Subgroups — #17952352

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 258 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×#11763672	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#3624	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	#11763672	C₄	1	4	yes	complemented
12	C₈×#3624	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	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×#107	—	2	2	no	—
36	#10587502	—	2	2	no	—
37	#10587489	—	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	2	no	—
46	C₂×#3624	—	2	2	no	—
47	C₂×#3624	—	2	2	no	—
48	C₂×#3624	—	2	1	no	—
49	C₂×#3624	—	2	1	no	—
50	C₂×#3624	—	2	4	no	—
51	C₂×#3624	—	4	2	no	—
52	C₂×#3624	—	4	2	no	—
53	C₂×#3624	C₄⋊C₄	1	2	yes	—
54	C₂×#3624	C₄⋊C₄	1	2	yes	—
55	C₂×#3624	D₈	1	2	yes	—
56	C₂×#3624	Q₁₆	1	2	yes	—
57	C₂×C₈×C₁₇	C₄²	1	2	yes	—
58	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
59	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
60	C₄⋊C₄×C₁₇	—	2	2	no	—
61	C₄⋊C₄×C₁₇	—	2	2	no	—
62	C₄⋊Dic₁₇	—	2	2	no	—
63	C₄⋊Dic₁₇	—	2	2	no	—
64	C₄×Dic₁₇	—	4	2	no	—
65	C₄×Dic₁₇	—	4	2	no	—
66	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
69	C₄×Dic₁₇	#41	1	1	yes	characteristic
70	C₂×#3622	—	2	1	no	—
71	C₂×#3622	—	2	1	no	—
72	C₂×#3622	C₄²	1	2	yes	—
73	#8282	—	17	1	no	maximal, Sylow
74	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	—	4	2	no	—
76	C₂×C₄×C₁₇	—	4	2	no	—
77	C₂×C₄×C₁₇	—	4	2	no	—
78	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
84	C₂×Dic₁₇	—	2	1	no	—
85	C₂×Dic₁₇	—	2	1	no	—
86	C₂×Dic₁₇	—	4	2	no	—
87	C₂×Dic₁₇	—	4	2	no	—
88	C₂×Dic₁₇	—	4	2	no	—
89	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×Dic₁₇	#117	1	1	yes	characteristic
91	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
92	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
93	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
94	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×Dic₁₇	#135	1	1	yes	characteristic
96	C₂×Dic₁₇	#135	1	1	yes	characteristic
97	#3624	—	2	4	no	—
98	#3624	—	2	2	no	—
99	#3624	—	4	4	no	—
100	#3624	#107	1	4	yes	complemented
101	C₈×C₁₇	#97	1	4	yes	complemented
102	#3622	—	2	2	no	—
103	C₄×#107	—	17	1	no	—
104	#1770	—	17	2	no	—
105	C₂×#335	—	17	1	no	—
106	C₂×C₈²	—	17	1	no	—
107	#1279	—	17	2	no	—
108	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₁₇	#436	1	1	yes	characteristic
110	C₂²×C₁₇	#436	1	1	yes	characteristic
111	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
112	C₂²×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
113	C₂²×C₁₇	#443	1	1	yes	characteristic
114	C₂²×C₁₇	#443	1	1	yes	characteristic
115	C₄×C₁₇	—	8	2	no	—
116	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
117	C₄×C₁₇	#405	1	1	yes	characteristic
118	C₄×C₁₇	#446	1	1	yes	characteristic
119	C₄×C₁₇	#446	1	1	yes	characteristic
120	Dic₁₇	—	2	1	no	—
121	Dic₁₇	—	2	1	no	—
122	Dic₁₇	—	8	2	no	—
123	Dic₁₇	C₂×#107	1	1	yes	characteristic
124	Dic₁₇	#427	1	1	yes	characteristic
125	Dic₁₇	C₄×D₈	1	1	yes	characteristic
126	Dic₁₇	C₄×Q₁₆	1	1	yes	characteristic
127	C₂×#107	—	17	1	no	—
128	C₂×#107	#595	1	1	yes	characteristic
129	C₂×C₄⋊C₈	—	17	2	no	—
130	C₂×#97	—	34	2	no	—
131	C₂×C₄×C₈	—	17	1	no	—
132	C₂×C₄×C₈	—	17	1	no	—
133	C₂×C₄×C₈	—	17	1	no	—
134	C₄×C₄⋊C₄	—	17	2	no	—
135	#340	—	17	2	no	—
136	#336	—	34	2	no	—
137	#335	—	17	4	no	—
138	C₈²	—	17	4	no	—
139	C₂×C₁₇	#1589	1	1	yes	characteristic
140	C₂×C₁₇	#1589	1	1	yes	characteristic
141	C₂×C₁₇	#1591	1	1	yes	characteristic
142	C₂×C₁₇	#1591	1	1	yes	characteristic
143	C₂×C₁₇	#1770	1	1	yes	characteristic
144	C₂×C₁₇	#1772	1	1	yes	characteristic
