Subgroups — #16950494

Order 4352.

438 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63214	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762636	C₂	1	2	yes	maximal, complemented
3	#14528813	C₂	1	1	yes	maximal, characteristic
4	#14526875	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1808	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63214	—	2	4	no	—
7	C₂²×#63214	C₄	1	4	yes	complemented
8	C₂²×#63214	C₂²	1	1	yes	characteristic
9	C₂²×#63214	C₂²	1	1	yes	characteristic
10	C₂×#10587489	C₂²	1	2	yes	—
11	#11762669	—	2	1	no	—
12	#11762636	—	2	4	no	—
13	#11762572	—	2	1	no	—
14	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×#107	C₂²	1	2	yes	—
16	C₁₇×#380	—	2	1	no	—
17	C₁₇×#340	—	2	1	no	—
18	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	—	2	2	no	—
20	C₂²×C₈×C₁₇	—	2	2	no	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
24	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
26	C₂×C₂²⋊C₄×C₁₇	—	4	1	no	—
27	C₂×C₂²⋊Dic₁₇	—	4	1	no	—
28	C₂×C₄⋊Dic₁₇	—	2	2	no	—
29	C₁₇×#107	—	2	2	no	—
30	C₁₇×#107	—	2	2	no	—
31	C₁₇×#95	—	4	1	no	—
32	#10587502	—	4	1	no	—
33	#10587489	—	2	2	no	—
34	#10587489	—	2	2	no	—
35	C₂×#63214	—	2	8	no	—
36	C₂×#63214	—	2	2	no	—
37	C₂×#63214	C₂×C₄	1	2	yes	—
38	C₂×#63214	C₂×C₄	1	2	yes	—
39	C₂×#63214	D₄	1	4	yes	—
40	C₂⁴×C₁₇	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	—	2	2	no	—
42	C₂²×C₄×C₁₇	—	2	2	no	—
43	C₂²×C₄×C₁₇	—	4	2	no	—
44	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
47	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₁₇	—	4	2	no	—
49	C₂×C₈×C₁₇	—	2	2	no	—
50	C₂×C₈×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	—	2	2	no	—
52	C₂×C₈×C₁₇	—	2	4	no	—
53	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₁₇	#41	1	2	yes	—
58	C₄⋊C₄×C₁₇	—	4	2	no	—
59	C₄⋊C₄×C₁₇	—	4	2	no	—
60	C₂²⋊C₄×C₁₇	—	8	2	no	—
61	C₂²⋊Dic₁₇	—	8	2	no	—
62	C₄⋊Dic₁₇	—	4	2	no	—
63	C₄⋊Dic₁₇	—	4	2	no	—
64	#63214	—	2	2	no	—
65	#63214	C₂²⋊C₄	1	4	yes	complemented
66	#12835	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	—	2	2	no	—
68	C₂³×C₁₇	—	2	2	no	—
69	C₂³×C₁₇	#107	1	2	yes	—
70	C₂³×C₁₇	SD₃₂	1	2	yes	—
71	C₂³×C₁₇	SD₃₂	1	2	yes	—
72	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	—	2	2	no	—
75	C₂×C₄×C₁₇	—	2	2	no	—
76	C₂×C₄×C₁₇	—	2	4	no	—
77	C₂×C₄×C₁₇	—	8	2	no	—
78	C₂×C₄×C₁₇	—	8	2	no	—
79	C₂×C₄×C₁₇	—	8	2	no	—
80	C₂×C₄×C₁₇	#107	1	2	yes	—
81	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#121	1	2	yes	—
83	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#135	1	2	yes	—
85	C₂×Dic₁₇	—	8	2	no	—
86	C₂×Dic₁₇	—	8	2	no	—
87	C₂×Dic₁₇	—	8	2	no	—
88	C₈×C₁₇	—	2	2	no	—
89	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₈×C₁₇	#117	1	1	yes	characteristic
91	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
92	C₂³×C₁₆	—	17	1	no	—
93	C₂×#367	—	17	2	no	—
94	#1808	—	17	1	no	—
95	#1808	D₁₇	1	1	yes	characteristic
96	#1380	—	17	2	no	—
97	C₂²×C₁₇	—	2	2	no	—
98	C₂²×C₁₇	—	2	2	no	—
99	C₂²×C₁₇	—	2	4	no	—
100	C₂²×C₁₇	—	2	4	no	—
101	C₂²×C₁₇	#367	1	4	yes	complemented
102	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
103	C₂²×C₁₇	#466	1	2	yes	—
104	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
105	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
106	C₂²×C₁₇	#508	1	2	yes	—
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	—	16	2	no	—
109	C₄×C₁₇	#380	1	1	yes	characteristic
110	C₄×C₁₇	#427	1	1	yes	characteristic
111	C₄×C₁₇	#467	1	2	yes	—
