Subgroups — #16945758

Order 4352.

330 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530583	C₂	1	1	yes	maximal, characteristic
2	#14527543	C₂	1	1	yes	maximal, characteristic
3	#14526842	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
5	#11762833	—	2	1	no	—
6	C₄⋊#63214	C₂²	1	2	yes	—
7	C₂²⋊#63214	—	2	1	no	—
8	C₄×#63214	—	2	1	no	—
9	#11762562	—	2	1	no	—
10	#11762562	C₂²	1	2	yes	—
11	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
12	C₁₇×#500	—	2	1	no	—
13	C₁₇×#500	—	2	1	no	—
14	C₁₇×#500	C₄	1	2	yes	—
15	C₁₇×#500	C₂²	1	2	yes	—
16	C₂×C₁₇×#107	—	2	1	no	—
17	C₂×C₁₇×#107	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₇	—	4	1	no	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₁₇×C₄⋊Q₈	—	2	2	no	—
23	C₁₇×C₄⋊Q₈	—	2	2	no	—
24	C₁₇×C₄⋊Q₈	—	4	1	no	—
25	C₁₇×C₄⋊Q₈	—	4	1	no	—
26	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
28	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
29	C₂×#63223	—	4	1	no	—
30	C₁₇×#107	—	2	2	no	—
31	C₁₇×#107	—	2	2	no	—
32	C₁₇×#107	—	2	2	no	—
33	C₁₇×#107	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	C₂×#63214	—	2	2	no	—
39	C₄⋊#3622	—	2	2	no	—
40	C₄⋊#3622	—	4	1	no	—
41	C₂×Q₈×C₁₇	—	4	2	no	—
42	C₂×Q₈×C₁₇	—	4	2	no	—
43	C₂×Q₈×C₁₇	—	8	4	no	—
44	C₂²×C₄×C₁₇	—	4	2	no	—
45	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₁₇	—	2	2	no	—
49	C₂×C₈×C₁₇	—	2	2	no	—
50	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₁₇	—	2	4	no	—
55	C₄⋊C₄×C₁₇	—	4	2	no	—
56	C₄⋊C₄×C₁₇	—	4	2	no	—
57	C₄⋊C₄×C₁₇	—	4	2	no	—
58	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₇	D₈	1	1	yes	characteristic
60	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
61	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
62	#63223	—	8	2	no	—
63	C₂×#3622	—	4	2	no	—
64	#63214	—	4	2	no	—
65	#11972	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	#102	1	1	yes	characteristic
67	Q₈×C₁₇	—	8	4	no	—
68	Q₈×C₁₇	—	8	4	no	—
69	C₂×C₄×C₁₇	—	2	2	no	—
70	C₂×C₄×C₁₇	—	2	1	no	—
71	C₂×C₄×C₁₇	—	2	1	no	—
72	C₂×C₄×C₁₇	—	4	4	no	—
73	C₂×C₄×C₁₇	—	8	4	no	—
74	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
78	C₂×C₄×C₁₇	Q₃₂	1	2	yes	—
79	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
82	C₈×C₁₇	—	2	1	no	—
83	C₈×C₁₇	—	2	1	no	—
84	C₈×C₁₇	#101	1	2	yes	—
85	C₈×C₁₇	#137	1	2	yes	—
86	#3622	—	8	2	no	—
87	C₂×#500	D₁₇	1	1	yes	characteristic
88	#2162	—	17	1	no	—
89	#1567	—	17	1	no	—
90	#1365	—	17	4	no	—
91	C₂²×C₁₇	—	2	1	no	—
92	C₂²×C₁₇	—	2	1	no	—
93	C₂²×C₁₇	#358	1	2	yes	—
94	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
95	C₄×C₁₇	—	2	1	no	—
96	C₄×C₁₇	—	2	1	no	—
97	C₄×C₁₇	—	8	4	no	—
98	C₄×C₁₇	#358	1	2	yes	—
99	C₄×C₁₇	#361	1	2	yes	—
100	C₄×C₁₇	C₂×#101	1	1	yes	characteristic
101	C₄×C₁₇	#419	1	1	yes	characteristic
102	C₄×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
103	C₄×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
104	C₂×C₄⋊Q₈	—	2	1	no	—
105	#500	—	2	1	no	—
106	#500	—	2	1	no	—
107	#500	Dic₁₇	1	2	yes	—
108	#500	C₂×D₁₇	1	2	yes	—
109	C₂×#107	—	2	1	no	—
110	C₂×#107	Dic₁₇	1	2	yes	—
111	#423	—	34	1	no	—
112	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
113	C₄⋊C₁₆	—	17	2	no	—
114	C₂²⋊C₁₆	—	34	1	no	—
115	C₄×C₁₆	—	34	1	no	—
116	#335	—	17	2	no	—
117	#335	—	34	1	no	—
118	C₂×C₁₇	—	2	2	no	—
119	C₂×C₁₇	#1546	1	1	yes	characteristic
120	C₂×C₁₇	C₂×#358	1	1	yes	characteristic
121	C₂×C₁₇	C₂×#361	1	1	yes	characteristic
122	C₂²×Q₈	—	4	1	no	—
123	C₂×M₁₆	—	68	1	no	—
124	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	4	1	no	—
129	C₄⋊Q₈	—	4	1	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
133	C₂×C₁₆	—	34	2	no	—
134	#107	—	2	2	no	—
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#107	C₂×Dic₁₇	1	4	yes	—
138	C₄⋊C₈	—	34	2	no	—
139	C₄⋊C₈	—	68	1	no	—
140	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
143	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
144	C₁₇	#11972	1	1	yes	characteristic, complemented, Sylow
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	8	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	#63228	1	1	yes	characteristic
150	C₂²×C₄	#63229	1	1	yes	characteristic
151	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
152	M₁₆	—	136	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	68	2	no	—
156	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
158	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
159	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄²	#63228	1	1	yes	characteristic
165	C₄²	#63229	1	1	yes	characteristic
166	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
168	C₁₆	—	68	2	no	—
169	C₂³	#10587503	1	1	yes	characteristic
170	Q₈	—	8	4	no	—
171	Q₈	—	8	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	#10587499	1	2	yes	—
178	C₂×C₄	#10587500	1	2	yes	—
179	C₂×C₄	#10587503	1	1	yes	characteristic
180	C₂×C₄	#10587503	1	1	yes	characteristic
181	C₂×C₄	#10587503	1	1	yes	characteristic
182	C₂×C₄	C₂×#63228	1	1	yes	characteristic
183	C₂×C₄	C₂×#63229	1	1	yes	characteristic
184	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	136	2	no	—
188	C₈	#10587507	1	2	yes	—
189	C₈	#10587628	1	2	yes	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#11762613	1	2	yes	—
193	C₂²	C₂×#10587503	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	8	4	no	—
197	C₄	#11762613	1	2	yes	—
198	C₄	#11762622	1	2	yes	—
199	C₄	C₂×#10587507	1	1	yes	characteristic
200	C₄	#11762818	1	1	yes	characteristic
201	C₄	C₂×#10587499	1	1	yes	characteristic
202	C₄	C₂×#10587500	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#14527456	1	1	yes	characteristic
205	C₂	C₂×#11762613	1	1	yes	characteristic
206	C₂	C₂×#11762622	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587503
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	#102
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	#11972
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #14526842	C₂	semidirect
144. C₁₇	#11972	semidirect

Maximal subgroups

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

subgroup	index
#14530583	2
#14527543	2
#14526842	2
C₂×C₁₇×#500	2
#11972	17