Subgroups — #17961643

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543823	C₂	1	2	yes	maximal
2	#14543823	C₂	1	2	yes	maximal
3	#14543822	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63216	C₂²	1	4	yes	—
6	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂³	1	1	yes	characteristic
24	C₂×#63216	—	2	4	no	—
25	C₂×#63216	—	2	4	no	—
26	C₂×#63216	—	2	4	no	—
27	C₄×C₈×C₁₇	—	2	2	no	—
28	C₈×Dic₁₇	—	2	2	no	—
29	C₈×Dic₁₇	—	2	2	no	—
30	C₈×Dic₁₇	—	2	2	no	—
31	C₈×Dic₁₇	—	2	2	no	—
32	C₄×#3622	—	2	2	no	—
33	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₁₇	—	2	1	no	—
41	C₂×C₈×C₁₇	—	2	1	no	—
42	C₂×C₈×C₁₇	—	2	1	no	—
43	C₂×C₈×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
45	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
46	C₄²×C₁₇	—	2	1	no	—
47	C₄²×C₁₇	—	2	1	no	—
48	C₄×Dic₁₇	—	2	2	no	—
49	C₄×Dic₁₇	—	2	1	no	—
50	C₄×Dic₁₇	—	2	1	no	—
51	C₄×Dic₁₇	—	2	1	no	—
52	C₄×Dic₁₇	—	2	1	no	—
53	C₄×Dic₁₇	—	2	2	no	—
54	C₄×Dic₁₇	—	2	2	no	—
55	C₄×Dic₁₇	C₁₆	1	4	yes	complemented
56	C₂×#3622	—	2	1	no	—
57	C₂×#3622	—	2	1	no	—
58	C₂×#3622	M₁₆	1	2	yes	—
59	C₂×#3622	#41	1	2	yes	—
60	C₂×#3622	#41	1	2	yes	—
61	C₂×#3622	#41	1	2	yes	—
62	#63216	—	4	4	no	—
63	#12894	—	17	1	no	maximal, Sylow
64	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	—	2	1	no	—
66	C₂×C₄×C₁₇	—	2	1	no	—
67	C₂×C₄×C₁₇	—	2	1	no	—
68	C₂×C₄×C₁₇	—	2	1	no	—
69	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
70	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
71	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	1	no	—
78	C₂×Dic₁₇	—	2	1	no	—
79	C₂×Dic₁₇	—	2	1	no	—
80	C₂×Dic₁₇	—	2	1	no	—
81	C₂×Dic₁₇	—	2	1	no	—
82	C₂×Dic₁₇	—	2	1	no	—
83	C₂×Dic₁₇	—	2	1	no	—
84	C₂×Dic₁₇	—	2	1	no	—
85	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
86	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
87	C₂×Dic₁₇	M₃₂	1	2	yes	—
88	C₂×Dic₁₇	M₃₂	1	2	yes	—
89	C₈×C₁₇	—	2	2	no	—
90	C₈×C₁₇	—	2	2	no	—
91	#3622	—	2	2	no	—
92	#3622	#126	1	4	yes	—
93	C₄²×C₈	—	17	1	no	—
94	#1374	—	17	2	no	—
95	#1374	—	17	2	no	—
96	#1374	—	17	2	no	—
97	C₂²×C₁₇	#431	1	1	yes	characteristic
98	C₂²×C₁₇	#431	1	1	yes	characteristic
99	C₂²×C₁₇	#431	1	1	yes	characteristic
100	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
101	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
102	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
103	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
104	C₄×C₁₇	—	2	2	no	—
105	C₄×C₁₇	—	2	2	no	—
106	C₄×C₁₇	#383	1	1	yes	characteristic
107	C₄×C₁₇	#432	1	1	yes	characteristic
108	C₄×C₁₇	#432	1	1	yes	characteristic
109	C₄×C₁₇	#432	1	1	yes	characteristic
110	Dic₁₇	—	2	2	no	—
111	Dic₁₇	—	2	2	no	—
112	Dic₁₇	—	2	2	no	—
113	Dic₁₇	—	2	2	no	—
114	C₂²×C₁₆	—	17	4	no	—
115	C₂×C₄×C₈	—	17	1	no	—
116	C₂×C₄×C₈	—	17	1	no	—
117	C₂×C₄×C₈	—	17	1	no	—
118	C₂×C₄×C₈	—	17	1	no	—
119	C₂×C₄×C₈	—	17	1	no	—
120	C₂×C₄×C₈	#595	1	1	yes	characteristic
121	C₄³	—	17	1	no	—
122	C₂×C₁₇	#1839	1	1	yes	characteristic
123	C₂×C₁₇	#2162	1	1	yes	characteristic
124	C₂×C₁₇	#2162	1	1	yes	characteristic
125	C₂×C₁₇	#2162	1	1	yes	characteristic
126	C₂×C₁₇	#2163	1	1	yes	characteristic
127	C₂×C₁₇	#2163	1	1	yes	characteristic