145	C₂×C₁₇	C₄×#107	1	1	yes	characteristic
146	C₂²×C₈	—	17	1	no	—
147	C₂²×C₈	—	17	1	no	—
148	C₂²×C₈	—	17	1	no	—
149	C₂²×C₈	—	34	1	no	—
150	C₂²×C₈	—	34	2	no	—
151	C₂²×C₈	C₂×#595	1	1	yes	characteristic
152	C₂×C₄⋊C₄	—	17	2	no	—
153	C₂×C₄⋊C₄	C₂×#595	1	2	yes	—
154	C₂×C₄²	—	17	1	no	—
155	C₂×C₄²	—	34	2	no	—
156	#107	—	2	2	no	—
157	#107	—	34	2	no	—
158	C₄⋊C₈	—	17	4	no	—
159	C₄⋊C₈	—	34	2	no	—
160	#97	—	68	4	no	—
161	C₄×C₈	—	17	2	no	—
162	C₄×C₈	—	17	2	no	—
163	C₄×C₈	—	17	2	no	—
164	C₄×C₈	—	17	2	no	—
165	C₄×C₈	—	17	2	no	—
166	C₄×C₈	—	34	1	no	—
167	#95	—	34	2	no	—
168	C₁₇	#8282	1	1	yes	characteristic, complemented, Sylow
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	17	1	no	—
171	C₂²×C₄	—	17	1	no	—
172	C₂²×C₄	—	34	2	no	—
173	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
174	C₂×C₈	—	17	2	no	—
175	C₂×C₈	—	17	2	no	—
176	C₂×C₈	—	17	2	no	—
177	C₂×C₈	—	17	2	no	—
178	C₂×C₈	—	17	2	no	—
179	C₂×C₈	—	34	2	no	—
180	C₂×C₈	—	34	1	no	—
181	C₂×C₈	—	34	1	no	—
182	C₂×C₈	—	34	2	no	—
183	C₂×C₈	—	34	2	no	—
184	C₂×C₈	—	34	1	no	—
185	C₂×C₈	—	34	1	no	—
186	C₂×C₈	—	34	4	no	—
187	C₂×C₈	—	68	2	no	—
188	C₂×C₈	—	68	2	no	—
189	C₂×C₈	#63243	1	2	yes	—
190	C₂×C₈	C₄×#595	1	2	yes	—
191	C₂×C₈	#63247	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	34	2	no	—
195	C₄⋊C₄	—	34	2	no	—
196	C₄²	—	17	1	no	—
197	C₄²	—	17	1	no	—
198	C₄²	—	17	1	no	—
199	C₄²	—	17	1	no	—
200	C₄²	—	68	2	no	—
201	C₄²	—	68	2	no	—
202	C₂³	D₄×#595	1	1	yes	characteristic
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	17	1	no	—
207	C₂×C₄	—	17	1	no	—
208	C₂×C₄	—	17	1	no	—
209	C₂×C₄	—	17	1	no	—
210	C₂×C₄	—	17	1	no	—
211	C₂×C₄	—	17	1	no	—
212	C₂×C₄	—	17	1	no	—
213	C₂×C₄	—	17	1	no	—
214	C₂×C₄	—	34	1	no	—
215	C₂×C₄	—	34	1	no	—
216	C₂×C₄	—	68	2	no	—
217	C₂×C₄	—	68	2	no	—
218	C₂×C₄	—	68	2	no	—
219	C₂×C₄	C₂×#63243	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
221	C₂×C₄	#10587678	1	1	yes	characteristic
222	C₂×C₄	#10587680	1	1	yes	characteristic
223	C₂×C₄	Q₈×#595	1	1	yes	characteristic
224	C₂×C₄	C₂×#63247	1	1	yes	characteristic
225	C₈	—	17	4	no	—
226	C₈	—	34	2	no	—
227	C₈	—	34	4	no	—
228	C₈	—	34	2	no	—
229	C₈	—	68	4	no	—
230	C₈	#10587542	1	4	yes	complemented
231	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
232	C₂²	#11763843	1	1	yes	characteristic
233	C₂²	#11763844	1	1	yes	characteristic
234	C₂²	D₈×#595	1	1	yes	characteristic
235	C₂²	Q₁₆×#595	1	1	yes	characteristic
236	C₂²	#11763851	1	1	yes	characteristic
237	C₂²	#11763853	1	1	yes	characteristic
238	C₄	—	8	2	no	—
239	C₄	—	17	1	no	—
240	C₄	—	17	1	no	—
241	C₄	—	17	1	no	—
242	C₄	—	17	1	no	—
243	C₄	—	34	1	no	—
244	C₄	—	34	1	no	—
245	C₄	—	136	2	no	—
246	C₄	C₂×#10587542	1	1	yes	characteristic
247	C₄	#11763774	1	1	yes	characteristic
248	C₄	#11763859	1	1	yes	characteristic
249	C₄	#11763860	1	1	yes	characteristic
250	C₂	#14544416	1	1	yes	characteristic
251	C₂	#14544417	1	1	yes	characteristic
252	C₂	#14544423	1	1	yes	characteristic
253	C₂	#14544424	1	1	yes	characteristic
254	C₂	#14544580	1	1	yes	characteristic
255	C₂	#14544588	1	1	yes	characteristic
256	C₂	#107×#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×#107	C₄
socle, prime radical	C₂³×C₁₇	C₄×D₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#107	#595
hypocenter	C₁₇	#8282
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. #11763672	C₄	semidirect
12. C₈×#3624	C₄	semidirect
100. #3624	#107	semidirect
101. C₈×C₁₇	#97	semidirect
168. C₁₇	#8282	semidirect
230. 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×#11763672	2
C₂×C₈×#3624	2
#14543620	2
#107×Dic₁₇	2
#8282	17