Subgroups — #16927265

Order 4352.

378 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528780	C₂	1	2	yes	maximal
2	#14527543	C₂	1	1	yes	maximal, characteristic
3	#14527540	C₂	1	1	yes	maximal, characteristic
4	#14526659	C₂	1	2	yes	maximal
5	C₂×C₁₇×#446	C₂	1	1	yes	maximal, characteristic
6	#11762833	C₂²	1	2	yes	—
7	#11762833	C₂²	1	2	yes	—
8	C₂×#10587507	—	2	2	no	—
9	#11762578	—	2	2	no	—
10	#11762562	—	2	1	no	—
11	#11762562	—	2	1	no	—
12	#11762560	—	2	1	no	—
13	#11762560	—	2	1	no	—
14	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#446	—	2	1	no	—
16	C₁₇×#446	—	2	1	no	—
17	C₁₇×#446	—	2	1	no	—
18	C₁₇×#446	—	2	1	no	—
19	C₂×C₁₇×C₄⋊C₈	C₄	1	2	yes	—
20	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₇×#97	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Q₈×C₁₇	—	2	1	no	—
28	C₄×Q₈×C₁₇	—	2	1	no	—
29	C₄×Q₈×C₁₇	—	2	1	no	—
30	C₄×Q₈×C₁₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₇	Q₈	1	2	yes	—
33	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×#63223	—	2	2	no	—
36	C₂×#63223	—	2	2	no	—
37	C₁₇×C₄⋊C₈	—	2	2	no	—
38	C₁₇×C₄⋊C₈	—	2	2	no	—
39	C₁₇×C₄⋊C₈	—	2	1	no	—
40	C₁₇×C₄⋊C₈	—	2	1	no	—
41	C₁₇×#97	—	2	2	no	—
42	C₁₇×#97	—	2	2	no	—
43	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	#10587507	—	4	4	no	—
48	C₄⋊#3622	—	2	2	no	—
49	C₄⋊#3622	—	2	2	no	—
50	C₄⋊#3622	—	2	2	no	—
51	C₄⋊#3622	—	2	2	no	—
52	C₂×Q₈×C₁₇	—	2	1	no	—
53	C₂×Q₈×C₁₇	—	2	1	no	—
54	C₂×Q₈×C₁₇	—	2	2	no	—
55	C₂×Q₈×C₁₇	—	2	2	no	—
56	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₁₇	—	2	2	no	—
62	C₂×C₈×C₁₇	—	2	2	no	—
63	C₂×C₈×C₁₇	—	4	1	no	—
64	C₂×C₈×C₁₇	—	4	2	no	—
65	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	—	2	2	no	—
72	C₄⋊C₄×C₁₇	—	2	2	no	—
73	C₄²×C₁₇	—	2	2	no	—
74	C₄²×C₁₇	—	2	2	no	—
75	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₇	#41	1	1	yes	characteristic
78	C₄²×C₁₇	#41	1	1	yes	characteristic
79	#63223	—	4	2	no	—
80	#63223	—	4	2	no	—
81	#63223	—	4	2	no	—
82	#63223	—	4	2	no	—
83	C₂×#3622	—	4	2	no	—
84	C₂×#3622	—	4	2	no	—
85	#8619	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	Q₈×C₁₇	—	4	2	no	—
88	Q₈×C₁₇	—	4	1	no	—
89	Q₈×C₁₇	—	4	1	no	—
90	C₂×C₄×C₁₇	—	2	1	no	—
91	C₂×C₄×C₁₇	—	2	1	no	—
92	C₂×C₄×C₁₇	—	2	2	no	—
93	C₂×C₄×C₁₇	—	2	2	no	—
94	C₂×C₄×C₁₇	—	2	4	no	—
95	C₂×C₄×C₁₇	—	4	1	no	—
96	C₂×C₄×C₁₇	—	4	2	no	—
97	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄×C₁₇	#137	1	2	yes	—
105	C₈×C₁₇	—	2	1	no	—
106	C₈×C₁₇	—	2	1	no	—
107	C₈×C₁₇	—	2	1	no	—
108	C₈×C₁₇	—	4	2	no	—
109	C₈×C₁₇	#101	1	2	yes	complemented
110	#3622	—	8	2	no	—
111	#3622	—	8	2	no	—
112	C₂×#446	D₁₇	1	1	yes	characteristic
113	#1801	—	17	2	no	—
114	#1567	—	17	1	no	—
115	#1566	—	17	1	no	—
116	#1292	—	17	2	no	—
117	C₂²×C₁₇	—	2	1	no	—
118	C₂²×C₁₇	—	2	1	no	—
119	C₂²×C₁₇	#380	1	1	yes	characteristic
120	C₂²×C₁₇	#428	1	2	yes	—
121	C₄×C₁₇	—	2	1	no	—
122	C₄×C₁₇	—	2	1	no	—
123	C₄×C₁₇	—	2	1	no	—
124	C₄×C₁₇	—	4	2	no	—
125	C₄×C₁₇	—	4	2	no	—
126	C₄×C₁₇	C₂×#101	1	1	yes	characteristic
127	C₄×C₁₇	#413	1	1	yes	characteristic
128	C₄×C₁₇	#468	1	2	yes	—
129	C₄×C₁₇	#469	1	1	yes	characteristic
130	C₄×C₁₇	#470	1	1	yes	characteristic
131	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
132	#446	—	2	1	no	—
133	#446	—	2	1	no	—
