Subgroups — #16925641

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587490	C₂	1	1	yes	maximal, characteristic
2	#14527074	C₂	1	1	yes	maximal, characteristic
3	#14527065	C₂	1	1	yes	maximal, characteristic
4	C₂×#11762531	C₂	1	1	yes	maximal, characteristic
5	#14526620	C₂	1	1	yes	maximal, characteristic
6	#14526620	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1564	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587490	C₄	1	2	yes	—
9	C₄×#63223	C₂²	1	1	yes	characteristic
10	C₄×#63223	C₂²	1	1	yes	characteristic
11	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
12	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
13	#11762574	C₄	1	2	yes	—
14	#11762531	—	2	1	no	—
15	#11762531	—	2	1	no	—
16	#11762531	C₄	1	2	yes	—
17	#11762531	C₄	1	2	yes	—
18	#11762530	—	2	1	no	—
19	C₈×#3622	—	2	1	no	—
20	C₁₇×#431	C₂²	1	1	yes	characteristic
21	C₁₇×#431	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₁₇×#335	C₄	1	2	yes	—
24	C₁₇×#334	C₄	1	2	yes	—
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×#63223	C₂×C₄	1	2	yes	—
29	C₂×#63223	C₂×C₄	1	2	yes	—
30	C₂²×#3622	—	2	1	no	—
31	C₂²×#3622	Q₈	1	2	yes	—
32	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
33	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
34	C₁₇×C₂²⋊C₈	—	2	1	no	—
35	C₁₇×C₂²⋊C₈	—	2	1	no	—
36	C₄×C₈×C₁₇	—	2	1	no	—
37	C₄×C₈×C₁₇	—	2	1	no	—
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	#10587490	—	2	4	no	—
43	#10587474	—	2	1	no	—
44	#10587474	—	2	1	no	—
45	C₄×#3622	—	2	1	no	—
46	C₄×#3622	—	2	1	no	—
47	C₄×#3622	—	2	1	no	—
48	C₄×#3622	C₂×C₄	1	2	yes	—
49	C₄×#3622	C₂×C₄	1	2	yes	—
50	C₄×#3622	D₄	1	1	yes	characteristic
51	C₄×#3622	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	—	2	2	no	—
57	C₂×C₈×C₁₇	—	2	2	no	—
58	C₂×C₈×C₁₇	—	2	2	no	—
59	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
60	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
63	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₇	#41	1	1	yes	characteristic
67	#63223	—	2	2	no	—
68	#63223	—	2	2	no	—
69	#63223	—	2	2	no	—
70	#63223	M₁₆	1	4	yes	—
71	C₂×#3622	—	2	4	no	—
72	C₂×#3622	—	2	2	no	—
73	C₂×#3622	—	2	2	no	—
74	C₂×#3622	—	2	2	no	—
75	C₂×#3622	—	4	2	no	—
76	C₂×#3622	C₄⋊C₄	1	2	yes	—
77	C₂×#3622	C₄⋊C₄	1	2	yes	—
78	#8296	—	17	1	no	maximal, Sylow
79	C₂³×C₁₇	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	—	2	2	no	—
81	C₂×C₄×C₁₇	—	2	2	no	—
82	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₁₇	#131	1	2	yes	—
90	C₈×C₁₇	—	2	1	no	—
91	C₈×C₁₇	—	2	1	no	—
92	C₈×C₁₇	—	2	1	no	—
93	C₈×C₁₇	—	2	1	no	—
94	C₈×C₁₇	—	4	2	no	—
95	C₈×C₁₇	—	4	2	no	—
96	#3622	—	2	4	no	—
97	#3622	—	2	2	no	—
98	#3622	—	2	2	no	—
99	#3622	—	2	4	no	—
100	#3622	—	4	2	no	—
101	C₄×#108	—	17	1	no	—
102	#1564	D₁₇	1	1	yes	characteristic
103	#1451	—	17	1	no	—
104	C₈×M₁₆	—	17	1	no	—
105	C₂×#322	—	17	1	no	—
106	#1278	—	17	1	no	—
107	#1278	—	17	1	no	—
108	C₂²×C₁₇	—	2	1	no	—
109	C₂²×C₁₇	—	2	1	no	—
110	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₁₇	#446	1	2	yes	—
112	C₄×C₁₇	—	2	1	no	—
113	C₄×C₁₇	—	2	1	no	—
114	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
115	C₄×C₁₇	#405	1	1	yes	characteristic
116	C₄×C₁₇	#430	1	2	yes	—
117	C₄×C₁₇	#435	1	2	yes	—
118	C₄×C₁₇	#443	1	2	yes	—
119	#431	C₂×D₁₇	1	1	yes	characteristic
120	#431	C₂×D₁₇	1	1	yes	characteristic
121	C₂×#108	—	17	2	no	—
122	C₄×M₁₆	—	17	1	no	—
123	C₄×M₁₆	—	17	1	no	—
124	C₂×C₄×C₈	—	17	1	no	—
125	C₂×C₄×C₈	—	17	1	no	—
126	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
127	#341	—	17	2	no	—
128	#335	Dic₁₇	1	2	yes	—
129	#334	Dic₁₇	1	2	yes	—
