Subgroups — #16944417

Order 4352.

334 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	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
2	#14527543	C₂	1	1	yes	maximal, characteristic
3	#14526748	C₂	1	4	yes	maximal, complemented
4	C₁₇×#3170	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	—	2	1	no	—
6	#11762833	—	2	1	no	—
7	C₄×#63214	—	2	1	no	—
8	C₄×#63214	C₂²	1	2	yes	—
9	#11762562	—	2	1	no	—
10	#11762562	C₂²	1	2	yes	—
11	C₁₇×#533	—	2	1	no	—
12	C₁₇×#500	C₄	1	1	yes	characteristic
13	C₁₇×#494	C₄	1	1	yes	characteristic
14	C₁₇×#487	—	2	1	no	—
15	C₁₇×#467	—	2	1	no	—
16	C₁₇×#467	C₄	1	2	yes	—
17	C₄×Q₁₆×C₁₇	—	2	1	no	—
18	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₁₇	—	2	1	no	—
21	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₁₇×C₄⋊Q₈	—	2	1	no	—
25	C₁₇×C₄⋊Q₈	—	2	1	no	—
26	C₁₇×#125	—	4	1	no	—
27	C₁₇×C₂²⋊Q₈	—	2	2	no	—
28	C₁₇×C₂²⋊Q₈	—	4	1	no	—
29	C₄×Q₈×C₁₇	—	2	2	no	—
30	C₄×Q₈×C₁₇	—	4	1	no	—
31	C₁₇×#117	—	4	1	no	—
32	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
33	C₂×#63223	—	4	1	no	—
34	C₁₇×#107	—	2	1	no	—
35	C₁₇×#107	C₂×C₄	1	2	yes	—
36	C₁₇×#103	—	2	2	no	—
37	C₁₇×#103	—	2	2	no	—
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
42	C₂×#63214	—	2	4	no	—
43	C₂×#63214	—	2	2	no	—
44	C₄⋊#3622	—	2	2	no	—
45	C₄⋊#3622	—	4	1	no	—
46	C₂×Q₈×C₁₇	—	2	2	no	—
47	C₂×Q₈×C₁₇	—	4	1	no	—
48	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₁₇	—	2	4	no	—
52	Q₁₆×C₁₇	—	4	2	no	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₁₇	—	2	2	no	—
60	C₄⋊C₄×C₁₇	—	4	2	no	—
61	C₄⋊C₄×C₁₇	—	4	1	no	—
62	C₄⋊C₄×C₁₇	—	4	1	no	—
63	C₄⋊C₄×C₁₇	—	4	1	no	—
64	C₄⋊C₄×C₁₇	—	4	2	no	—
65	C₂²⋊C₄×C₁₇	—	4	2	no	—
66	C₄²×C₁₇	—	4	2	no	—
67	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₇	D₈	1	1	yes	characteristic
69	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
70	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
71	#63223	—	8	2	no	—
72	C₂×#3622	—	4	2	no	—
73	#63214	—	2	4	no	—
74	#11672	—	17	1	no	maximal, Sylow
75	C₂³×C₁₇	#102	1	1	yes	characteristic
76	Q₈×C₁₇	—	4	2	no	—
77	Q₈×C₁₇	—	4	2	no	—
78	Q₈×C₁₇	—	8	2	no	—
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	2	1	no	—
81	C₂×C₄×C₁₇	—	2	1	no	—
82	C₂×C₄×C₁₇	—	4	2	no	—
83	C₂×C₄×C₁₇	—	4	2	no	—
84	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
88	C₂×C₄×C₁₇	Q₃₂	1	2	yes	—
89	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₁₇	—	2	1	no	—
93	C₈×C₁₇	—	2	1	no	—
94	C₈×C₁₇	#104	1	2	yes	—
95	C₈×C₁₇	#131	1	2	yes	—
96	#3622	—	8	2	no	—
97	#3170	D₁₇	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	17	1	no	—
99	#1567	—	17	1	no	—
100	#1332	—	17	4	no	—
101	C₂²×C₁₇	—	2	1	no	—
102	C₂²×C₁₇	—	2	1	no	—
103	C₂²×C₁₇	#358	1	2	yes	—
104	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
105	C₄×C₁₇	—	2	1	no	—
106	C₄×C₁₇	—	2	1	no	—
107	C₄×C₁₇	—	8	2	no	—
108	C₄×C₁₇	—	8	2	no	—
109	C₄×C₁₇	#358	1	2	yes	—
110	C₄×C₁₇	#359	1	2	yes	—
111	C₄×C₁₇	#408	1	1	yes	characteristic
112	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
113	C₄×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
114	C₄×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
115	#533	—	2	1	no	—