134	#446	—	2	1	no	—
135	#446	—	2	1	no	—
136	#423	—	17	2	no	—
137	#423	—	17	2	no	—
138	C₂×C₄⋊C₈	Dic₁₇	1	2	yes	—
139	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
140	C₂×#101	—	34	2	no	—
141	C₂×#97	Dic₁₇	1	2	yes	—
142	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
143	#343	—	34	2	no	—
144	#335	—	34	1	no	—
145	#335	—	34	1	no	—
146	#334	—	34	1	no	—
147	#334	—	34	1	no	—
148	C₂×C₁₇	—	2	2	no	—
149	C₂×C₁₇	#1801	1	1	yes	characteristic
150	C₂×C₁₇	#1809	1	1	yes	characteristic
151	C₂×C₁₇	#1813	1	1	yes	characteristic
152	C₂²×Q₈	D₆₈	1	1	yes	characteristic
153	C₂×M₁₆	—	34	2	no	—
154	C₂×M₁₆	—	34	2	no	—
155	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
156	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
157	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
158	C₄×Q₈	—	2	1	no	—
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₂×C₄⋊C₄	Dic₃₄	1	2	yes	—
163	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
165	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	34	2	no	—
171	C₄⋊C₈	—	34	2	no	—
172	C₄⋊C₈	—	34	2	no	—
173	C₄⋊C₈	—	34	2	no	—
174	#101	—	68	4	no	—
175	#97	—	2	2	no	—
176	#97	—	2	2	no	—
177	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
181	C₁₇	#8619	1	1	yes	characteristic, complemented, Sylow
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	—	2	1	no	—
186	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
188	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
191	M₁₆	—	68	2	no	—
192	M₁₆	—	68	2	no	—
193	M₁₆	—	68	2	no	—
194	M₁₆	—	68	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	68	2	no	—
200	C₂×C₈	—	68	2	no	—
201	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	#63252	1	1	yes	characteristic
212	C₄²	#63252	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
215	C₂³	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
216	Q₈	—	4	2	no	—
217	Q₈	—	4	1	no	—
218	Q₈	—	4	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	#10587594	1	2	yes	—
227	C₂×C₄	#10587595	1	2	yes	—
228	C₂×C₄	#10587610	1	1	yes	characteristic
229	C₂×C₄	#10587610	1	1	yes	characteristic
230	C₂×C₄	#10587612	1	1	yes	characteristic
231	C₂×C₄	#10587616	1	1	yes	characteristic
232	C₂×C₄	#10587616	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	4	2	no	—
238	C₈	—	136	2	no	—
239	C₈	—	136	2	no	—
240	C₈	#10587507	1	2	yes	complemented
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#11762669	1	1	yes	characteristic
244	C₂²	#11762851	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	C₂×#10587507	1	1	yes	characteristic
251	C₄	#11762793	1	1	yes	characteristic
252	C₄	#11763055	1	2	yes	—
253	C₄	#11763062	1	1	yes	characteristic
254	C₄	#11763067	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	#14528780	1	1	yes	characteristic
257	C₂	#14528817	1	1	yes	characteristic
258	C₂	#14528834	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	#8619
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
109. C₈×C₁₇	#101	semidirect
181. C₁₇	#8619	semidirect
240. C₈	#10587507	semidirect

Maximal subgroups

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

subgroup	index
#14528780	2
#14527543	2
#14527540	2
#14526659	2
C₂×C₁₇×#446	2
#8619	17