Subgroups — #17960079

Order 4352.

286 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	#14545114	C₂	1	2	yes	maximal
2	#14543814	C₂	1	2	yes	maximal, complemented
3	#14543813	C₂	1	2	yes	maximal, complemented
4	C₈²×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587529	—	2	2	no	—
6	#11763730	—	2	2	no	—
7	C₄⋊#63216	C₂²	1	4	yes	—
8	C₄×C₈×D₁₇	C₄	1	1	yes	characteristic
9	C₄×C₈×D₁₇	C₄	1	1	yes	characteristic
10	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
11	C₈×#3622	C₄	1	1	yes	characteristic
12	C₈×#3622	C₂²	1	1	yes	characteristic
13	C₈×#3622	C₂²	1	1	yes	characteristic
14	C₈²×C₁₇	C₄	1	1	yes	characteristic
15	C₂×C₈×D₁₇	—	2	1	no	—
16	C₂×C₈×D₁₇	C₈	1	1	yes	characteristic
17	C₂×C₈×D₁₇	C₈	1	1	yes	characteristic
18	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
19	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
20	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×#63216	—	2	4	no	—
22	#10587529	—	4	4	no	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₁₇	—	2	1	no	—
27	C₈×Dic₁₇	C₈	1	1	yes	characteristic
28	C₈×Dic₁₇	C₈	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
31	C₄×#3622	C₂×C₄	1	1	yes	characteristic
32	C₄×#3622	C₂×C₄	1	1	yes	characteristic
33	C₄×#3622	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₇	—	2	1	no	—
38	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
45	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
46	C₂×#3622	—	2	1	no	—
47	C₂×#3622	C₂×C₈	1	1	yes	characteristic
48	C₂×#3622	C₂×C₈	1	1	yes	characteristic
49	C₂×#3622	C₂×D₄	1	1	yes	characteristic
50	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
51	C₈×D₁₇	—	2	4	no	—
52	C₈×D₁₇	—	2	2	no	—
53	C₈×D₁₇	—	2	2	no	—
54	C₈×D₁₇	C₁₆	1	4	yes	complemented
55	C₈×D₁₇	D₈	1	2	yes	—
56	C₈×D₁₇	Q₁₆	1	2	yes	—
57	#63216	—	4	4	no	—
58	C₂×#1370	—	17	1	no	maximal, Sylow
59	C₂²×D₁₇	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
63	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
65	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
66	C₄×D₁₇	—	2	2	no	—
67	C₄×D₁₇	#107	1	2	yes	—
68	C₄×D₁₇	#108	1	2	yes	—
69	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
70	C₄×D₁₇	M₃₂	1	2	yes	—
71	C₈×C₁₇	—	2	2	no	—
72	C₈×C₁₇	—	2	1	no	—
73	C₈×C₁₇	—	2	1	no	—
74	C₈×C₁₇	#107	1	2	yes	—
75	C₈×C₁₇	C₂×C₁₆	1	2	yes	complemented
76	#3622	—	2	2	no	—
77	#3622	—	2	1	no	—
78	#3622	—	2	1	no	—
79	#3622	C₂×C₁₆	1	2	yes	complemented
80	#3622	C₂×D₈	1	1	yes	characteristic
81	#3622	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄⋊C₁₆	—	17	2	no	—
83	C₂×C₈²	—	17	1	no	—
84	#1370	—	17	2	no	—
85	#1370	—	17	2	no	—
86	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂×D₁₇	#335	1	2	yes	—
88	C₂×D₁₇	C₄⋊C₁₆	1	2	yes	—
89	C₂×D₁₇	#364	1	2	yes	—
90	C₄×C₁₇	—	2	1	no	—
91	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
92	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
93	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
94	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
95	Dic₁₇	—	2	1	no	—
96	Dic₁₇	#335	1	2	yes	—
97	Dic₁₇	C₄⋊C₁₆	1	2	yes	—
98	C₂²×C₁₆	—	34	2	no	—
99	C₂×C₄×C₈	—	17	1	no	—
100	C₂×C₄×C₈	—	17	1	no	—
101	C₂×C₄×C₈	—	17	1	no	—
102	C₄⋊C₁₆	—	17	4	no	—
