Subgroups — #17952363

Order 4352.

430 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×#3624	C₂	1	2	yes	maximal
2	C₂×#11763672	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763650	C₂	1	1	yes	maximal, characteristic
4	#14543620	C₂	1	2	yes	maximal
5	#106×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3624	C₂²	1	2	yes	—
7	C₂×C₄×#3624	—	2	2	no	—
8	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
9	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
10	#11763675	—	2	2	no	—
11	#11763672	C₄	1	4	yes	complemented
12	#11763650	C₄	1	4	yes	complemented
13	C₂×#10587488	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₁₇	C₂²	1	2	yes	—
16	#11762571	C₄	1	2	yes	—
17	C₂×C₁₇×#106	C₄	1	1	yes	characteristic
18	C₂²×#3624	—	2	2	no	—
19	C₂²×#3624	—	2	1	no	—
20	C₂²×#3624	D₄	1	2	yes	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×Dic₁₇	—	2	2	no	—
25	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
26	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
27	C₄⋊#3624	—	2	2	no	—
28	C₄⋊#3624	C₂×C₄	1	4	yes	—
29	C₄×#3624	—	2	1	no	—
30	C₄×#3624	—	4	4	no	—
31	C₄×#3624	C₂×C₄	1	2	yes	—
32	C₄×#3624	C₂×C₄	1	2	yes	—
33	C₄×#3624	Q₈	1	2	yes	—
34	C₁₇×#106	C₈	1	4	yes	complemented
35	#10587502	—	2	2	no	—
36	#10587488	C₈	1	4	yes	complemented
37	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
38	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₇	—	2	2	no	—
40	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₇	—	2	2	no	—
42	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₇	#41	1	1	yes	characteristic
44	C₂×#3624	—	2	2	no	—
45	C₂×#3624	—	2	2	no	—
46	C₂×#3624	—	2	2	no	—
47	C₂×#3624	—	2	4	no	—
48	C₂×#3624	—	4	2	no	—
49	C₂×#3624	—	4	2	no	—
50	C₂×#3624	—	4	2	no	—
51	C₂×#3624	C₄⋊C₄	1	4	yes	—
52	C₂×C₈×C₁₇	C₄²	1	2	yes	—
53	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
55	C₄⋊C₄×C₁₇	—	2	2	no	—
56	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
57	C₄⋊Dic₁₇	—	2	2	no	—
58	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
59	C₄×Dic₁₇	—	4	2	no	—
60	C₄×Dic₁₇	—	4	2	no	—
61	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
62	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
64	C₄×Dic₁₇	#41	1	1	yes	characteristic
65	C₂×#3622	—	2	1	no	—
66	C₂×#3622	—	2	1	no	—
67	C₂×#3622	C₄²	1	2	yes	—
68	#8285	—	17	1	no	maximal, Sylow
69	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	—	2	4	no	—
71	C₂×C₄×C₁₇	—	4	2	no	—
72	C₂×C₄×C₁₇	—	4	2	no	—
73	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
79	C₂×Dic₁₇	—	2	2	no	—
80	C₂×Dic₁₇	—	2	4	no	—
81	C₂×Dic₁₇	—	4	2	no	—
82	C₂×Dic₁₇	—	4	2	no	—
83	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×Dic₁₇	#117	1	1	yes	characteristic
85	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
86	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
87	C₂×Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×Dic₁₇	#137	1	2	yes	—
89	#3624	—	2	4	no	—
90	#3624	—	4	2	no	—
91	#3624	—	4	4	no	—
92	C₈×C₁₇	C₄×C₈	1	4	yes	complemented
93	#3622	—	2	2	no	—
94	C₄×#106	—	17	1	no	—
95	C₈×C₄⋊C₄	—	17	2	no	—
96	C₂×#335	—	17	1	no	—
97	C₂×#322	—	17	1	no	—
98	#1279	—	17	2	no	—
99	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
101	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
102	C₂²×C₁₇	#440	1	2	yes	—
103	C₂²×C₁₇	#444	1	2	yes	—
104	C₄×C₁₇	—	4	4	no	—
105	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
106	C₄×C₁₇	#405	1	1	yes	characteristic
107	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
108	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
109	Dic₁₇	—	2	1	no	—
110	Dic₁₇	—	2	1	no	—
111	Dic₁₇	—	4	4	no	—
112	Dic₁₇	#428	1	2	yes	—
113	Dic₁₇	#441	1	1	yes	characteristic
114	Dic₁₇	#442	1	1	yes	characteristic
115	C₂×#106	—	17	1	no	—
