Subgroups — #16925624

Order 4352.

350 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	C₄×#10587490	C₂	1	1	yes	maximal, characteristic
2	#14527074	C₂	1	2	yes	maximal
3	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
4	#14526620	C₂	1	2	yes	maximal
5	C₁₇×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587490	C₄	1	2	yes	—
7	C₄×#63223	C₂²	1	2	yes	—
8	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	#11762574	C₄	1	2	yes	—
11	#11762531	—	2	2	no	—
12	#11762530	—	2	2	no	—
13	C₈×#3622	C₄	1	2	yes	—
14	C₈×#3622	C₄	1	2	yes	—
15	C₁₇×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#335	C₄	1	2	yes	—
18	C₁₇×#334	C₄	1	2	yes	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×#63223	C₂×C₄	1	4	yes	—
23	C₂²×#3622	—	2	1	no	—
24	C₂²×#3622	Q₈	1	2	yes	—
25	C₁₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
26	C₁₇×C₂²⋊C₈	—	2	2	no	—
27	C₄×C₈×C₁₇	—	2	2	no	—
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	#10587490	—	2	4	no	—
33	#10587474	—	2	2	no	—
34	C₄×#3622	—	2	2	no	—
35	C₄×#3622	—	2	1	no	—
36	C₄×#3622	C₂×C₄	1	2	yes	—
37	C₄×#3622	C₂×C₄	1	2	yes	—
38	C₄×#3622	D₄	1	1	yes	characteristic
39	C₄×#3622	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	—	2	4	no	—
45	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
50	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₇	#41	1	1	yes	characteristic
54	#63223	—	2	4	no	—
55	#63223	—	2	4	no	—
56	C₂×#3622	—	2	4	no	—
57	C₂×#3622	—	2	2	no	—
58	C₂×#3622	—	2	4	no	—
59	C₂×#3622	—	2	4	no	—
60	C₂×#3622	C₄⋊C₄	1	2	yes	—
61	C₂×#3622	C₄⋊C₄	1	2	yes	—
62	#8293	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₄×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₁₇	#131	1	2	yes	—
74	C₈×C₁₇	—	2	1	no	—
75	C₈×C₁₇	—	2	1	no	—
76	C₈×C₁₇	—	4	4	no	—
77	C₈×C₁₇	#97	1	2	yes	—
78	C₈×C₁₇	#97	1	2	yes	—
79	#3622	—	2	4	no	—
80	#3622	—	2	2	no	—
81	#3622	—	4	4	no	—
82	#3622	#108	1	4	yes	complemented
83	C₄×#108	—	17	1	no	—
84	#1564	D₁₇	1	1	yes	characteristic
85	#1451	—	17	2	no	—
86	C₂×C₈²	—	17	1	no	—
87	#1278	—	17	2	no	—
88	C₂²×C₁₇	—	2	1	no	—
89	C₂²×C₁₇	—	2	1	no	—
90	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₁₇	#446	1	2	yes	—
92	C₄×C₁₇	—	2	1	no	—
93	C₄×C₁₇	—	2	1	no	—
94	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
95	C₄×C₁₇	#405	1	1	yes	characteristic
96	C₄×C₁₇	C₂×#108	1	2	yes	—
97	C₄×C₁₇	#436	1	2	yes	—
98	C₄×C₁₇	#443	1	2	yes	—
99	#431	C₂×D₁₇	1	2	yes	—
100	C₂×#108	—	17	2	no	—
101	C₄×M₁₆	—	17	2	no	—
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	#341	—	17	2	no	—
106	#335	Dic₁₇	1	2	yes	—
107	#334	Dic₁₇	1	2	yes	—
108	#322	—	34	2	no	—
109	#322	—	34	2	no	—
110	C₈²	—	17	2	no	—
111	C₈²	—	17	2	no	—
112	C₂×C₁₇	—	2	2	no	—
113	C₂×C₁₇	#1770	1	1	yes	characteristic
114	C₂×C₁₇	#1772	1	1	yes	characteristic
115	C₂×C₁₇	C₄×#108	1	1	yes	characteristic
116	C₂×M₁₆	—	17	4	no	—
117	C₂²×C₈	—	17	2	no	—
118	C₂²×C₈	—	34	1	no	—
119	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
120	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
121	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
122	#108	—	34	4	no	—
123	C₄⋊C₈	C₂×Dic₁₇	1	4	yes	—
124	C₂²⋊C₈	—	2	2	no	—
125	#97	—	34	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	17	2	no	—
128	C₄×C₈	—	17	2	no	—
129	C₄×C₈	—	17	1	no	—
130	C₄×C₈	—	17	1	no	—
131	C₄×C₈	—	34	2	no	—
132	C₄×C₈	—	34	1	no	—
133	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
134	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
135	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
137	C₁₇	#8293	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
139	C₂²×C₄	#63254	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
141	M₁₆	—	34	4	no	—
142	M₁₆	—	34	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	17	2	no	—
146	C₂×C₈	—	17	2	no	—
147	C₂×C₈	—	34	4	no	—
148	C₂×C₈	—	34	4	no	—
149	C₂×C₈	—	34	2	no	—
150	C₂×C₈	—	34	4	no	—
151	C₂×C₈	#63218	1	4	yes	—
152	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
155	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
157	C₄²	D₄×D₁₇	1	1	yes	characteristic
158	C₄²	#63256	1	1	yes	characteristic
159	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
160	C₂³	#10587573	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
164	C₂×C₄	#10587578	1	1	yes	characteristic
165	C₂×C₄	#10587579	1	1	yes	characteristic
166	C₂×C₄	C₂×#63218	1	2	yes	—
167	C₂×C₄	#10587593	1	2	yes	—
168	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
169	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
170	C₂×C₄	Q₈×Dic₁₇	1	1	yes	characteristic
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	4	no	—
174	C₈	—	17	4	no	—
175	C₈	—	34	4	no	—
176	C₈	—	34	2	no	—
177	C₈	—	68	4	no	—
178	C₈	#10587486	1	2	yes	—
179	C₈	#10587486	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄⋊C₄×Dic₁₇	1	1	yes	characteristic
183	C₂²	#11762936	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×#10587486	1	1	yes	characteristic
187	C₄	#11762767	1	1	yes	characteristic
188	C₄	#108×D₁₇	1	2	yes	—
189	C₄	#11762886	1	2	yes	—
190	C₄	#11762926	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	#14528654	1	1	yes	characteristic
193	C₂	#14528665	1	1	yes	characteristic
194	C₂	#108×Dic₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1564	C₂
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₇	C₄×Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1564	D₁₇
hypocenter	C₁₇	#8293
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
82. #3622	#108	semidirect
137. C₁₇	#8293	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587490	2
#14527074	2
C₂×C₈×#3622	2
#14526620	2
C₁₇×#1564	2
#8293	17