Subgroups — #16934461

Order 4352.

430 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537147	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
3	#14526668	C₂	1	2	yes	maximal
4	#14526668	C₂	1	2	yes	maximal
5	C₁₇×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587556	C₄	1	2	yes	—
7	C₂×C₄⋊#3622	C₂²	1	2	yes	—
8	C₂×C₄×#3622	—	2	1	no	—
9	#11762707	—	2	1	no	—
10	#11762707	—	2	1	no	—
11	#11762707	—	2	1	no	—
12	#11762707	—	2	1	no	—
13	#11762679	C₂²	1	1	yes	characteristic
14	#11762679	C₂²	1	1	yes	characteristic
15	#11762675	C₄	1	2	yes	—
16	#11762564	—	2	1	no	—
17	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
18	C₄³×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#336	—	2	1	no	—
20	C₁₇×#336	—	2	1	no	—
21	C₂²×C₈×C₁₇	—	2	2	no	—
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₇	—	2	1	no	—
30	C₂×C₄⋊Dic₁₇	—	2	1	no	—
31	C₂×C₄⋊Dic₁₇	—	2	1	no	—
32	C₂×C₄⋊Dic₁₇	—	2	1	no	—
33	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
35	C₂×#63225	—	2	2	no	—
36	C₂×#63225	—	2	2	no	—
37	C₂²×#3622	—	2	2	no	—
38	#10587556	—	4	4	no	—
39	C₁₇×C₄⋊C₈	—	2	4	no	—
40	#10587502	—	2	2	no	—
41	#10587502	—	2	2	no	—
42	C₄⋊#3622	—	2	2	no	—
43	C₄⋊#3622	Q₈	1	4	yes	—
44	C₄×#3622	—	4	2	no	—
45	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₁₇	—	2	2	no	—
53	C₂²×Dic₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	—	4	2	no	—
55	C₂×C₈×C₁₇	—	4	2	no	—
56	C₂×C₈×C₁₇	—	4	2	no	—
57	C₄²×C₁₇	—	2	2	no	—
58	C₄²×C₁₇	—	2	2	no	—
59	C₄²×C₁₇	—	2	2	no	—
60	C₄²×C₁₇	—	2	2	no	—
61	C₄²×C₁₇	—	2	2	no	—
62	C₄²×C₁₇	—	2	1	no	—
63	C₄²×C₁₇	—	2	1	no	—
64	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
65	C₄²×C₁₇	#41	1	2	yes	—
66	C₄⋊Dic₁₇	—	4	2	no	—
67	C₄⋊Dic₁₇	—	4	2	no	—
68	C₄⋊Dic₁₇	—	4	2	no	—
69	C₄⋊Dic₁₇	—	4	1	no	—
70	C₄⋊Dic₁₇	—	4	1	no	—
71	C₄⋊Dic₁₇	—	4	2	no	—
72	C₄⋊Dic₁₇	—	4	2	no	—
73	#63225	—	4	4	no	—
74	#63225	—	4	4	no	—
75	C₂×#3622	—	2	4	no	—
76	C₂×#3622	—	4	2	no	—
77	C₂×#3622	—	4	2	no	—
78	#9901	—	17	1	no	maximal, Sylow
79	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	—	2	1	no	—
81	C₂×C₄×C₁₇	—	2	1	no	—
82	C₂×C₄×C₁₇	—	2	1	no	—
83	C₂×C₄×C₁₇	—	2	1	no	—
84	C₂×C₄×C₁₇	—	2	1	no	—
85	C₂×C₄×C₁₇	—	2	1	no	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	2	1	no	—
89	C₂×C₄×C₁₇	—	2	1	no	—
90	C₂×C₄×C₁₇	—	2	1	no	—
91	C₂×C₄×C₁₇	—	2	1	no	—
92	C₂×C₄×C₁₇	#100	1	2	yes	—
93	C₂×C₄×C₁₇	#101	1	2	yes	—
94	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
