Subgroups — #16949166

Order 4352.

614 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587483	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762764	C₂	1	1	yes	maximal, characteristic
3	#14526861	C₂	1	4	yes	maximal, complemented
4	C₂³×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63223	C₄	1	1	yes	characteristic
6	C₂×#10587612	C₄	1	1	yes	characteristic
7	C₂×#10587483	—	2	2	no	—
8	C₂×#10587483	—	2	2	no	—
9	C₂×#10587483	C₄	1	4	yes	complemented
10	C₂²×#63214	C₂²	1	2	yes	—
11	#11762764	—	2	2	no	—
12	#11762764	—	2	2	no	—
13	C₂×#10587486	C₄	1	2	yes	—
14	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
15	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₁₇	—	2	2	no	—
17	C₂²×C₁₆×C₁₇	—	2	2	no	—
18	C₂²×C₁₆×C₁₇	C₄	1	4	yes	complemented
19	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₇	D₄	1	2	yes	—
25	C₂²×C₈×C₁₇	D₄	1	2	yes	—
26	C₂²×C₈×C₁₇	D₄	1	2	yes	—
27	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₁₇	C₈	1	2	yes	—
30	#10587612	—	2	2	no	—
31	#10587612	—	2	2	no	—
32	C₂×C₄⋊Dic₁₇	C₈	1	2	yes	—
33	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
34	C₂×#63223	—	2	2	no	—
35	C₂×#63223	—	2	2	no	—
36	C₂×#63223	C₈	1	4	yes	—
37	C₂²×#3622	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₁₇	—	2	8	no	—
39	C₂×C₁₆×C₁₇	—	2	2	no	—
40	C₂×C₁₆×C₁₇	—	2	2	no	—
41	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
42	#10587486	—	2	4	no	—
43	C₈×Dic₁₇	—	2	2	no	—
44	C₈×Dic₁₇	—	2	2	no	—
45	#10587483	—	2	8	no	—
46	C₂×#63214	—	2	2	no	—
47	C₂×#63214	—	2	2	no	—
48	C₂×#63214	C₂×C₄	1	4	yes	—
49	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
51	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
56	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
57	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₁₇	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₁₇	C₄²	1	2	yes	—
60	C₂×C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
61	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
63	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
65	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²⋊Dic₁₇	—	2	4	no	—
73	C₄⋊Dic₁₇	—	2	4	no	—
74	C₄×Dic₁₇	—	2	2	no	—
75	C₄×Dic₁₇	—	2	2	no	—
76	#63223	—	2	8	no	—
77	C₂×#3622	—	2	2	no	—
78	C₂×#3622	—	2	2	no	—
79	C₂×#3622	C₂×C₈	1	4	yes	—
80	C₁₆×C₁₇	—	2	4	no	—
81	#63214	—	2	4	no	—
82	#12545	—	17	1	no	maximal, Sylow
83	C₂³×C₁₇	—	2	4	no	—
84	C₂³×C₁₇	C₄×C₈	1	2	yes	—
85	C₂³×C₁₇	C₂²⋊C₈	1	2	yes	—
86	C₂³×C₁₇	C₄⋊C₈	1	2	yes	—
87	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₇	#95	1	4	yes	—
89	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
90	C₂×C₄×C₁₇	#97	1	4	yes	—
91	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
92	C₂×C₄×C₁₇	C₂²⋊C₈	1	4	yes	—
93	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₁₇	C₄⋊C₈	1	4	yes	—
95	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₁₇	—	2	4	no	—
102	C₂×Dic₁₇	—	2	8	no	—
103	C₈×C₁₇	#95	1	4	yes	—
104	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	#3622	—	2	4	no	—
109	C₂²×M₃₂	—	17	1	no	—
110	C₂³×C₁₆	D₁₇	1	1	yes	characteristic
111	C₂×#405	—	17	1	no	—
112	#1374	—	17	4	no	—
113	C₂²×C₁₇	—	2	4	no	—
114	C₂²×C₁₇	—	2	4	no	—
115	C₂²×C₁₇	—	2	4	no	—
116	C₂²×C₁₇	#336	1	4	yes	—
117	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₂²×C₁₇	#504	1	4	yes	—
121	C₄×C₁₇	#336	1	4	yes	—
122	C₄×C₁₇	C₂×#95	1	1	yes	characteristic
123	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
124	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
126	Dic₁₇	—	4	4	no	—
127	C₂²×M₁₆	—	17	1	no	—
128	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
129	C₂×#117	—	17	1	no	—
130	C₂×M₃₂	—	17	4	no	—
131	C₂×M₃₂	—	34	2	no	—
132	C₂×M₃₂	—	34	2	no	—
133	C₂²×C₁₆	—	2	2	no	—
134	C₂²×C₁₆	—	2	2	no	—
135	C₂²×C₁₆	—	17	2	no	—
136	C₂²×C₁₆	Dic₁₇	1	4	yes	complemented
137	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
138	#405	—	34	2	no	—
139	#405	—	34	2	no	—
140	C₂×#97	—	17	2	no	—
141	C₂×C₄×C₈	—	17	2	no	—
142	C₂×C₁₇	—	2	4	no	—
143	C₂×C₁₇	C₂×#336	1	1	yes	characteristic
