Subgroups — #16927245

Order 4352.

398 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic
2	#14528777	C₂	1	2	yes	maximal
3	#14527540	C₂	1	1	yes	maximal, characteristic
4	#14526658	C₂	1	2	yes	maximal
5	C₁₇×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₃₄	C₂²	1	1	yes	characteristic
7	C₈×Dic₃₄	—	2	1	no	—
8	C₈×Dic₃₄	—	2	1	no	—
9	C₈×Dic₃₄	—	2	1	no	—
10	C₈×Dic₃₄	—	2	1	no	—
11	#11762833	C₂²	1	2	yes	—
12	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
13	C₂×#10587485	C₄	1	2	yes	—
14	C₂×#10587495	—	2	2	no	—
15	C₂×C₈×Dic₁₇	C₄	1	2	yes	—
16	#11762577	—	2	2	no	—
17	#11762560	—	2	1	no	—
18	#11762560	—	2	1	no	—
19	C₁₇×#423	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₁₇×#334	—	2	1	no	—
22	C₁₇×#334	—	2	1	no	—
23	C₂²×Dic₃₄	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₁₇	—	2	2	no	—
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
29	C₂×#63225	Q₈	1	2	yes	—
30	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
31	C₂×#63223	—	2	2	no	—
32	C₂²×#3622	C₂×C₄	1	2	yes	—
33	C₄×Dic₃₄	—	2	1	no	—
34	C₄×Dic₃₄	—	2	1	no	—
35	C₄×Dic₃₄	—	2	1	no	—
36	C₄×Dic₃₄	—	2	1	no	—
37	C₁₇×C₄⋊C₈	—	2	2	no	—
38	C₁₇×C₄⋊C₈	—	2	2	no	—
39	C₄×C₈×C₁₇	D₄	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	D₄	1	1	yes	characteristic
43	#10587495	—	4	4	no	—
44	#10587485	—	2	2	no	—
45	#10587485	—	2	2	no	—
46	C₈×Dic₁₇	—	2	2	no	—
47	C₈×Dic₁₇	—	2	2	no	—
48	C₄⋊#3622	—	2	1	no	—
49	C₄⋊#3622	—	2	1	no	—
50	C₄⋊#3622	—	2	2	no	—
51	C₄⋊#3622	—	2	2	no	—
52	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×Dic₁₇	C₄⋊C₄	1	2	yes	—
56	C₂²×Dic₁₇	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₃₄	—	2	1	no	—
58	C₂×Dic₃₄	—	2	1	no	—
59	C₂×Dic₃₄	—	2	2	no	—
60	C₂×Dic₃₄	SD₁₆	1	4	yes	—
61	M₁₆×C₁₇	—	4	2	no	—
62	M₁₆×C₁₇	—	4	2	no	—
63	C₂×C₈×C₁₇	—	2	2	no	—
64	C₂×C₈×C₁₇	—	4	1	no	—
65	C₂×C₈×C₁₇	—	4	2	no	—
66	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₇	#41	1	1	yes	characteristic
73	C₄²×C₁₇	#41	1	1	yes	characteristic
74	C₄⋊Dic₁₇	—	2	1	no	—
75	C₄⋊Dic₁₇	—	2	1	no	—
76	#63225	—	2	2	no	—
77	#63225	—	2	2	no	—
78	C₄×Dic₁₇	—	2	2	no	—
79	C₄×Dic₁₇	—	2	2	no	—
80	#63223	—	4	2	no	—
81	#63223	—	4	2	no	—
82	C₂×#3622	—	2	2	no	—
83	C₂×#3622	—	4	2	no	—
84	C₂×#3622	—	4	2	no	—
85	#8615	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	2	1	no	—
89	C₂×C₄×C₁₇	—	4	1	no	—
90	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
92	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₁₇	#135	1	2	yes	—
98	C₂×Dic₁₇	—	2	2	no	—
99	C₂×Dic₁₇	—	2	2	no	—
100	C₂×Dic₁₇	—	2	2	no	—
101	C₂×Dic₁₇	—	4	2	no	—
102	C₂×Dic₁₇	#106	1	4	yes	—
103	Dic₃₄	—	2	4	no	—
104	Dic₃₄	—	2	4	no	—
105	C₈×C₁₇	—	2	1	no	—
106	C₈×C₁₇	—	2	1	no	—
107	C₈×C₁₇	—	2	1	no	—
108	C₈×C₁₇	—	8	2	no	—
109	C₈×C₁₇	#101	1	2	yes	complemented
110	#3622	—	4	2	no	—
111	#3622	—	8	2	no	—
112	C₂×C₈×Q₈	—	17	1	no	—
113	#1801	—	17	2	no	—
114	#1566	—	17	1	no	—
115	#1566	D₁₇	1	1	yes	characteristic
116	#1292	—	17	2	no	—
117	C₂²×C₁₇	—	2	1	no	—
118	C₂²×C₁₇	—	2	1	no	—
119	C₂²×C₁₇	#380	1	1	yes	characteristic
120	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
121	C₂²×C₁₇	#427	1	1	yes	characteristic
122	C₄×C₁₇	—	2	1	no	—
123	C₄×C₁₇	—	2	1	no	—
124	C₄×C₁₇	—	2	1	no	—
125	C₄×C₁₇	C₂×#101	1	1	yes	characteristic
126	C₄×C₁₇	#413	1	1	yes	characteristic
127	C₄×C₁₇	#465	1	2	yes	—
128	C₄×C₁₇	#465	1	2	yes	—
129	Dic₁₇	—	2	4	no	—
130	Dic₁₇	—	4	2	no	—
131	C₂×C₄×Q₈	—	17	1	no	—
132	C₈×Q₈	—	34	1	no	—
133	C₈×Q₈	—	34	1	no	—
134	C₈×Q₈	—	34	1	no	—
135	C₈×Q₈	—	34	1	no	—
136	#423	—	17	2	no	—
137	#423	C₂×D₁₇	1	2	yes	—
138	C₂×C₄⋊C₈	—	17	2	no	—
139	C₂×C₄⋊C₈	—	17	1	no	—
140	C₂×#101	—	34	2	no	—
141	C₂×C₄×C₈	—	17	2	no	—
142	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
