Subgroups — #17952372

Order 4352.

398 conjugacy classes of subgroups.

The subgroup lattice has 240 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×#11763650	C₂	1	1	yes	maximal, characteristic
4	#14543620	C₂	1	2	yes	maximal
5	#107×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	#11763650	C₄	1	4	yes	complemented
13	C₂×#10587489	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×#107	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	#10587542	—	4	4	no	—
28	C₄⋊#3624	—	2	2	no	—
29	C₄⋊#3624	C₂×C₄	1	4	yes	—
30	C₄×#3624	—	2	1	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×#107	—	2	2	no	—
35	#10587502	—	2	2	no	—
36	#10587489	—	2	2	no	—
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	1	no	—
49	C₂×#3624	—	4	1	no	—
50	C₂×#3624	—	4	2	no	—
51	C₂×#3624	—	4	2	no	—
52	C₂×#3624	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₁₇	C₄²	1	2	yes	—
54	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
55	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
56	C₄⋊C₄×C₁₇	—	2	2	no	—
57	C₄⋊C₄×C₁₇	—	2	2	no	—
58	C₄⋊Dic₁₇	—	2	2	no	—
59	C₄⋊Dic₁₇	—	2	2	no	—
60	C₄×Dic₁₇	—	4	2	no	—
61	C₄×Dic₁₇	—	4	2	no	—
62	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₁₇	#41	1	1	yes	characteristic
66	C₂×#3622	—	2	1	no	—
67	C₂×#3622	—	2	1	no	—
68	C₂×#3622	C₄²	1	2	yes	—
69	#8287	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	—	4	2	no	—
72	C₂×C₄×C₁₇	—	4	2	no	—
73	C₂×C₄×C₁₇	—	4	2	no	—
74	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
80	C₂×Dic₁₇	—	2	1	no	—
81	C₂×Dic₁₇	—	2	1	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	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×Dic₁₇	#137	1	2	yes	—
91	#3624	—	2	4	no	—
92	#3624	—	4	2	no	—
93	#3624	—	4	4	no	—
94	C₈×C₁₇	#97	1	4	yes	complemented
95	#3622	—	2	2	no	—
96	C₄×#107	—	17	1	no	—
97	#1770	—	17	2	no	—
98	C₂×#334	—	17	1	no	—
99	C₂×#322	—	17	1	no	—
100	#1279	—	17	2	no	—
101	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₁₇	#436	1	1	yes	characteristic
103	C₂²×C₁₇	#436	1	1	yes	characteristic
104	C₂²×C₁₇	#441	1	1	yes	characteristic
105	C₂²×C₁₇	#442	1	1	yes	characteristic
106	C₂²×C₁₇	#444	1	1	yes	characteristic
107	C₂²×C₁₇	#444	1	1	yes	characteristic
108	C₄×C₁₇	—	8	2	no	—
109	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
110	C₄×C₁₇	#405	1	1	yes	characteristic
111	C₄×C₁₇	#446	1	1	yes	characteristic
112	C₄×C₁₇	#446	1	1	yes	characteristic
113	Dic₁₇	—	2	1	no	—
114	Dic₁₇	—	2	1	no	—
115	Dic₁₇	—	8	2	no	—
116	Dic₁₇	#428	1	2	yes	—
117	Dic₁₇	#440	1	2	yes	—
118	C₂×#107	—	17	1	no	—
119	C₂×#107	#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	#322	—	17	4	no	—
130	C₂×C₁₇	#1593	1	1	yes	characteristic
131	C₂×C₁₇	#1593	1	1	yes	characteristic
132	C₂×C₁₇	#1594	1	1	yes	characteristic
133	C₂×C₁₇	#1594	1	1	yes	characteristic
134	C₂×C₁₇	#1770	1	1	yes	characteristic
135	C₂×C₁₇	#1773	1	1	yes	characteristic
136	C₂×C₁₇	#1776	1	1	yes	characteristic
137	C₂²×C₈	—	17	2	no	—
138	C₂²×C₈	—	17	1	no	—
139	C₂²×C₈	—	34	2	no	—
140	C₂²×C₈	—	34	1	no	—
141	C₂²×C₈	C₂×#595	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	17	2	no	—
