Subgroups — #16945926

Order 4352.

250 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530587	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
3	#14526853	C₂	1	2	yes	maximal, complemented
4	#14526847	C₂	1	2	yes	maximal, complemented
5	C₁₇×#2163	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#3622	C₄	1	1	yes	characteristic
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₄⋊#63214	C₂²	1	2	yes	—
9	C₂²⋊#63214	—	2	1	no	—
10	#11762585	—	2	1	no	—
11	C₈×#3622	C₄	1	1	yes	characteristic
12	C₈×#3622	C₄	1	1	yes	characteristic
13	C₈×#3622	C₂²	1	1	yes	characteristic
14	C₈×#3622	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₁₇×C₂²⋊C₁₆	—	2	1	no	—
18	C₁₇×#346	—	2	1	no	—
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	—	2	1	no	—
23	C₂²×#3622	C₈	1	1	yes	characteristic
24	C₂²×#3622	C₈	1	1	yes	characteristic
25	C₂×C₁₆×C₁₇	—	2	2	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×#63214	—	2	2	no	—
31	C₄×#3622	—	2	1	no	—
32	C₄×#3622	C₈	1	1	yes	characteristic
33	C₄×#3622	C₈	1	1	yes	characteristic
34	C₄×#3622	C₂×C₄	1	1	yes	characteristic
35	C₄×#3622	C₂×C₄	1	1	yes	characteristic
36	C₄×#3622	C₂×C₄	1	1	yes	characteristic
37	C₄×#3622	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
41	C₂×C₈×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	—	2	2	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
50	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×#3622	—	2	4	no	—
52	C₂×#3622	—	2	1	no	—
53	C₂×#3622	—	2	1	no	—
54	C₂×#3622	—	2	2	no	—
55	C₂×#3622	—	2	2	no	—
56	C₂×#3622	C₂×C₈	1	1	yes	characteristic
57	C₂×#3622	C₂×C₈	1	1	yes	characteristic
58	C₂×#3622	C₂×C₈	1	1	yes	characteristic
59	C₂×#3622	C₂×C₈	1	1	yes	characteristic
60	C₁₆×C₁₇	—	4	2	no	—
61	#63214	—	4	2	no	—
62	#12013	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	1	no	—
66	C₂×C₄×C₁₇	—	2	1	no	—
67	C₂×C₄×C₁₇	—	2	1	no	—
68	C₂×C₄×C₁₇	—	2	1	no	—
69	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₈×C₁₇	—	2	1	no	—
76	C₈×C₁₇	—	2	1	no	—
77	C₈×C₁₇	#108	1	2	yes	—
78	C₈×C₁₇	#135	1	2	yes	—
79	#3622	—	2	2	no	—
80	#3622	—	2	2	no	—
81	#3622	—	2	1	no	—
82	#3622	—	2	1	no	—
83	#3622	M₃₂	1	2	yes	—
84	#3622	M₃₂	1	2	yes	—
85	#2163	—	17	1	no	—
86	#2163	D₁₇	1	1	yes	characteristic
87	C₂×C₈²	—	17	1	no	—
88	#1370	—	17	2	no	—
89	#1367	—	17	2	no	—
90	C₂²×C₁₇	—	2	1	no	—
91	C₂²×C₁₇	—	2	1	no	—
92	C₂²×C₁₇	#364	1	2	yes	—
93	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₄×C₁₇	—	2	1	no	—
95	C₄×C₁₇	—	2	1	no	—
96	C₄×C₁₇	—	2	1	no	—
97	C₄×C₁₇	—	2	1	no	—
98	C₄×C₁₇	#427	1	1	yes	characteristic
99	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
100	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
101	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
102	C₂×C₄×C₈	—	17	1	no	—
103	C₂×C₄×C₈	—	17	1	no	—
104	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
105	C₄⋊C₁₆	—	17	2	no	—
106	C₄⋊C₁₆	C₂×D₁₇	1	2	yes	—
107	C₂²⋊C₁₆	—	2	1	no	—
108	C₂²⋊C₁₆	—	34	1	no	—
109	#346	—	2	1	no	—
110	#346	—	34	1	no	—
111	C₈²	—	17	1	no	—
112	C₈²	—	17	1	no	—
113	C₈²	—	17	1	no	—
114	C₈²	—	17	1	no	—
115	C₂×C₁₇	—	2	2	no	—
116	C₂×C₁₇	#1564	1	1	yes	characteristic
117	C₂×C₁₇	#2150	1	1	yes	characteristic
118	C₂×C₁₇	C₂×#364	1	1	yes	characteristic
119	C₂²×C₈	—	17	1	no	—
120	C₂²×C₈	—	17	1	no	—
121	C₂²×C₈	—	34	1	no	—
122	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
124	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	34	2	no	—
127	C₄×C₈	—	17	1	no	—
128	C₄×C₈	—	17	1	no	—
129	C₄×C₈	—	17	1	no	—
130	C₄×C₈	—	17	1	no	—
131	C₄×C₈	—	17	1	no	—
132	C₄×C₈	—	17	1	no	—
133	C₄×C₈	—	34	1	no	—
134	C₄×C₈	Dic₃₄	1	1	yes	characteristic
135	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
137	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
138	C₁₇	#12013	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
140	C₂²×C₄	#63218	1	1	yes	characteristic
141	C₂²×C₄	#63225	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	17	1	no	—
145	C₂×C₈	—	17	1	no	—
146	C₂×C₈	—	17	1	no	—
147	C₂×C₈	—	17	1	no	—
148	C₂×C₈	—	34	2	no	—
149	C₂×C₈	—	34	2	no	—
150	C₂×C₈	—	34	4	no	—
151	C₂×C₈	—	34	1	no	—
152	C₂×C₈	—	34	1	no	—
153	C₂×C₈	#63225	1	1	yes	characteristic
154	C₂×C₈	#63225	1	1	yes	characteristic
155	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
157	C₄²	C₈×D₁₇	1	1	yes	characteristic
158	C₄²	#63218	1	1	yes	characteristic
159	C₄²	#63225	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
161	C₁₆	—	4	2	no	—
162	C₁₆	—	68	2	no	—
163	C₂³	#10587485	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	#10587485	1	1	yes	characteristic
170	C₂×C₄	#10587485	1	1	yes	characteristic
171	C₂×C₄	#10587485	1	1	yes	characteristic
172	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
173	C₂×C₄	C₂×#63218	1	1	yes	characteristic
174	C₂×C₄	C₂×#63225	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	17	2	no	—
178	C₈	—	17	2	no	—
179	C₈	—	34	2	no	—
180	C₈	—	34	2	no	—
181	C₈	—	34	1	no	—
182	C₈	—	34	1	no	—
183	C₈	#10587493	1	2	yes	—
184	C₈	#10587636	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#11762629	1	2	yes	—
188	C₂²	C₂×#10587485	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	#11762845	1	1	yes	characteristic
194	C₄	C₂×#10587493	1	1	yes	characteristic
195	C₄	M₃₂×D₁₇	1	1	yes	characteristic
196	C₄	M₃₂×D₁₇	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	#14527530	1	1	yes	characteristic
199	C₂	#14530539	1	1	yes	characteristic
200	C₂	C₂×#11762629	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#2163	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#2163	D₁₇
hypocenter	C₁₇	#12013
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #14526853	C₂	semidirect
4. #14526847	C₂	semidirect
138. C₁₇	#12013	semidirect

Maximal subgroups

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

subgroup	index
#14530587	2
C₂×C₈×#3622	2
#14526853	2
#14526847	2
C₁₇×#2163	2
#12013	17