Subgroups — #16932960

Order 4352.

382 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	#14529547	C₂	1	2	yes	maximal
2	#14528941	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762560	C₂	1	1	yes	maximal, characteristic
4	#14526668	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587488	—	2	1	no	—
7	C₂×#10587488	—	2	1	no	—
8	C₂×#10587479	—	2	2	no	—
9	C₂×C₄⋊#3622	C₂²	1	2	yes	—
10	#11762679	C₂²	1	2	yes	—
11	#11762571	—	2	2	no	—
12	#11762560	C₄	1	4	yes	complemented
13	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₇	—	2	1	no	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#336	—	2	1	no	—
18	C₂²×C₈×C₁₇	—	2	1	no	—
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₇	D₄	1	2	yes	—
22	C₂×C₄²×C₁₇	—	2	1	no	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×C₄⋊Dic₁₇	—	2	2	no	—
26	C₂×C₄⋊Dic₁₇	—	2	2	no	—
27	C₂²×#3622	—	2	2	no	—
28	C₁₇×C₄⋊C₈	—	2	1	no	—
29	C₁₇×C₄⋊C₈	Q₈	1	2	yes	—
30	C₄×C₈×C₁₇	—	4	2	no	—
31	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₇	Q₈	1	2	yes	—
33	C₁₇×#95	—	2	1	no	—
34	#10587488	—	4	2	no	—
35	#10587488	—	4	2	no	—
36	#10587479	—	4	4	no	—
37	C₄⋊#3622	—	2	2	no	—
38	C₄⋊#3622	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₁₇	—	2	2	no	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂²×Dic₁₇	—	4	2	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	2	1	no	—
46	C₂×C₈×C₁₇	—	2	2	no	—
47	C₂×C₈×C₁₇	—	4	1	no	—
48	C₂×C₈×C₁₇	—	4	1	no	—
49	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
52	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
53	C₄⋊C₄×C₁₇	—	2	2	no	—
54	C₄⋊C₄×C₁₇	SD₁₆	1	4	yes	—
55	C₄²×C₁₇	—	4	1	no	—
56	C₄²×C₁₇	—	4	1	no	—
57	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₁₇	#41	1	1	yes	characteristic
61	C₄⋊Dic₁₇	—	4	2	no	—
62	C₄⋊Dic₁₇	—	4	2	no	—
63	C₄⋊Dic₁₇	—	4	2	no	—
64	C₄⋊Dic₁₇	—	4	2	no	—
65	C₄⋊Dic₁₇	—	8	2	no	—
66	C₂×#3622	—	2	4	no	—
67	C₂×#3622	—	4	2	no	—
68	C₂×#3622	—	4	2	no	—
69	#9612	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	#121	1	1	yes	characteristic
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	2	4	no	—
73	C₂×C₄×C₁₇	—	4	2	no	—
74	C₂×C₄×C₁₇	—	4	2	no	—
75	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#125	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#135	1	2	yes	—
85	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
87	C₂×Dic₁₇	—	8	2	no	—
88	C₂×Dic₁₇	—	8	2	no	—
89	C₂×Dic₁₇	—	8	2	no	—
90	C₈×C₁₇	—	2	2	no	—
91	C₈×C₁₇	—	4	2	no	—
92	C₈×C₁₇	#106	1	4	yes	complemented
93	#3622	—	4	4	no	—
94	#1948	—	17	2	no	—
95	#1844	—	17	1	no	—
96	C₈×C₄⋊C₄	D₁₇	1	1	yes	characteristic
97	C₂×#334	—	17	1	no	—
98	#1296	—	17	2	no	—
99	C₂²×C₁₇	#389	1	1	yes	characteristic
100	C₂²×C₁₇	#460	1	1	yes	characteristic
101	C₂²×C₁₇	#461	1	1	yes	characteristic
102	C₂²×C₁₇	#465	1	2	yes	—
103	C₂²×C₁₇	#471	1	2	yes	—
104	C₄×C₁₇	—	2	1	no	—
105	C₄×C₁₇	—	2	1	no	—
106	C₄×C₁₇	—	4	4	no	—
107	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
108	C₄×C₁₇	#428	1	1	yes	characteristic
109	C₄×C₁₇	C₄×SD₁₆	1	2	yes	—
110	C₄×C₁₇	#475	1	1	yes	characteristic
111	C₄×C₁₇	#476	1	1	yes	characteristic
112	C₄×C₁₇	#498	1	1	yes	characteristic
113	C₄×C₁₇	#499	1	1	yes	characteristic
114	Dic₁₇	—	16	2	no	—
115	C₂×#106	—	34	1	no	—
116	C₂×#106	—	34	1	no	—
117	C₂×#106	—	34	2	no	—
118	C₂×C₄⋊C₈	—	17	2	no	—
119	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
120	C₂×C₄×C₈	—	2	1	no	—
121	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
122	#384	—	17	2	no	—
123	C₄×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
124	#340	—	34	2	no	—
125	#336	—	2	1	no	—
126	#334	—	17	4	no	—
