Subgroups — #17952356

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544580	C₂	1	2	yes	maximal
2	C₂×#11763670	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#3624	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×#10587542	—	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	#11763670	C₄	1	4	yes	complemented
12	C₈×#3624	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	1	no	—
19	C₂²×#3624	—	2	2	no	—
20	C₂²×#3624	D₄	1	1	yes	characteristic
21	C₂²×#3624	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₁₇	—	2	2	no	—
26	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
27	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
28	#10587542	—	4	4	no	—
29	C₄⋊#3624	—	2	2	no	—
30	C₄⋊#3624	C₂×C₄	1	4	yes	—
31	C₄×#3624	—	2	1	no	—
32	C₄×#3624	C₂×C₄	1	2	yes	—
33	C₄×#3624	C₂×C₄	1	2	yes	—
34	C₄×#3624	Q₈	1	2	yes	—
35	C₁₇×#106	—	2	2	no	—
36	#10587502	—	2	2	no	—
37	#10587488	—	2	2	no	—
38	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
39	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₁₇	—	2	2	no	—
41	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	—	2	2	no	—
43	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	#41	1	1	yes	characteristic
45	C₂×#3624	—	2	4	no	—
46	C₂×#3624	—	2	2	no	—
47	C₂×#3624	—	2	2	no	—
48	C₂×#3624	—	2	4	no	—
49	C₂×#3624	—	4	2	no	—
50	C₂×#3624	—	4	2	no	—
51	C₂×#3624	C₄⋊C₄	1	2	yes	—
52	C₂×#3624	C₄⋊C₄	1	2	yes	—
53	C₂×#3624	SD₁₆	1	4	yes	—
54	C₂×C₈×C₁₇	C₄²	1	2	yes	—
55	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
56	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
57	C₄⋊C₄×C₁₇	—	2	2	no	—
58	C₄⋊C₄×C₁₇	—	2	2	no	—
59	C₄⋊Dic₁₇	—	2	2	no	—
60	C₄⋊Dic₁₇	—	2	2	no	—
61	C₄×Dic₁₇	—	4	2	no	—
62	C₄×Dic₁₇	—	4	2	no	—
63	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₁₇	#41	1	1	yes	characteristic
67	C₂×#3622	—	2	1	no	—
68	C₂×#3622	—	2	1	no	—
69	C₂×#3622	C₄²	1	2	yes	—
70	#8283	—	17	1	no	maximal, Sylow
71	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	—	4	2	no	—
73	C₂×C₄×C₁₇	—	4	2	no	—
74	C₂×C₄×C₁₇	—	4	2	no	—
75	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
81	C₂×Dic₁₇	—	2	2	no	—
82	C₂×Dic₁₇	—	4	2	no	—
83	C₂×Dic₁₇	—	4	2	no	—
84	C₂×Dic₁₇	—	4	2	no	—
85	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×Dic₁₇	#117	1	1	yes	characteristic
87	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
88	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
89	C₂×Dic₁₇	C₂×SD₁₆	1	2	yes	—
90	C₂×Dic₁₇	#135	1	2	yes	—
91	#3624	—	2	4	no	—
92	#3624	—	2	2	no	—
93	#3624	—	4	4	no	—
94	#3624	#106	1	4	yes	complemented
95	C₈×C₁₇	#97	1	4	yes	complemented
96	#3622	—	2	2	no	—
97	C₄×#106	—	17	1	no	—
98	#1770	—	17	2	no	—
99	C₂×#334	—	17	1	no	—
100	C₂×C₈²	—	17	1	no	—
101	#1279	—	17	2	no	—
102	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₁₇	#436	1	1	yes	characteristic
104	C₂²×C₁₇	#436	1	1	yes	characteristic
105	C₂²×C₁₇	C₄×SD₁₆	1	2	yes	—
106	C₂²×C₁₇	#443	1	2	yes	—
107	C₄×C₁₇	—	8	2	no	—
108	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
109	C₄×C₁₇	#405	1	1	yes	characteristic
110	C₄×C₁₇	#446	1	1	yes	characteristic
111	C₄×C₁₇	#446	1	1	yes	characteristic
112	Dic₁₇	—	2	1	no	—
113	Dic₁₇	—	2	1	no	—
114	Dic₁₇	—	8	2	no	—
115	Dic₁₇	C₂×#106	1	1	yes	characteristic
116	Dic₁₇	#427	1	1	yes	characteristic
117	Dic₁₇	C₄×SD₁₆	1	2	yes	—
118	C₂×#106	—	17	1	no	—
119	C₂×#106	#595	1	1	yes	characteristic
120	C₂×C₄⋊C₈	—	17	2	no	—
121	C₂×#97	—	34	2	no	—
122	C₂×C₄×C₈	—	17	1	no	—