130	#322	—	17	2	no	—
131	#322	—	17	2	no	—
132	#322	—	34	1	no	—
133	#322	—	34	1	no	—
134	#322	—	34	1	no	—
135	C₈²	—	34	1	no	—
136	C₂×C₁₇	—	2	2	no	—
137	C₂×C₁₇	#1771	1	1	yes	characteristic
138	C₂×C₁₇	#1772	1	1	yes	characteristic
139	C₂×C₁₇	#1778	1	1	yes	characteristic
140	C₂×M₁₆	—	17	2	no	—
141	C₂×M₁₆	—	17	2	no	—
142	C₂²×C₈	—	17	2	no	—
143	C₂²×C₈	—	34	1	no	—
144	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
145	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
146	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
147	#108	—	34	4	no	—
148	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
149	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
150	C₂²⋊C₈	—	2	1	no	—
151	C₂²⋊C₈	—	2	1	no	—
152	#97	—	34	1	no	—
153	#97	—	34	1	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	17	2	no	—
157	C₄×C₈	—	17	2	no	—
158	C₄×C₈	—	17	1	no	—
159	C₄×C₈	—	17	1	no	—
160	C₄×C₈	—	34	1	no	—
161	C₄×C₈	—	34	1	no	—
162	C₄×C₈	—	34	1	no	—
163	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
164	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
165	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
166	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
167	C₁₇	#8296	1	1	yes	characteristic, complemented, Sylow
168	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
169	C₂²×C₄	#63254	1	1	yes	characteristic
170	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
171	M₁₆	—	17	4	no	—
172	M₁₆	—	34	2	no	—
173	M₁₆	—	34	2	no	—
174	M₁₆	—	34	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	17	2	no	—
180	C₂×C₈	—	17	2	no	—
181	C₂×C₈	—	34	4	no	—
182	C₂×C₈	—	34	2	no	—
183	C₂×C₈	—	34	2	no	—
184	C₂×C₈	—	34	2	no	—
185	C₂×C₈	—	68	2	no	—
186	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
187	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
188	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
189	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
190	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
191	C₄²	D₄×D₁₇	1	1	yes	characteristic
192	C₄²	#63256	1	1	yes	characteristic
193	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
194	C₂³	#10587573	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
198	C₂×C₄	#10587578	1	1	yes	characteristic
199	C₂×C₄	#10587579	1	1	yes	characteristic
200	C₂×C₄	#10587587	1	2	yes	—
201	C₂×C₄	M₁₆×D₁₇	1	2	yes	—
202	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
203	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
204	C₂×C₄	Q₈×Dic₁₇	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	4	2	no	—
210	C₈	—	4	2	no	—
211	C₈	—	34	4	no	—
212	C₈	—	34	2	no	—
213	C₈	—	34	2	no	—
214	C₈	—	34	4	no	—
215	C₈	—	68	2	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₄⋊C₄×Dic₁₇	1	1	yes	characteristic
219	C₂²	#11762938	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	M₁₆×Dic₁₇	1	1	yes	characteristic
223	C₄	#11762764	1	1	yes	characteristic
224	C₄	#11762856	1	2	yes	—
225	C₄	#11762879	1	2	yes	—
226	C₄	#11762926	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	#14528661	1	1	yes	characteristic
229	C₂	#14528665	1	1	yes	characteristic
230	C₂	#14528692	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1564	C₂
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₇	C₄×Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1564	D₁₇
hypocenter	C₁₇	#8296
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
167. C₁₇	#8296	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587490	2
#14527074	2
#14527065	2
C₂×#11762531	2
#14526620	2
C₁₇×#1564	2
#8296	17