128	C₂×C₁₇	#2163	1	1	yes	characteristic
129	C₂²×C₈	—	17	1	no	—
130	C₂²×C₈	—	17	1	no	—
131	C₂²×C₈	C₂×#595	1	1	yes	characteristic
132	C₂²×C₈	C₂×#595	1	1	yes	characteristic
133	C₂×C₄²	—	17	1	no	—
134	C₂×C₄²	—	17	1	no	—
135	C₂×C₄²	—	17	1	no	—
136	C₂×C₄²	—	17	1	no	—
137	C₂×C₄²	—	17	1	no	—
138	C₂×C₄²	—	17	1	no	—
139	C₂×C₄²	C₂×#595	1	1	yes	characteristic
140	C₂×C₁₆	—	34	4	no	—
141	C₂×C₁₆	—	34	4	no	—
142	C₂×C₁₆	—	34	4	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	34	2	no	—
145	C₄×C₈	—	34	2	no	—
146	C₄×C₈	—	34	2	no	—
147	C₄×C₈	—	34	2	no	—
148	C₄×C₈	—	34	2	no	—
149	C₁₇	#12894	1	1	yes	characteristic, complemented, Sylow
150	C₂²×C₄	—	17	1	no	—
151	C₂²×C₄	—	17	1	no	—
152	C₂²×C₄	—	17	1	no	—
153	C₂²×C₄	—	17	1	no	—
154	C₂²×C₄	#63242	1	1	yes	characteristic
155	C₂²×C₄	#63242	1	1	yes	characteristic
156	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	17	2	no	—
162	C₂×C₈	—	17	2	no	—
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	17	2	no	—
165	C₂×C₈	—	34	1	no	—
166	C₂×C₈	—	34	1	no	—
167	C₂×C₈	#63243	1	2	yes	—
168	C₂×C₈	#63247	1	2	yes	—
169	C₄²	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	17	4	no	—
172	C₄²	—	34	2	no	—
173	C₄²	—	34	2	no	—
174	C₄²	—	34	1	no	—
175	C₄²	—	34	1	no	—
176	C₄²	—	34	1	no	—
177	C₄²	—	34	1	no	—
178	C₄²	—	34	2	no	—
179	C₁₆	—	68	4	no	—
180	C₂³	C₂×#63242	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	17	2	no	—
186	C₂×C₄	—	17	2	no	—
187	C₂×C₄	—	17	2	no	—
188	C₂×C₄	—	17	2	no	—
189	C₂×C₄	—	34	1	no	—
190	C₂×C₄	—	34	1	no	—
191	C₂×C₄	—	34	1	no	—
192	C₂×C₄	—	34	1	no	—
193	C₂×C₄	—	34	1	no	—
194	C₂×C₄	—	34	1	no	—
195	C₂×C₄	—	34	1	no	—
196	C₂×C₄	—	34	1	no	—
197	C₂×C₄	#10587529	1	2	yes	—
198	C₂×C₄	#10587530	1	2	yes	—
199	C₂×C₄	C₂×#63243	1	1	yes	characteristic
200	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
201	C₂×C₄	#10587677	1	1	yes	characteristic
202	C₂×C₄	#10587677	1	1	yes	characteristic
203	C₂×C₄	#10587677	1	1	yes	characteristic
204	C₂×C₄	C₂×#63247	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	17	4	no	—
208	C₈	—	34	2	no	—
209	C₂²	#11763826	1	1	yes	characteristic
210	C₂²	#11763827	1	1	yes	characteristic
211	C₂²	#11763828	1	1	yes	characteristic
212	C₂²	C₂×#10587529	1	1	yes	characteristic
213	C₂²	#11763865	1	1	yes	characteristic
214	C₂²	#11763865	1	1	yes	characteristic
215	C₂²	C₂×#10587530	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	34	2	no	—
219	C₄	—	34	2	no	—
220	C₄	—	34	2	no	—
221	C₄	—	34	2	no	—
222	C₄	#11763754	1	1	yes	characteristic
223	C₄	#11763829	1	1	yes	characteristic
224	C₄	#11763830	1	1	yes	characteristic
225	C₄	#11763831	1	1	yes	characteristic
226	C₂	#14544712	1	1	yes	characteristic
227	C₂	#14545142	1	1	yes	characteristic
228	C₂	#14545143	1	1	yes	characteristic
229	C₂	#14545144	1	1	yes	characteristic
230	C₂	#14545146	1	1	yes	characteristic
231	C₂	#14545147	1	1	yes	characteristic
232	C₂	#14545148	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#12894
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
55. C₄×Dic₁₇	C₁₆	semidirect
149. C₁₇	#12894	semidirect

Maximal subgroups

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

subgroup	index
#14543823	2
#14543822	2
C₄×C₈×Dic₁₇	2
#12894	17