127	C₂×C₁₇	#1666	1	2	yes	—
128	C₂×C₁₇	#1671	1	2	yes	—
129	C₂×C₁₇	#1941	1	1	yes	characteristic
130	C₂×C₁₇	#1948	1	1	yes	characteristic
131	C₂×C₁₇	#1951	1	1	yes	characteristic
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	—	34	2	no	—
134	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	34	2	no	—
137	C₂×C₄⋊C₄	—	34	2	no	—
138	C₂×C₄⋊C₄	—	34	2	no	—
139	C₂×C₄⋊C₄	D₆₈	1	2	yes	—
140	C₂×C₄²	—	2	1	no	—
141	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
142	#106	—	68	2	no	—
143	#106	—	68	2	no	—
144	#106	—	68	4	no	—
145	C₄⋊C₈	—	2	1	no	—
146	C₄⋊C₈	—	17	4	no	—
147	C₄⋊C₈	—	34	2	no	—
148	C₄⋊C₈	Dic₃₄	1	2	yes	—
149	C₄×C₈	—	4	2	no	—
150	C₄×C₈	Dic₃₄	1	2	yes	—
151	C₄×C₈	C₄×D₁₇	1	2	yes	—
152	#95	—	2	1	no	—
153	C₁₇	#9612	1	1	yes	characteristic, complemented, Sylow
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	68	2	no	—
156	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
157	C₂²×C₄	#63252	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	34	4	no	—
165	C₂×C₈	—	68	2	no	—
166	C₂×C₈	—	68	2	no	—
167	C₂×C₈	#63225	1	2	yes	—
168	C₂×C₈	#63225	1	2	yes	—
169	C₂×C₈	#63229	1	2	yes	—
170	C₂×C₈	#63230	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	68	2	no	—
173	C₄⋊C₄	—	68	2	no	—
174	C₄⋊C₄	—	68	2	no	—
175	C₄⋊C₄	—	68	2	no	—
176	C₄⋊C₄	—	136	2	no	—
177	C₄⋊C₄	#63219	1	4	yes	—
178	C₄²	—	4	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
181	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
182	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
183	C₄²	#63252	1	1	yes	characteristic
184	C₂³	#10587616	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	136	2	no	—
190	C₂×C₄	—	136	2	no	—
191	C₂×C₄	—	136	2	no	—
192	C₂×C₄	C₄×Dic₃₄	1	1	yes	characteristic
193	C₂×C₄	#10587555	1	1	yes	characteristic
194	C₂×C₄	#10587556	1	1	yes	characteristic
195	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
196	C₂×C₄	C₂×#63219	1	2	yes	—
197	C₂×C₄	#10587590	1	2	yes	—
198	C₂×C₄	C₂×#63225	1	1	yes	characteristic
199	C₂×C₄	#10587610	1	1	yes	characteristic
200	C₂×C₄	C₂×#63229	1	1	yes	characteristic
201	C₂×C₄	C₂×#63230	1	1	yes	characteristic
202	C₂×C₄	#10587635	1	1	yes	characteristic
203	C₂×C₄	#10587637	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	4	2	no	—
206	C₈	—	68	4	no	—
207	C₈	#10587479	1	4	yes	complemented
208	C₂²	#11762707	1	1	yes	characteristic
209	C₂²	#11763009	1	1	yes	characteristic
210	C₂²	#11763014	1	1	yes	characteristic
211	C₂²	#11763038	1	2	yes	—
212	C₂²	#11763072	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	4	no	—
216	C₄	—	272	2	no	—
217	C₄	C₂×#10587479	1	1	yes	characteristic
218	C₄	#11762852	1	1	yes	characteristic
219	C₄	C₄×#63219	1	2	yes	—
220	C₄	#11763094	1	1	yes	characteristic
221	C₄	#11763099	1	1	yes	characteristic
222	C₄	#11763207	1	1	yes	characteristic
223	C₄	#11763211	1	1	yes	characteristic
224	C₂	#14528132	1	2	yes	—
225	C₂	#14528159	1	2	yes	—
226	C₂	#14529512	1	1	yes	characteristic
227	C₂	#14529547	1	1	yes	characteristic
228	C₂	#14529564	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587616
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₈×C₄⋊C₄×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	#121
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₁₇
hypocenter	C₁₇	#9612
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. #11762560	C₄	semidirect
92. C₈×C₁₇	#106	semidirect
153. C₁₇	#9612	semidirect
207. C₈	#10587479	semidirect

Maximal subgroups

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

subgroup	index
#14529547	2
#14528941	2
C₂×#11762560	2
#14526668	2
C₈×C₄⋊C₄×C₁₇	2
#9612	17