95	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
96	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₇	#135	1	2	yes	—
99	C₂×C₄×C₁₇	#135	1	2	yes	—
100	C₂×C₄×C₁₇	#135	1	2	yes	—
101	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
103	C₂×Dic₁₇	—	4	2	no	—
104	C₂×Dic₁₇	—	4	2	no	—
105	C₂×Dic₁₇	—	4	2	no	—
106	C₂×Dic₁₇	—	4	2	no	—
107	C₂×Dic₁₇	—	4	2	no	—
108	C₂×Dic₁₇	—	4	2	no	—
109	C₈×C₁₇	—	8	2	no	—
110	#3622	—	4	4	no	—
111	#2859	—	17	1	no	—
112	#1831	D₁₇	1	1	yes	characteristic
113	C₄×C₄⋊C₈	—	17	1	no	—
114	#1296	—	17	2	no	—
115	#1296	—	17	2	no	—
116	C₂²×C₁₇	#391	1	1	yes	characteristic
117	C₂²×C₁₇	C₂×#100	1	1	yes	characteristic
118	C₂²×C₁₇	C₂×#101	1	1	yes	characteristic
119	C₂²×C₁₇	#416	1	1	yes	characteristic
120	C₂²×C₁₇	#416	1	1	yes	characteristic
121	C₂²×C₁₇	#416	1	1	yes	characteristic
122	C₂²×C₁₇	#417	1	1	yes	characteristic
123	C₄×C₁₇	—	2	2	no	—
124	C₄×C₁₇	—	2	2	no	—
125	C₄×C₁₇	—	2	2	no	—
126	C₄×C₁₇	—	4	1	no	—
127	C₄×C₁₇	—	4	1	no	—
128	C₄×C₁₇	#478	1	2	yes	—
129	C₄×C₁₇	#479	1	2	yes	—
130	C₄×C₁₇	#488	1	2	yes	—
131	C₄×C₁₇	#499	1	2	yes	—
132	Dic₁₇	—	8	2	no	—
133	Dic₁₇	—	8	2	no	—
134	C₂×#125	—	17	2	no	—
135	C₂×C₄⋊C₈	—	17	2	no	—
136	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
137	C₂×C₄×C₈	—	34	1	no	—
138	#389	—	34	1	no	—
139	#389	—	34	1	no	—
140	#389	—	34	1	no	—
141	#389	—	34	1	no	—
142	#384	—	17	1	no	—
143	#384	—	17	1	no	—
144	#382	—	17	2	no	—
145	C₄³	C₂×D₁₇	1	1	yes	characteristic
146	#336	—	2	1	no	—
147	#336	—	2	1	no	—
148	#336	—	34	1	no	—
149	C₂×C₁₇	#1543	1	1	yes	characteristic
150	C₂×C₁₇	#1544	1	1	yes	characteristic
151	C₂×C₁₇	#1551	1	1	yes	characteristic
152	C₂×C₁₇	#1552	1	1	yes	characteristic
153	C₂×C₁₇	#1989	1	1	yes	characteristic
154	C₂×C₁₇	#1991	1	1	yes	characteristic
155	C₂×C₁₇	#1997	1	1	yes	characteristic
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	34	2	no	—
158	#125	—	68	4	no	—
159	C₂×C₄⋊C₄	—	17	2	no	—
160	C₂×C₄⋊C₄	—	17	2	no	—
161	C₂×C₄⋊C₄	—	34	1	no	—
162	C₂×C₄⋊C₄	—	34	1	no	—
163	C₂×C₄⋊C₄	—	34	1	no	—
164	C₂×C₄⋊C₄	—	34	1	no	—
165	C₂×C₄⋊C₄	—	34	2	no	—
166	C₂×C₄⋊C₄	—	34	2	no	—
167	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
168	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
169	C₂×C₄²	D₆₈	1	1	yes	characteristic
170	C₂×C₄²	D₆₈	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	—	17	4	no	—
176	C₄⋊C₈	—	34	2	no	—
177	C₄×C₈	—	68	2	no	—
178	#95	—	34	2	no	—
179	#95	—	34	2	no	—