116	C₂×#106	#595	1	1	yes	characteristic
117	C₂×C₄⋊C₈	—	17	2	no	—
118	C₂×C₄×C₈	—	17	1	no	—
119	C₂×C₄×C₈	—	17	1	no	—
120	C₂×C₄×C₈	—	17	1	no	—
121	C₂×C₄×C₈	—	34	2	no	—
122	C₄×C₄⋊C₄	—	17	2	no	—
123	#340	—	17	2	no	—
124	#336	—	34	2	no	—
125	#335	—	17	4	no	—
126	#322	—	17	4	no	—
127	C₂×C₁₇	SD₁₆⋊C₈	1	2	yes	—
128	C₂×C₁₇	SD₁₆⋊C₈	1	2	yes	—
129	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₁₇	#1773	1	1	yes	characteristic
131	C₂×C₁₇	#1776	1	1	yes	characteristic
132	C₂²×C₈	—	17	2	no	—
133	C₂²×C₈	—	17	1	no	—
134	C₂²×C₈	—	34	2	no	—
135	C₂²×C₈	—	34	1	no	—
136	C₂²×C₈	C₂×#595	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	17	2	no	—
138	C₂×C₄⋊C₄	C₂×#595	1	2	yes	—
139	C₂×C₄²	—	17	1	no	—
140	C₂×C₄²	—	34	2	no	—
141	#106	—	17	4	no	—
142	#106	#3624	1	4	yes	complemented
143	C₄⋊C₈	—	17	4	no	—
144	C₄⋊C₈	—	34	2	no	—
145	C₄×C₈	—	17	2	no	—
146	C₄×C₈	—	17	2	no	—
147	C₄×C₈	—	17	2	no	—
148	C₄×C₈	—	17	2	no	—
149	C₄×C₈	—	17	2	no	—
150	C₄×C₈	—	34	1	no	—
151	C₄×C₈	—	68	4	no	—
152	#95	—	34	2	no	—
153	C₁₇	#8285	1	1	yes	characteristic, complemented, Sylow
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	17	1	no	—
156	C₂²×C₄	—	17	1	no	—
157	C₂²×C₄	—	34	2	no	—
158	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
159	C₂×C₈	—	17	4	no	—
160	C₂×C₈	—	17	2	no	—
161	C₂×C₈	—	34	2	no	—
162	C₂×C₈	—	34	2	no	—
163	C₂×C₈	—	34	1	no	—
164	C₂×C₈	—	34	1	no	—
165	C₂×C₈	—	34	2	no	—
166	C₂×C₈	—	34	4	no	—
167	C₂×C₈	—	68	2	no	—
168	C₂×C₈	—	68	2	no	—
169	C₂×C₈	—	68	2	no	—
170	C₂×C₈	#63242	1	2	yes	—
171	C₂×C₈	C₄×#595	1	2	yes	—
172	C₂×C₈	C₂×#3624	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	17	4	no	—
175	C₄⋊C₄	—	34	2	no	—
176	C₄⋊C₄	C₂×#3624	1	4	yes	—
177	C₄²	—	17	1	no	—
178	C₄²	—	17	1	no	—
179	C₄²	—	17	1	no	—
180	C₄²	—	17	1	no	—
181	C₄²	—	68	2	no	—
182	C₄²	—	68	2	no	—
183	C₂³	D₄×#595	1	1	yes	characteristic
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	17	1	no	—
188	C₂×C₄	—	17	1	no	—
189	C₂×C₄	—	17	2	no	—
190	C₂×C₄	—	17	2	no	—
191	C₂×C₄	—	17	1	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	34	2	no	—
194	C₂×C₄	—	34	4	no	—
195	C₂×C₄	—	68	2	no	—
196	C₂×C₄	—	68	2	no	—
197	C₂×C₄	C₂×#63242	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
199	C₂×C₄	#10587678	1	1	yes	characteristic
200	C₂×C₄	#10587680	1	1	yes	characteristic
201	C₂×C₄	Q₈×#595	1	1	yes	characteristic
202	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
203	C₈	—	34	4	no	—
204	C₈	—	34	2	no	—
205	C₈	—	68	4	no	—
206	C₈	—	68	2	no	—
207	C₈	C₄×#3624	1	4	yes	complemented
208	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
209	C₂²	D₄×#3624	1	1	yes	characteristic
210	C₂²	#11763837	1	1	yes	characteristic
211	C₂²	#11763848	1	2	yes	—
212	C₂²	#11763854	1	2	yes	—
213	C₄	—	4	4	no	—
214	C₄	—	17	2	no	—
215	C₄	—	17	1	no	—
216	C₄	—	17	1	no	—
217	C₄	—	34	1	no	—
218	C₄	—	34	1	no	—
219	C₄	—	68	4	no	—
220	C₄	C₂×C₄×#3624	1	1	yes	characteristic
221	C₄	#11763774	1	1	yes	characteristic
222	C₄	Q₈×#3624	1	1	yes	characteristic
223	C₄	#11763858	1	1	yes	characteristic
224	C₂	SD₁₆⋊#3624	1	2	yes	—
225	C₂	#14544376	1	2	yes	—
226	C₂	C₄⋊C₄×#3624	1	1	yes	characteristic
227	C₂	#14544592	1	1	yes	characteristic
228	C₂	#14544600	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₁₇×#106	C₄
socle, prime radical	C₂³×C₁₇	C₄×D₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#106	#595
hypocenter	C₁₇	#8285
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
11. #11763672	C₄	semidirect
12. #11763650	C₄	semidirect
34. C₁₇×#106	C₈	semidirect
36. #10587488	C₈	semidirect
92. C₈×C₁₇	C₄×C₈	semidirect
142. #106	#3624	semidirect
153. C₁₇	#8285	semidirect
207. C₈	C₄×#3624	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×#3624	2
C₂×#11763672	2
C₂×#11763650	2
#14543620	2
#106×Dic₁₇	2
#8285	17