Subgroups — #16927291

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14527547	C₂	1	1	yes	maximal, characteristic
2	#14527547	C₂	1	1	yes	maximal, characteristic
3	#14527128	C₂	1	2	yes	maximal
4	#14526654	C₂	1	2	yes	maximal
5	C₂×C₁₇×#446	C₂	1	1	yes	maximal, characteristic
6	#11762866	C₂²	1	2	yes	—
7	#11762866	C₂²	1	2	yes	—
8	#11762562	—	2	1	no	—
9	#11762562	—	2	1	no	—
10	#11762560	—	2	1	no	—
11	#11762560	—	2	1	no	—
12	#11762535	—	2	4	no	—
13	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#446	—	2	1	no	—
15	C₁₇×#446	—	2	1	no	—
16	C₁₇×#446	—	2	1	no	—
17	C₁₇×#446	—	2	1	no	—
18	C₂×C₁₇×C₄⋊C₈	C₄	1	2	yes	—
19	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₇×#97	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₁₇	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₇	—	2	1	no	—
27	C₄×Q₈×C₁₇	—	2	1	no	—
28	C₄×Q₈×C₁₇	—	2	1	no	—
29	C₄×Q₈×C₁₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₇	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₇	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	C₁₇×C₄⋊C₈	—	2	2	no	—
35	C₁₇×C₄⋊C₈	—	2	2	no	—
36	C₁₇×C₄⋊C₈	—	2	1	no	—
37	C₁₇×C₄⋊C₈	—	2	1	no	—
38	C₁₇×#97	—	2	2	no	—
39	C₁₇×#97	—	2	2	no	—
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	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
44	C₂²⋊#3622	—	2	2	no	—
45	C₂²⋊#3622	—	2	2	no	—
46	C₄⋊#3622	—	2	2	no	—
47	C₄⋊#3622	—	2	2	no	—
48	#10587474	—	2	2	no	—
49	#10587474	—	2	2	no	—
50	C₂×Q₈×C₁₇	—	2	1	no	—
51	C₂×Q₈×C₁₇	—	2	1	no	—
52	C₂×Q₈×C₁₇	—	4	2	no	—
53	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₁₇	—	2	2	no	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	—	4	1	no	—
61	C₂×C₈×C₁₇	—	4	2	no	—
62	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₇	—	2	2	no	—
67	C₄⋊C₄×C₁₇	—	2	2	no	—
68	C₄⋊C₄×C₁₇	—	2	1	no	—
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄²×C₁₇	—	2	2	no	—
71	C₄²×C₁₇	—	2	2	no	—
72	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₇	#41	1	1	yes	characteristic
75	C₄²×C₁₇	#41	1	1	yes	characteristic
76	C₂×#3622	—	4	2	no	—
77	C₂×#3622	—	4	2	no	—
78	#8625	—	17	1	no	maximal, Sylow
79	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	Q₈×C₁₇	—	4	2	no	—
81	Q₈×C₁₇	—	4	2	no	—
82	C₂×C₄×C₁₇	—	2	2	no	—
83	C₂×C₄×C₁₇	—	2	1	no	—
84	C₂×C₄×C₁₇	—	2	1	no	—
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	—	2	2	no	—
87	C₂×C₄×C₁₇	—	2	4	no	—
88	C₂×C₄×C₁₇	—	4	1	no	—
89	C₂×C₄×C₁₇	—	4	2	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	C₈×C₁₇	—	2	1	no	—
97	C₈×C₁₇	—	2	1	no	—
98	C₈×C₁₇	—	2	1	no	—
99	C₈×C₁₇	—	4	2	no	—
100	C₈×C₁₇	#101	1	2	yes	—
101	#3622	—	8	2	no	—
102	#3622	—	8	2	no	—
103	C₂×#446	D₁₇	1	1	yes	characteristic
104	#1568	—	17	1	no	—
105	#1568	—	17	1	no	—
106	#1467	—	17	2	no	—
107	#1290	—	17	2	no	—
108	C₂²×C₁₇	—	2	1	no	—
109	C₂²×C₁₇	—	2	1	no	—
110	C₂²×C₁₇	#380	1	1	yes	characteristic
111	C₂²×C₁₇	#430	1	2	yes	—
112	C₄×C₁₇	—	2	1	no	—
113	C₄×C₁₇	—	2	1	no	—
114	C₄×C₁₇	—	2	1	no	—
115	C₄×C₁₇	—	2	1	no	—
116	C₄×C₁₇	—	2	1	no	—
117	C₄×C₁₇	—	4	2	no	—
118	C₄×C₁₇	—	4	2	no	—
119	C₄×C₁₇	C₂×#101	1	1	yes	characteristic
120	C₄×C₁₇	#413	1	1	yes	characteristic
