Subgroups — #16945811

Order 4352.

314 conjugacy classes of subgroups.

The subgroup lattice has 238 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	#14526842	C₂	1	2	yes	maximal, complemented
4	#14526833	C₂	1	2	yes	maximal, complemented
5	C₁₇×#3170	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63214	—	2	1	no	—
7	#11762833	—	2	1	no	—
8	C₄⋊#63214	—	2	1	no	—
9	C₄⋊#63214	C₂²	1	2	yes	—
10	#11762562	—	2	1	no	—
11	#11762562	C₂²	1	2	yes	—
12	C₁₇×#533	—	2	1	no	—
13	C₁₇×#500	C₂²	1	1	yes	characteristic
14	C₁₇×#494	C₂²	1	1	yes	characteristic
15	C₁₇×#487	—	2	1	no	—
16	C₁₇×#467	—	2	1	no	—
17	C₁₇×#467	C₄	1	2	yes	—
18	C₄×Q₁₆×C₁₇	—	2	1	no	—
19	C₄×Q₁₆×C₁₇	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆×C₁₇	—	2	1	no	—
22	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₁₇×C₄⋊Q₈	—	2	1	no	—
26	C₁₇×C₄⋊Q₈	—	2	1	no	—
27	C₁₇×#125	—	4	1	no	—
28	C₁₇×C₂²⋊Q₈	—	2	2	no	—
29	C₁₇×C₂²⋊Q₈	—	4	1	no	—
30	C₄×Q₈×C₁₇	—	2	2	no	—
31	C₄×Q₈×C₁₇	—	4	1	no	—
32	C₁₇×#117	—	4	1	no	—
33	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
34	C₂×#63223	—	4	1	no	—
35	C₁₇×#107	—	2	1	no	—
36	C₁₇×#107	C₂×C₄	1	2	yes	—
37	C₁₇×#103	—	2	2	no	—
38	C₁₇×#103	—	2	2	no	—
39	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
43	C₂×#63214	—	2	2	no	—
44	C₂×#63214	—	4	2	no	—
45	C₄⋊#3622	—	2	2	no	—
46	C₄⋊#3622	—	4	1	no	—
47	C₂×Q₈×C₁₇	—	2	2	no	—
48	C₂×Q₈×C₁₇	—	4	1	no	—
49	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
52	Q₁₆×C₁₇	—	4	2	no	—
53	Q₁₆×C₁₇	—	4	1	no	—
54	Q₁₆×C₁₇	—	4	1	no	—
55	C₂×C₈×C₁₇	—	2	1	no	—
56	C₂×C₈×C₁₇	—	2	1	no	—
57	C₂×C₈×C₁₇	—	4	1	no	—
58	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₁₇	—	2	2	no	—
63	C₄⋊C₄×C₁₇	—	4	2	no	—
64	C₄⋊C₄×C₁₇	—	4	2	no	—
65	C₄⋊C₄×C₁₇	—	4	1	no	—
66	C₄⋊C₄×C₁₇	—	4	1	no	—
67	C₄⋊C₄×C₁₇	—	4	1	no	—
68	C₂²⋊C₄×C₁₇	—	4	2	no	—
69	C₄²×C₁₇	—	4	2	no	—
70	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₁₇	D₈	1	1	yes	characteristic
72	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
73	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
74	#63223	—	8	2	no	—
75	C₂×#3622	—	4	2	no	—
76	#63214	—	4	2	no	—
77	#11983	—	17	1	no	maximal, Sylow
78	C₂³×C₁₇	#102	1	1	yes	characteristic
79	Q₈×C₁₇	—	4	2	no	—
80	Q₈×C₁₇	—	4	2	no	—
81	Q₈×C₁₇	—	8	2	no	—
82	C₂×C₄×C₁₇	—	2	1	no	—
83	C₂×C₄×C₁₇	—	2	1	no	—
84	C₂×C₄×C₁₇	—	4	1	no	—
85	C₂×C₄×C₁₇	—	4	2	no	—
86	C₂×C₄×C₁₇	—	4	2	no	—
87	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
89	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
91	C₂×C₄×C₁₇	Q₃₂	1	2	yes	—
92	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₁₇	—	2	1	no	—
96	C₈×C₁₇	—	2	1	no	—
97	C₈×C₁₇	—	2	1	no	—
98	C₈×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₈×C₁₇	#137	1	1	yes	characteristic
100	#3622	—	8	2	no	—
101	#3170	D₁₇	1	1	yes	characteristic
102	C₂×C₄⋊C₁₆	—	17	1	no	—
103	#1567	—	17	1	no	—
104	#1365	—	17	2	no	—
105	#1362	—	17	2	no	—
106	C₂²×C₁₇	—	2	1	no	—
107	C₂²×C₁₇	—	2	1	no	—
108	C₂²×C₁₇	#358	1	2	yes	—
109	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
110	C₄×C₁₇	—	2	1	no	—
111	C₄×C₁₇	—	2	1	no	—
112	C₄×C₁₇	—	2	1	no	—
113	C₄×C₁₇	—	8	2	no	—
114	C₄×C₁₇	—	8	2	no	—
115	C₄×C₁₇	#358	1	2	yes	—
116	C₄×C₁₇	#413	1	1	yes	characteristic
117	C₄×C₁₇	#417	1	1	yes	characteristic
118	C₄×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
119	C₄×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
120	#533	—	2	1	no	—
