Subgroups — #16927147

Order 4352.

476 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587555	C₂	1	1	yes	maximal, characteristic
2	#14528901	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762554	C₂	1	2	yes	maximal
4	#14526668	C₂	1	2	yes	maximal
5	C₁₇×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587555	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₃₄	—	2	1	no	—
8	C₂×C₄⋊#3622	C₂²	1	2	yes	—
9	#11762715	—	2	1	no	—
10	#11762707	—	2	1	no	—
11	#11762707	C₄	1	2	yes	—
12	#11762679	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₁₇	—	2	1	no	—
14	C₄×C₄⋊Dic₁₇	C₄	1	2	yes	—
15	#11762564	—	2	1	no	—
16	#11762564	—	2	1	no	—
17	#11762554	—	2	4	no	—
18	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
19	C₄³×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×#336	—	2	1	no	—
21	C₁₇×#336	—	2	1	no	—
22	C₂²×Dic₃₄	—	2	1	no	—
23	C₂²×C₈×C₁₇	—	2	2	no	—
24	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	C₂×C₄	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	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₇	—	2	1	no	—
32	C₂×C₄⋊Dic₁₇	—	2	1	no	—
33	C₂×C₄⋊Dic₁₇	—	2	1	no	—
34	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
36	C₂×#63225	—	2	2	no	—
37	C₂×C₄×Dic₁₇	—	2	2	no	—
38	C₂²×#3622	—	2	2	no	—
39	#10587555	—	2	1	no	—
40	#10587555	—	2	1	no	—
41	#10587555	D₄	1	2	yes	—
42	#10587555	Q₈	1	2	yes	—
43	C₄×Dic₃₄	—	4	4	no	—
44	C₁₇×C₄⋊C₈	—	2	2	no	—
45	C₁₇×C₄⋊C₈	—	2	2	no	—
46	#10587502	—	2	2	no	—
47	#10587502	—	2	2	no	—
48	C₄⋊#3622	—	2	2	no	—
49	C₄⋊#3622	—	2	2	no	—
50	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₇	—	2	2	no	—
58	C₂²×Dic₁₇	—	2	2	no	—
59	C₂×Dic₃₄	—	2	2	no	—
60	C₂×Dic₃₄	—	2	2	no	—
61	C₂×Dic₃₄	—	4	2	no	—
62	C₂×Dic₃₄	—	4	2	no	—
63	C₂×C₈×C₁₇	—	4	2	no	—
64	C₂×C₈×C₁₇	—	4	2	no	—
65	C₂×C₈×C₁₇	—	4	2	no	—
66	C₄²×C₁₇	—	2	2	no	—
67	C₄²×C₁₇	—	2	2	no	—
68	C₄²×C₁₇	—	2	2	no	—
69	C₄²×C₁₇	—	2	2	no	—
70	C₄²×C₁₇	—	2	2	no	—
71	C₄²×C₁₇	—	4	1	no	—
72	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₁₇	#41	1	1	yes	characteristic
75	C₄²×C₁₇	#41	1	1	yes	characteristic
76	C₄⋊Dic₁₇	—	2	2	no	—
77	C₄⋊Dic₁₇	—	2	2	no	—
78	C₄⋊Dic₁₇	—	2	2	no	—
79	C₄⋊Dic₁₇	—	4	2	no	—
80	C₄⋊Dic₁₇	—	4	2	no	—
81	C₄⋊Dic₁₇	—	4	2	no	—
82	C₄⋊Dic₁₇	C₄⋊C₄	1	4	yes	—
83	#63225	—	4	4	no	—
84	C₄×Dic₁₇	—	4	4	no	—
85	C₂×#3622	—	4	2	no	—
86	C₂×#3622	—	4	2	no	—
87	C₂×#3622	—	4	2	no	—
88	#8593	—	17	1	no	maximal, Sylow
89	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	—	2	1	no	—