143	C₂×C₄⋊C₄	C₂×#595	1	2	yes	—
144	C₂×C₄²	—	17	1	no	—
145	C₂×C₄²	—	34	2	no	—
146	#107	—	2	2	no	—
147	#107	—	34	2	no	—
148	C₄⋊C₈	—	17	4	no	—
149	C₄⋊C₈	—	34	2	no	—
150	#97	—	68	4	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	—	17	2	no	—
156	C₄×C₈	—	34	1	no	—
157	#95	—	34	2	no	—
158	C₁₇	#8287	1	1	yes	characteristic, complemented, Sylow
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	17	1	no	—
161	C₂²×C₄	—	17	1	no	—
162	C₂²×C₄	—	34	2	no	—
163	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
164	C₂×C₈	—	17	4	no	—
165	C₂×C₈	—	17	2	no	—
166	C₂×C₈	—	34	2	no	—
167	C₂×C₈	—	34	2	no	—
168	C₂×C₈	—	34	1	no	—
169	C₂×C₈	—	34	1	no	—
170	C₂×C₈	—	34	2	no	—
171	C₂×C₈	—	34	4	no	—
172	C₂×C₈	—	68	2	no	—
173	C₂×C₈	—	68	2	no	—
174	C₂×C₈	—	68	1	no	—
175	C₂×C₈	—	68	1	no	—
176	C₂×C₈	#63243	1	2	yes	—
177	C₂×C₈	C₄×#595	1	2	yes	—
178	C₂×C₈	#63247	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	34	2	no	—
182	C₄⋊C₄	—	34	2	no	—
183	C₄²	—	17	1	no	—
184	C₄²	—	17	1	no	—
185	C₄²	—	17	1	no	—
186	C₄²	—	17	1	no	—
187	C₄²	—	68	2	no	—
188	C₄²	—	68	2	no	—
189	C₂³	D₄×#595	1	1	yes	characteristic
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	17	2	no	—
196	C₂×C₄	—	17	2	no	—
197	C₂×C₄	—	17	1	no	—
198	C₂×C₄	—	17	1	no	—
199	C₂×C₄	—	34	1	no	—
200	C₂×C₄	—	34	1	no	—
201	C₂×C₄	—	68	2	no	—
202	C₂×C₄	—	68	2	no	—
203	C₂×C₄	—	68	2	no	—
204	C₂×C₄	C₂×#63243	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
206	C₂×C₄	#10587678	1	1	yes	characteristic
207	C₂×C₄	#10587680	1	1	yes	characteristic
208	C₂×C₄	Q₈×#595	1	1	yes	characteristic
209	C₂×C₄	C₂×#63247	1	1	yes	characteristic
210	C₈	—	34	4	no	—
211	C₈	—	34	2	no	—
212	C₈	—	68	4	no	—
213	C₈	—	68	2	no	—
214	C₈	#10587542	1	4	yes	complemented
215	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
216	C₂²	#11763843	1	1	yes	characteristic
217	C₂²	#11763844	1	1	yes	characteristic
218	C₂²	#11763849	1	1	yes	characteristic
219	C₂²	#11763850	1	1	yes	characteristic
220	C₂²	#11763855	1	1	yes	characteristic
221	C₂²	#11763856	1	1	yes	characteristic
222	C₄	—	8	2	no	—
223	C₄	—	17	2	no	—
224	C₄	—	17	2	no	—
225	C₄	—	34	1	no	—
226	C₄	—	34	1	no	—
227	C₄	—	136	2	no	—
228	C₄	C₂×#10587542	1	1	yes	characteristic
229	C₄	#11763774	1	1	yes	characteristic
230	C₄	#11763859	1	1	yes	characteristic
231	C₄	#11763860	1	1	yes	characteristic
232	C₂	#14544430	1	1	yes	characteristic
233	C₂	#14544431	1	1	yes	characteristic
234	C₂	#14544433	1	1	yes	characteristic
235	C₂	#14544434	1	1	yes	characteristic
236	C₂	#14544580	1	1	yes	characteristic
237	C₂	#14544591	1	1	yes	characteristic
238	C₂	#14544599	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×#107	C₄
socle, prime radical	C₂³×C₁₇	C₄×D₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#107	#595
hypocenter	C₁₇	#8287
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
11. #11763670	C₄	semidirect
12. #11763650	C₄	semidirect
94. C₈×C₁₇	#97	semidirect
158. C₁₇	#8287	semidirect
214. 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×#11763650	2
#14543620	2
#107×Dic₁₇	2
#8287	17