180	C₁₇	#9901	1	1	yes	characteristic, complemented, Sylow
181	C₂²×C₄	—	34	2	no	—
182	C₂²×C₄	—	34	2	no	—
183	C₂²×C₄	#63227	1	1	yes	characteristic
184	C₂²×C₄	#63227	1	1	yes	characteristic
185	C₂²×C₄	#63227	1	1	yes	characteristic
186	C₂²×C₄	#63227	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	34	4	no	—
194	C₂×C₈	—	68	2	no	—
195	C₂×C₈	—	68	2	no	—
196	C₄⋊C₄	—	68	2	no	—
197	C₄⋊C₄	—	68	2	no	—
198	C₄⋊C₄	—	68	4	no	—
199	C₄⋊C₄	—	68	4	no	—
200	C₄⋊C₄	—	68	2	no	—
201	C₄⋊C₄	—	68	1	no	—
202	C₄⋊C₄	—	68	1	no	—
203	C₄⋊C₄	—	68	2	no	—
204	C₄⋊C₄	—	68	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	#63254	1	2	yes	—
213	C₄²	Q₈×D₁₇	1	2	yes	—
214	C₂³	C₂×#63227	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	68	2	no	—
228	C₂×C₄	—	68	2	no	—
229	C₂×C₄	—	68	2	no	—
230	C₂×C₄	—	68	2	no	—
231	C₂×C₄	—	68	2	no	—
232	C₂×C₄	—	68	2	no	—
233	C₂×C₄	#10587494	1	2	yes	—
234	C₂×C₄	#10587495	1	2	yes	—
235	C₂×C₄	#10587573	1	2	yes	—
236	C₂×C₄	#10587583	1	2	yes	—
237	C₂×C₄	#10587590	1	2	yes	—
238	C₂×C₄	#10587594	1	1	yes	characteristic
239	C₂×C₄	#10587595	1	1	yes	characteristic
240	C₂×C₄	#10587622	1	1	yes	characteristic
241	C₂×C₄	#10587630	1	1	yes	characteristic
242	C₂×C₄	#10587636	1	2	yes	—
243	C₂×C₄	#10587636	1	2	yes	—
244	C₈	—	8	2	no	—
245	C₈	—	68	4	no	—
246	C₂²	#11762714	1	1	yes	characteristic
247	C₂²	C₂×#10587494	1	1	yes	characteristic
248	C₂²	C₂×#10587495	1	1	yes	characteristic
249	C₂²	#11762803	1	1	yes	characteristic
250	C₂²	#11762803	1	1	yes	characteristic
251	C₂²	#11762806	1	1	yes	characteristic
252	C₂²	#11762811	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	136	2	no	—
259	C₄	—	136	2	no	—
260	C₄	#11763110	1	2	yes	—
261	C₄	#11763117	1	2	yes	—
262	C₄	#11763163	1	2	yes	—
263	C₄	#11763208	1	2	yes	—
264	C₂	#14527444	1	1	yes	characteristic
265	C₂	#14527448	1	1	yes	characteristic
266	C₂	#14527480	1	1	yes	characteristic
267	C₂	#14527485	1	1	yes	characteristic
268	C₂	#14529739	1	1	yes	characteristic
269	C₂	#14529748	1	1	yes	characteristic
270	C₂	#14529772	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#63227
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1831	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1831	D₁₇
hypocenter	C₁₇	#9901
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
180. C₁₇	#9901	semidirect

Maximal subgroups

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

subgroup	index
#14537147	2
C₄×C₄⋊#3622	2
#14526668	2
C₁₇×#1831	2
#9901	17