121	C₂²×C₁₆	—	34	1	no	—
122	#500	C₂×D₁₇	1	1	yes	characteristic
123	#494	C₂×D₁₇	1	1	yes	characteristic
124	#487	—	2	1	no	—
125	#467	—	2	1	no	—
126	#467	Dic₁₇	1	2	yes	—
127	C₄×Q₁₆	—	2	1	no	—
128	C₄×Q₁₆	Dic₁₇	1	2	yes	—
129	#423	—	34	1	no	—
130	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
131	C₄⋊C₁₆	—	17	2	no	—
132	C₄⋊C₁₆	—	34	1	no	—
133	#335	—	17	2	no	—
134	#335	—	34	1	no	—
135	C₂×C₁₇	—	2	2	no	—
136	C₂×C₁₇	#1547	1	1	yes	characteristic
137	C₂×C₁₇	C₂×#358	1	1	yes	characteristic
138	C₂×C₁₇	#2148	1	1	yes	characteristic
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×Q₁₆	C₂×Dic₁₇	1	2	yes	—
141	C₂×M₁₆	—	68	1	no	—
142	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
144	C₄⋊Q₈	—	2	1	no	—
145	C₄⋊Q₈	—	2	1	no	—
146	#125	—	4	1	no	—
147	C₂²⋊Q₈	—	2	2	no	—
148	C₂²⋊Q₈	—	4	1	no	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	4	1	no	—
151	#117	—	4	1	no	—
152	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
153	C₂×C₁₆	—	34	2	no	—
154	C₂×C₁₆	—	68	2	no	—
155	#107	—	2	1	no	—
156	#107	C₂×Dic₁₇	1	2	yes	—
157	C₄⋊C₈	—	34	2	no	—
158	C₄⋊C₈	—	68	1	no	—
159	#103	—	2	2	no	—
160	#103	—	2	2	no	—
161	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
164	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
165	C₁₇	#11983	1	1	yes	characteristic, complemented, Sylow
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂²×C₄	#63228	1	1	yes	characteristic
169	C₂²×C₄	#63229	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
171	Q₁₆	—	4	2	no	—
172	Q₁₆	—	4	1	no	—
173	Q₁₆	—	4	1	no	—
174	M₁₆	—	136	2	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	68	2	no	—
179	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
180	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
181	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
182	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	#63228	1	1	yes	characteristic
192	C₄²	#63229	1	1	yes	characteristic
193	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
195	C₁₆	—	68	2	no	—
196	C₂³	#10587503	1	1	yes	characteristic
197	Q₈	—	4	2	no	—
198	Q₈	—	4	2	no	—
199	Q₈	—	8	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	#10587499	1	2	yes	—
206	C₂×C₄	#10587500	1	2	yes	—
207	C₂×C₄	#10587503	1	1	yes	characteristic
208	C₂×C₄	#10587503	1	1	yes	characteristic
209	C₂×C₄	#10587503	1	1	yes	characteristic
210	C₂×C₄	C₂×#63228	1	1	yes	characteristic
211	C₂×C₄	C₂×#63229	1	1	yes	characteristic
212	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	136	2	no	—
217	C₈	#10587635	1	1	yes	characteristic
218	C₈	#10587637	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#11762613	1	2	yes	—
222	C₂²	C₂×#10587503	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	#11762613	1	2	yes	—
229	C₄	#11762793	1	1	yes	characteristic
230	C₄	#11762810	1	1	yes	characteristic
231	C₄	C₂×#10587499	1	1	yes	characteristic
232	C₄	C₂×#10587500	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	#14527461	1	1	yes	characteristic
235	C₂	C₂×#11762613	1	1	yes	characteristic
236	C₂	#14530533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587503
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	#11983
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #14526842	C₂	semidirect
4. #14526833	C₂	semidirect
165. C₁₇	#11983	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
#14526842	2
#14526833	2
C₁₇×#3170	2
#11983	17