116	C₂²×C₁₆	—	34	1	no	—
117	#500	Dic₁₇	1	1	yes	characteristic
118	#494	Dic₁₇	1	1	yes	characteristic
119	#487	—	2	1	no	—
120	#467	—	2	1	no	—
121	#467	Dic₁₇	1	2	yes	—
122	C₄×Q₁₆	—	2	1	no	—
123	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
124	#423	—	34	1	no	—
125	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
126	C₄×C₁₆	—	17	2	no	—
127	C₄×C₁₆	—	34	1	no	—
128	#335	—	17	2	no	—
129	#335	—	34	1	no	—
130	C₂×C₁₇	—	2	2	no	—
131	C₂×C₁₇	#1492	1	1	yes	characteristic
132	C₂×C₁₇	C₂×#358	1	1	yes	characteristic
133	C₂×C₁₇	C₂×#359	1	1	yes	characteristic
134	C₂×Q₁₆	—	2	1	no	—
135	C₂×Q₁₆	C₂×Dic₁₇	1	2	yes	—
136	C₂×M₁₆	—	68	1	no	—
137	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	—	2	1	no	—
141	#125	—	4	1	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	—	4	1	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	4	1	no	—
146	#117	—	4	1	no	—
147	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
148	C₂×C₁₆	—	34	4	no	—
149	C₂×C₁₆	—	34	2	no	—
150	#107	—	2	1	no	—
151	#107	C₂×Dic₁₇	1	2	yes	—
152	C₄⋊C₈	—	34	2	no	—
153	C₄⋊C₈	—	68	1	no	—
154	#103	—	2	2	no	—
155	#103	—	2	2	no	—
156	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
160	C₁₇	#11672	1	1	yes	characteristic, complemented, Sylow
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	#63228	1	1	yes	characteristic
164	C₂²×C₄	#63229	1	1	yes	characteristic
165	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	2	no	—
168	M₁₆	—	136	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	68	2	no	—
172	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
174	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	#63228	1	1	yes	characteristic
185	C₄²	#63229	1	1	yes	characteristic
186	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
188	C₁₆	—	34	4	no	—
189	C₂³	#10587503	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#10587499	1	2	yes	—
199	C₂×C₄	#10587500	1	2	yes	—
200	C₂×C₄	#10587503	1	1	yes	characteristic
201	C₂×C₄	#10587503	1	1	yes	characteristic
202	C₂×C₄	#10587503	1	1	yes	characteristic
203	C₂×C₄	C₂×#63228	1	1	yes	characteristic
204	C₂×C₄	C₂×#63229	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	136	2	no	—
209	C₈	#10587508	1	2	yes	—
210	C₈	#10587634	1	2	yes	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#11762613	1	2	yes	—
214	C₂²	C₂×#10587503	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	#11762613	1	2	yes	—
220	C₄	#11762616	1	2	yes	—
221	C₄	#11762776	1	1	yes	characteristic
222	C₄	C₂×#10587508	1	1	yes	characteristic
223	C₄	C₂×#10587499	1	1	yes	characteristic
224	C₄	C₂×#10587500	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	#14527235	1	1	yes	characteristic
227	C₂	C₂×#11762613	1	1	yes	characteristic
228	C₂	C₂×#11762616	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#3170	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	#102
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#3170	D₁₇
hypocenter	C₁₇	#11672
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #14526748	C₂	semidirect
160. C₁₇	#11672	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#63214	2
#14527543	2
#14526748	2
C₁₇×#3170	2
#11672	17