144	C₂×C₁₇	#2108	1	2	yes	—
145	C₂×C₁₇	#2113	1	2	yes	—
146	C₂×C₁₇	#2151	1	2	yes	—
147	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
148	C₂×M₁₆	—	17	4	no	—
149	C₂×M₁₆	—	34	2	no	—
150	C₂×M₁₆	—	34	2	no	—
151	C₂²×C₈	—	17	2	no	—
152	C₂²×C₈	Dic₃₄	1	2	yes	—
153	C₂²×C₈	C₄×D₁₇	1	2	yes	—
154	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
155	C₂²×C₈	D₆₈	1	2	yes	—
156	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
157	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
158	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
159	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
160	#117	—	34	2	no	—
161	#117	—	34	2	no	—
162	C₂×C₄⋊C₄	—	17	2	no	—
163	C₂×C₂²⋊C₄	—	17	2	no	—
164	C₂×C₄²	—	17	2	no	—
165	M₃₂	—	34	8	no	—
166	C₂×C₁₆	—	2	8	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	17	4	no	—
170	C₂×C₁₆	—	34	2	no	—
171	C₂×C₁₆	—	34	2	no	—
172	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
173	#97	—	34	4	no	—
174	C₄×C₈	—	34	2	no	—
175	C₄×C₈	—	34	2	no	—
176	C₁₇	#12545	1	1	yes	characteristic, complemented, Sylow
177	C₂⁴	C₈×D₁₇	1	1	yes	characteristic
178	C₂²×C₄	—	17	4	no	—
179	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
180	C₂²×C₄	C₈×D₁₇	1	2	yes	—
181	C₂²×C₄	#63218	1	2	yes	—
182	C₂²×C₄	#63218	1	2	yes	—
183	C₂²×C₄	C₄×Dic₁₇	1	2	yes	—
184	C₂²×C₄	#63225	1	2	yes	—
185	C₂²×C₄	#63227	1	2	yes	—
186	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
187	M₁₆	—	34	8	no	—
188	C₂×C₈	—	17	4	no	—
189	C₂×C₈	—	34	2	no	—
190	C₂×C₈	—	34	2	no	—
191	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
192	C₂×C₈	#63225	1	2	yes	—
193	C₂×C₈	#63225	1	4	yes	—
194	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
195	C₂×C₈	#63227	1	2	yes	—
196	C₂×C₈	#63227	1	4	yes	—
197	C₂×C₈	C₂²⋊Dic₁₇	1	4	yes	—
198	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
200	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
204	C₄⋊C₄	—	34	4	no	—
205	C₂²⋊C₄	—	34	4	no	—
206	C₄²	—	34	2	no	—
207	C₄²	—	34	2	no	—
208	C₁₆	—	2	4	no	—
209	C₁₆	—	34	4	no	—
210	C₂³	—	2	4	no	—
211	C₂³	C₈×Dic₁₇	1	2	yes	—
212	C₂³	#10587485	1	2	yes	—
213	C₂³	#10587491	1	2	yes	—
214	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
215	C₂×C₄	—	34	8	no	—
216	C₂×C₄	—	34	4	no	—
217	C₂×C₄	C₈×Dic₁₇	1	2	yes	—
218	C₂×C₄	#10587485	1	2	yes	—
219	C₂×C₄	#10587485	1	4	yes	—
220	C₂×C₄	#10587486	1	4	yes	—
221	C₂×C₄	#10587491	1	4	yes	—
222	C₂×C₄	#10587491	1	2	yes	—
223	C₂×C₄	#10587502	1	4	yes	—
224	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
225	C₂×C₄	C₂×#63218	1	1	yes	characteristic
226	C₂×C₄	C₂×#63218	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
228	C₂×C₄	C₂×#63225	1	1	yes	characteristic
229	C₂×C₄	C₂×#63227	1	1	yes	characteristic
230	C₈	—	34	4	no	—
231	C₈	#10587502	1	4	yes	—
232	C₈	C₂×#63225	1	1	yes	characteristic
233	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
234	C₈	C₂×#63227	1	1	yes	characteristic
235	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	#11762563	1	4	yes	—
240	C₂²	C₂×C₈×Dic₁₇	1	1	yes	characteristic
241	C₂²	C₂×#10587491	1	1	yes	characteristic
242	C₂²	C₂×#10587485	1	1	yes	characteristic
243	C₂²	#11763225	1	4	yes	—
244	C₄	—	68	4	no	—
245	C₄	#11762563	1	4	yes	—
246	C₄	C₂×#10587502	1	1	yes	characteristic
247	C₄	C₂×#10587486	1	1	yes	characteristic
248	C₄	C₂×#10587491	1	1	yes	characteristic
249	C₄	C₂×#10587485	1	1	yes	characteristic
250	C₂	—	2	4	no	—
251	C₂	C₂×#11762563	1	1	yes	characteristic
252	C₂	#14530380	1	2	yes	—
253	C₂	#14530397	1	2	yes	—
254	C₂	#14530542	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₂³×C₁₆×C₁₇	C₂
socle	C₂³×C₁₇	C₂²×C₈
prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂³×C₁₆	D₁₇
hypocenter	C₁₇	#12545
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. #14526861	C₂	semidirect
9. C₂×#10587483	C₄	semidirect
18. C₂²×C₁₆×C₁₇	C₄	semidirect
136. C₂²×C₁₆	Dic₁₇	semidirect
176. C₁₇	#12545	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10587483	2
C₂×#11762764	2
#14526861	2
C₂³×C₁₆×C₁₇	2
#12545	17