123	C₂×C₄×C₈	—	17	1	no	—
124	C₂×C₄×C₈	—	17	1	no	—
125	C₄×C₄⋊C₄	—	17	2	no	—
126	#340	—	17	2	no	—
127	#336	—	34	2	no	—
128	#334	—	17	4	no	—
129	C₈²	—	17	4	no	—
130	C₂×C₁₇	#1590	1	2	yes	—
131	C₂×C₁₇	#1590	1	2	yes	—
132	C₂×C₁₇	#1770	1	1	yes	characteristic
133	C₂×C₁₇	#1772	1	1	yes	characteristic
134	C₂×C₁₇	C₄×#106	1	1	yes	characteristic
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	17	1	no	—
137	C₂²×C₈	—	17	1	no	—
138	C₂²×C₈	—	34	1	no	—
139	C₂²×C₈	—	34	2	no	—
140	C₂²×C₈	C₂×#595	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	17	2	no	—
142	C₂×C₄⋊C₄	C₂×#595	1	2	yes	—
143	C₂×C₄²	—	17	1	no	—
144	C₂×C₄²	—	34	2	no	—
145	#106	—	2	2	no	—
146	#106	—	34	2	no	—
147	C₄⋊C₈	—	17	4	no	—
148	C₄⋊C₈	—	34	2	no	—
149	#97	—	68	4	no	—
150	C₄×C₈	—	17	2	no	—
151	C₄×C₈	—	17	2	no	—
152	C₄×C₈	—	17	2	no	—
153	C₄×C₈	—	17	2	no	—
154	C₄×C₈	—	17	2	no	—
155	C₄×C₈	—	34	1	no	—
156	#95	—	34	2	no	—
157	C₁₇	#8283	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	17	1	no	—
160	C₂²×C₄	—	17	1	no	—
161	C₂²×C₄	—	34	2	no	—
162	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	17	4	no	—
165	C₂×C₈	—	17	2	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	—	34	2	no	—
168	C₂×C₈	—	34	2	no	—
169	C₂×C₈	—	34	4	no	—
170	C₂×C₈	—	34	1	no	—
171	C₂×C₈	—	34	1	no	—
172	C₂×C₈	—	34	4	no	—
173	C₂×C₈	—	68	2	no	—
174	C₂×C₈	—	68	2	no	—
175	C₂×C₈	#63243	1	2	yes	—
176	C₂×C₈	C₄×#595	1	2	yes	—
177	C₂×C₈	#63247	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	34	2	no	—
181	C₄⋊C₄	—	34	2	no	—
182	C₄²	—	17	1	no	—
183	C₄²	—	17	1	no	—
184	C₄²	—	17	1	no	—
185	C₄²	—	17	1	no	—
186	C₄²	—	68	2	no	—
187	C₄²	—	68	2	no	—
188	C₂³	D₄×#595	1	1	yes	characteristic
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	2	no	—
195	C₂×C₄	—	17	2	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	17	1	no	—
198	C₂×C₄	—	34	2	no	—
199	C₂×C₄	—	68	2	no	—
200	C₂×C₄	—	68	2	no	—
201	C₂×C₄	—	68	2	no	—
202	C₂×C₄	C₂×#63243	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
204	C₂×C₄	#10587678	1	1	yes	characteristic
205	C₂×C₄	#10587680	1	1	yes	characteristic
206	C₂×C₄	Q₈×#595	1	1	yes	characteristic
207	C₂×C₄	C₂×#63247	1	1	yes	characteristic
208	C₈	—	17	4	no	—
209	C₈	—	34	2	no	—
210	C₈	—	34	4	no	—
211	C₈	—	34	2	no	—
212	C₈	—	68	4	no	—
213	C₈	#10587542	1	4	yes	complemented
214	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
215	C₂²	#11763843	1	1	yes	characteristic
216	C₂²	#11763844	1	1	yes	characteristic
217	C₂²	SD₁₆×#595	1	2	yes	—
218	C₂²	#11763852	1	2	yes	—
219	C₄	—	8	2	no	—
220	C₄	—	17	1	no	—
221	C₄	—	17	1	no	—
222	C₄	—	17	2	no	—
223	C₄	—	34	1	no	—
224	C₄	—	34	1	no	—
225	C₄	—	136	2	no	—
226	C₄	C₂×#10587542	1	1	yes	characteristic
227	C₄	#11763774	1	1	yes	characteristic
228	C₄	#11763859	1	1	yes	characteristic
229	C₄	#11763860	1	1	yes	characteristic
230	C₂	#14544419	1	2	yes	—
231	C₂	#14544420	1	2	yes	—
232	C₂	#14544580	1	1	yes	characteristic
233	C₂	#14544587	1	1	yes	characteristic
234	C₂	#106×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
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	#8283
LCS	G ▷ C₂×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. #11763670	C₄	semidirect
12. C₈×#3624	C₄	semidirect
94. #3624	#106	semidirect
95. C₈×C₁₇	#97	semidirect
157. C₁₇	#8283	semidirect
213. C₈	#10587542	semidirect

Maximal subgroups

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

subgroup	index
#14544580	2
C₂×#11763670	2
C₂×C₈×#3624	2
#14543620	2
#106×Dic₁₇	2
#8283	17