121	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
122	#446	—	2	1	no	—
123	#446	—	2	1	no	—
124	#446	—	2	1	no	—
125	#446	—	2	1	no	—
126	#432	—	17	2	no	—
127	#432	—	17	2	no	—
128	C₂×C₄⋊C₈	Dic₁₇	1	2	yes	—
129	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
130	C₂×#97	Dic₁₇	1	2	yes	—
131	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
132	#335	—	34	1	no	—
133	#335	—	34	1	no	—
134	#334	—	34	1	no	—
135	#334	—	34	1	no	—
136	#324	—	34	4	no	—
137	C₂×C₁₇	—	2	2	no	—
138	C₂×C₁₇	#1801	1	1	yes	characteristic
139	C₂×C₁₇	#1811	1	1	yes	characteristic
140	C₂×C₁₇	#1812	1	1	yes	characteristic
141	C₂²×Q₈	Dic₃₄	1	1	yes	characteristic
142	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
144	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	—	2	1	no	—
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	—	2	1	no	—
149	C₂×C₄⋊C₄	Dic₃₄	1	1	yes	characteristic
150	C₂×C₄⋊C₄	D₆₈	1	2	yes	—
151	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
152	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	34	2	no	—
158	C₄⋊C₈	—	34	2	no	—
159	C₂²⋊C₈	—	34	2	no	—
160	C₂²⋊C₈	—	34	2	no	—
161	#97	—	2	2	no	—
162	#97	—	2	2	no	—
163	#97	—	34	2	no	—
164	#97	—	34	2	no	—
165	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
169	C₁₇	#8625	1	1	yes	characteristic, complemented, Sylow
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
174	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
175	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
177	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	68	2	no	—
183	C₂×C₈	—	68	2	no	—
184	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	#63252	1	1	yes	characteristic
195	C₄²	#63252	1	1	yes	characteristic
196	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
198	C₂³	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
199	Q₈	—	4	2	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	#10587610	1	1	yes	characteristic
210	C₂×C₄	#10587610	1	1	yes	characteristic
211	C₂×C₄	#10587612	1	1	yes	characteristic
212	C₂×C₄	#10587616	1	1	yes	characteristic
213	C₂×C₄	#10587616	1	1	yes	characteristic
214	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	4	2	no	—
219	C₈	—	136	2	no	—
220	C₈	—	136	2	no	—
221	C₈	#10587507	1	2	yes	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	#11762669	1	1	yes	characteristic
225	C₂²	#11762858	1	2	yes	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₂×#10587507	1	1	yes	characteristic
234	C₄	#11762793	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	#14528780	1	1	yes	characteristic
237	C₂	#14528826	1	1	yes	characteristic
238	C₂	#14528830	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762669
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#446	C₂
socle	C₂²×C₁₇	#380
prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#446	D₁₇
hypocenter	C₁₇	#8625
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
169. C₁₇	#8625	semidirect

Maximal subgroups

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

subgroup	index
#14527547	2
#14527128	2
#14526654	2
C₂×C₁₇×#446	2
#8625	17