143	#343	—	34	2	no	—
144	#334	—	2	1	no	—
145	#334	—	2	1	no	—
146	#334	—	34	1	no	—
147	#334	—	34	1	no	—
148	C₂×C₁₇	—	2	1	no	—
149	C₂×C₁₇	—	2	1	no	—
150	C₂×C₁₇	#1801	1	1	yes	characteristic
151	C₂×C₁₇	#1807	1	1	yes	characteristic
152	C₂×C₁₇	#1813	1	1	yes	characteristic
153	C₂²×Q₈	—	17	1	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	34	2	no	—
156	C₂²×C₈	—	17	2	no	—
157	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
158	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
159	C₄×Q₈	—	34	1	no	—
160	C₄×Q₈	—	34	1	no	—
161	C₄×Q₈	—	34	1	no	—
162	C₄×Q₈	—	34	1	no	—
163	C₂×C₄⋊C₄	—	17	1	no	—
164	C₂×C₄⋊C₄	—	17	2	no	—
165	C₂×C₄²	—	17	2	no	—
166	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	34	2	no	—
170	C₄⋊C₈	—	34	2	no	—
171	C₄⋊C₈	—	34	1	no	—
172	C₄⋊C₈	—	34	2	no	—
173	C₄⋊C₈	—	34	2	no	—
174	C₄⋊C₈	—	34	1	no	—
175	#101	—	68	4	no	—
176	C₄×C₈	—	34	2	no	—
177	C₄×C₈	—	34	2	no	—
178	C₄×C₈	D₆₈	1	1	yes	characteristic
179	C₄×C₈	D₆₈	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
182	C₁₇	#8615	1	1	yes	characteristic, complemented, Sylow
183	C₂×Q₈	—	17	4	no	—
184	C₂×Q₈	—	34	1	no	—
185	C₂×Q₈	—	34	2	no	—
186	C₂×Q₈	—	34	1	no	—
187	C₂²×C₄	—	17	2	no	—
188	C₂²×C₄	—	17	2	no	—
189	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
190	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
191	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	68	2	no	—
195	M₁₆	—	68	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	34	2	no	—
200	C₂×C₈	—	68	2	no	—
201	C₂×C₈	—	68	2	no	—
202	C₂×C₈	#63227	1	1	yes	characteristic
203	C₂×C₈	#63227	1	1	yes	characteristic
204	C₂×C₈	#63227	1	1	yes	characteristic
205	C₂×C₈	#63227	1	1	yes	characteristic
206	C₄⋊C₄	—	34	1	no	—
207	C₄⋊C₄	—	34	2	no	—
208	C₄⋊C₄	—	34	1	no	—
209	C₄⋊C₄	—	34	2	no	—
210	C₄²	—	34	2	no	—
211	C₄²	—	34	2	no	—
212	C₄²	C₂×D₆₈	1	1	yes	characteristic
213	C₄²	#63254	1	1	yes	characteristic
214	C₄²	#63256	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
216	C₂³	C₄⋊C₄×D₁₇	1	1	yes	characteristic
217	Q₈	—	34	4	no	—
218	Q₈	—	34	4	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	17	4	no	—
223	C₂×C₄	—	34	2	no	—
224	C₂×C₄	—	34	2	no	—
225	C₂×C₄	—	34	2	no	—
226	C₂×C₄	—	68	2	no	—
227	C₂×C₄	#10587578	1	1	yes	characteristic
228	C₂×C₄	#10587581	1	1	yes	characteristic
229	C₂×C₄	#10587583	1	1	yes	characteristic
230	C₂×C₄	SD₁₆×D₁₇	1	2	yes	—
231	C₂×C₄	#10587602	1	2	yes	—
232	C₂×C₄	C₂×#63227	1	1	yes	characteristic
233	C₂×C₄	#10587624	1	1	yes	characteristic
234	C₂×C₄	#10587632	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	68	2	no	—
240	C₈	—	136	2	no	—
241	C₈	#10587495	1	2	yes	complemented
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#11762666	1	1	yes	characteristic
245	C₂²	#106×D₁₇	1	1	yes	characteristic
246	C₂²	#11762844	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	34	4	no	—
251	C₄	—	68	2	no	—
252	C₄	C₂×#10587495	1	1	yes	characteristic
253	C₄	#11762791	1	1	yes	characteristic
254	C₄	#11763035	1	2	yes	—
255	C₄	#11763039	1	2	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	#14528777	1	1	yes	characteristic
259	C₂	#14528806	1	1	yes	characteristic
260	C₂	#14528831	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762666
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1566	C₂
socle	C₂²×C₁₇	#380
prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1566	D₁₇
hypocenter	C₁₇	#8615
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
109. C₈×C₁₇	#101	semidirect
182. C₁₇	#8615	semidirect
241. C₈	#10587495	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₃₄	2
#14528777	2
#14527540	2
#14526658	2
C₁₇×#1566	2
#8615	17