semidirect
165. C₄	#11763672	semidirect

Maximal subgroups

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

subgroup	index
#14544716	2
C₂×#11763672	2
C₄×C₈×Dic₁₇	2
#8806	17