103	C₄⋊C₁₆	—	34	2	no	—
104	C₈²	—	17	1	no	—
105	C₈²	—	17	1	no	—
106	C₈²	—	17	1	no	—
107	C₈²	#595	1	1	yes	characteristic
108	C₂×C₁₇	C₂×#335	1	1	yes	characteristic
109	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
110	C₂×C₁₇	C₂×#364	1	1	yes	characteristic
111	D₁₇	#1370	1	4	yes	complemented
112	C₂²×C₈	—	17	1	no	—
113	C₂²×C₈	—	17	1	no	—
114	C₂²×C₈	—	17	1	no	—
115	C₂²×C₈	—	17	1	no	—
116	C₂²×C₈	—	34	1	no	—
117	C₂×C₄²	—	17	1	no	—
118	C₂×C₁₆	—	34	4	no	—
119	C₂×C₁₆	—	68	4	no	—
120	C₄×C₈	—	17	1	no	—
121	C₄×C₈	—	17	1	no	—
122	C₄×C₈	—	17	1	no	—
123	C₄×C₈	—	17	1	no	—
124	C₄×C₈	—	17	1	no	—
125	C₄×C₈	—	17	1	no	—
126	C₄×C₈	—	17	1	no	—
127	C₄×C₈	—	34	1	no	—
128	C₄×C₈	C₂×#595	1	1	yes	characteristic
129	C₄×C₈	C₂×#595	1	1	yes	characteristic
130	C₄×C₈	C₂×#595	1	1	yes	characteristic
131	C₁₇	C₂×#1370	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₄	—	17	1	no	—
133	C₂²×C₄	—	17	1	no	—
134	C₂²×C₄	—	17	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	1	no	—
138	C₂×C₈	—	17	4	no	—
139	C₂×C₈	—	17	2	no	—
140	C₂×C₈	—	17	2	no	—
141	C₂×C₈	—	17	1	no	—
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	34	2	no	—
144	C₂×C₈	—	34	2	no	—
145	C₂×C₈	—	34	4	no	—
146	C₂×C₈	—	34	1	no	—
147	C₂×C₈	#63242	1	1	yes	characteristic
148	C₂×C₈	#63242	1	1	yes	characteristic
149	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
150	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
151	C₄²	—	17	1	no	—
152	C₄²	—	17	1	no	—
153	C₄²	—	17	1	no	—
154	C₄²	C₂²×#595	1	1	yes	characteristic
155	C₁₆	—	68	4	no	—
156	C₂³	—	17	1	no	—
157	C₂×C₄	—	17	1	no	—
158	C₂×C₄	—	17	1	no	—
159	C₂×C₄	—	17	2	no	—
160	C₂×C₄	—	17	2	no	—
161	C₂×C₄	—	17	2	no	—
162	C₂×C₄	—	17	2	no	—
163	C₂×C₄	—	17	1	no	—
164	C₂×C₄	—	34	2	no	—
165	C₂×C₄	C₂×#63242	1	1	yes	characteristic
166	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	17	1	no	—
172	C₈	—	17	1	no	—
173	C₈	—	17	2	no	—
174	C₈	—	34	2	no	—
175	C₈	—	34	1	no	—
176	C₈	—	34	1	no	—
177	C₈	#10587529	1	2	yes	complemented
178	C₈	#10587534	1	2	yes	—
179	C₂²	—	17	2	no	—
180	C₂²	—	17	2	no	—
181	C₂²	—	17	2	no	—
182	C₂²	#11763810	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	17	2	no	—
185	C₄	—	17	2	no	—
186	C₄	—	34	1	no	—
187	C₄	C₂×#10587534	1	1	yes	characteristic
188	C₄	D₁₇⋊#108	1	1	yes	characteristic
189	C₄	C₂×#10587529	1	1	yes	characteristic
190	C₄	#11763865	1	1	yes	characteristic
191	C₂	—	17	4	no	—
192	C₂	#14544329	1	1	yes	characteristic
193	C₂	#14545114	1	1	yes	characteristic
194	C₂	#14545123	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊#595
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄²	C₂²×#595
Fitting	C₈²×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂²×D₁₇	C₄⋊C₈
cosocle	C₄×#3622	C₂³
hypercenter	C₈²	#595
hypocenter	C₁₇	C₂×#1370
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #14543814	C₂	semidirect
3. #14543813	C₂	semidirect
54. C₈×D₁₇	C₁₆	semidirect
75. C₈×C₁₇	C₂×C₁₆	semidirect
79. #3622	C₂×C₁₆	semidirect
111. D₁₇	#1370	semidirect
131. C₁₇	C₂×#1370	semidirect
177. C₈	#10587529	semidirect

Maximal subgroups

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

subgroup	index
#14545114	2
#14543814	2
#14543813	2
C₈²×D₁₇	2
C₂×#1370	17