112	Dic₁₇	—	16	2	no	—
113	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
114	C₂²×C₁₆	—	17	1	no	—
115	C₂²×C₁₆	—	17	4	no	—
116	C₂²×C₁₆	—	17	1	no	—
117	C₂²×C₁₆	—	34	4	no	—
118	C₂×#107	—	17	2	no	—
119	C₂×#107	C₂×D₁₇	1	2	yes	—
120	#380	—	2	1	no	—
121	#380	—	34	1	no	—
122	#367	—	34	4	no	—
123	#340	—	2	1	no	—
124	#340	—	34	1	no	—
125	C₂×C₁₇	—	2	4	no	—
126	C₂×C₁₇	#1808	1	1	yes	characteristic
127	C₂×C₁₇	C₂×#367	1	1	yes	characteristic
128	C₂×C₁₇	#2158	1	1	yes	characteristic
129	C₂×C₁₇	#2217	1	2	yes	—
130	C₂×C₁₇	#2217	1	2	yes	—
131	C₂³×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	Dic₃₄	1	2	yes	—
135	C₂²×C₈	C₄×D₁₇	1	2	yes	—
136	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	34	2	no	—
140	C₂×C₂²⋊C₄	—	4	1	no	—
141	C₂×C₂²⋊C₄	—	68	1	no	—
142	C₂×C₁₆	—	17	2	no	—
143	C₂×C₁₆	—	17	4	no	—
144	C₂×C₁₆	—	17	2	no	—
145	C₂×C₁₆	—	34	8	no	—
146	C₂×C₁₆	—	34	2	no	—
147	#107	—	2	2	no	—
148	#107	—	2	2	no	—
149	#107	—	34	2	no	—
150	#107	—	34	2	no	—
151	#95	—	4	1	no	—
152	#95	—	68	1	no	—
153	C₁₇	#12835	1	1	yes	characteristic, complemented, Sylow
154	C₂⁴	#63228	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	68	2	no	—
159	C₂²×C₄	#63225	1	2	yes	—
160	C₂²×C₄	#63228	1	1	yes	characteristic
161	C₂²×C₄	#63231	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	#63225	1	2	yes	—
168	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
169	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
170	C₂×C₈	D₄×D₁₇	1	2	yes	—
171	C₂×C₈	#63254	1	2	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	68	2	no	—
175	C₄⋊C₄	—	68	2	no	—
176	C₂²⋊C₄	—	8	2	no	—
177	C₂²⋊C₄	—	136	2	no	—
178	C₁₆	—	17	4	no	—
179	C₁₆	—	34	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	#10587478	1	2	yes	—
183	C₂³	#10587498	1	2	yes	—
184	C₂³	#10587499	1	2	yes	—
185	C₂³	C₂×#63228	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	136	2	no	—
194	C₂×C₄	—	136	2	no	—
195	C₂×C₄	—	136	2	no	—
196	C₂×C₄	#10587478	1	2	yes	—
197	C₂×C₄	C₂×#63225	1	1	yes	characteristic
198	C₂×C₄	#10587625	1	2	yes	—
199	C₂×C₄	C₂×#63231	1	1	yes	characteristic
200	C₂×C₄	#10587636	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	#10587563	1	1	yes	characteristic
203	C₈	#10587564	1	2	yes	—
204	C₈	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	#11762636	1	4	yes	complemented
210	C₂²	C₂×#10587478	1	1	yes	characteristic
211	C₂²	#11763043	1	2	yes	—
212	C₂²	C₂×#10587499	1	1	yes	characteristic
213	C₂²	C₂×#10587498	1	1	yes	characteristic
214	C₂²	#11763239	1	2	yes	—
215	C₄	—	2	2	no	—
216	C₄	—	16	2	no	—
217	C₄	—	272	2	no	—
218	C₄	#11762667	1	1	yes	characteristic
219	C₄	#11762845	1	1	yes	characteristic
220	C₄	#11763048	1	2	yes	—
221	C₂	—	2	4	no	—
222	C₂	#14528811	1	1	yes	characteristic
223	C₂	C₂×#11762636	1	1	yes	characteristic
224	C₂	#14530566	1	1	yes	characteristic
225	C₂	#14530878	1	2	yes	—
226	C₂	#14530882	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1808	C₂
socle	C₂³×C₁₇	C₂×D₈
prime radical	C₂⁴×C₁₇	D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1808	D₁₇
hypocenter	C₁₇	#12835
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
2. C₂×#11762636	C₂	semidirect
4. #14526875	C₂	semidirect
7. C₂²×#63214	C₄	semidirect
65. #63214	C₂²⋊C₄	semidirect
101. C₂²×C₁₇	#367	semidirect
153. C₁₇	#12835	semidirect
209. C₂²	#11762636	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#63214	2
C₂×#11762636	2
#14528813	2
#14526875	2
C₁₇×#1808	2
#12835	17