91	C₂×C₄×C₁₇	—	2	1	no	—
92	C₂×C₄×C₁₇	—	2	1	no	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	—	2	1	no	—
98	C₂×C₄×C₁₇	—	2	1	no	—
99	C₂×C₄×C₁₇	—	2	1	no	—
100	C₂×C₄×C₁₇	—	2	1	no	—
101	C₂×C₄×C₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	#101	1	2	yes	—
104	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#135	1	2	yes	—
111	C₂×C₄×C₁₇	#135	1	2	yes	—
112	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₁₇	#137	1	2	yes	—
114	C₂×Dic₁₇	—	2	4	no	—
115	C₂×Dic₁₇	—	4	2	no	—
116	C₂×Dic₁₇	—	4	2	no	—
117	C₂×Dic₁₇	—	4	2	no	—
118	C₂×Dic₁₇	—	4	2	no	—
119	C₂×Dic₁₇	—	4	2	no	—
120	Dic₃₄	—	4	4	no	—
121	Dic₃₄	—	4	4	no	—
122	C₈×C₁₇	—	8	2	no	—
123	#3622	—	8	2	no	—
124	C₄×C₄⋊Q₈	—	17	1	no	—
125	#1831	—	17	1	no	—
126	#1831	D₁₇	1	1	yes	characteristic
127	C₂×#332	—	17	2	no	—
128	#1296	—	17	2	no	—
129	C₂²×C₁₇	#380	1	1	yes	characteristic
130	C₂²×C₁₇	C₂×#101	1	1	yes	characteristic
131	C₂²×C₁₇	#413	1	1	yes	characteristic
132	C₂²×C₁₇	#416	1	1	yes	characteristic
133	C₂²×C₁₇	#416	1	1	yes	characteristic
134	C₂²×C₁₇	#418	1	1	yes	characteristic
135	C₂²×C₁₇	#419	1	1	yes	characteristic
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	2	no	—
139	C₄×C₁₇	—	4	1	no	—
140	C₄×C₁₇	—	4	1	no	—
141	C₄×C₁₇	—	4	1	no	—
142	C₄×C₁₇	#463	1	2	yes	—
143	C₄×C₁₇	#464	1	2	yes	—
144	C₄×C₁₇	#478	1	2	yes	—
145	C₄×C₁₇	#479	1	2	yes	—
146	Dic₁₇	—	4	4	no	—
147	Dic₁₇	—	8	2	no	—
148	C₂×C₄⋊Q₈	—	17	1	no	—
149	C₂×C₄×Q₈	—	34	1	no	—
150	C₂×C₄⋊C₈	—	17	2	no	—
151	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
152	#391	—	34	1	no	—
153	#389	—	17	2	no	—
154	#389	—	34	1	no	—
155	#384	—	17	1	no	—
156	C₄×C₄⋊C₄	—	17	2	no	—
157	C₄×C₄⋊C₄	—	34	1	no	—
158	C₄³	C₂×D₁₇	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	#336	—	2	1	no	—
161	#336	—	34	1	no	—
162	#336	—	34	1	no	—
163	#332	—	34	4	no	—
164	C₂×C₁₇	#1544	1	1	yes	characteristic
165	C₂×C₁₇	#1553	1	1	yes	characteristic
166	C₂×C₁₇	#1554	1	1	yes	characteristic
167	C₂×C₁₇	#1556	1	1	yes	characteristic
168	C₂×C₁₇	#1801	1	1	yes	characteristic
169	C₂×C₁₇	#1804	1	1	yes	characteristic
170	C₂×C₁₇	#1805	1	1	yes	characteristic
171	C₂²×Q₈	—	34	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	34	2	no	—
174	C₄⋊Q₈	—	17	2	no	—
175	C₄⋊Q₈	—	17	2	no	—
176	C₄⋊Q₈	—	34	1	no	—
177	C₄⋊Q₈	—	34	1	no	—
178	C₄×Q₈	—	68	4	no	—
179	C₂×C₄⋊C₄	—	17	2	no	—
180	C₂×C₄⋊C₄	—	17	2	no	—
181	C₂×C₄⋊C₄	—	34	1	no	—
182	C₂×C₄⋊C₄	—	34	1	no	—
183	C₂×C₄⋊C₄	—	34	1	no	—
184	C₂×C₄⋊C₄	—	34	2	no	—
185	C₂×C₄²	—	34	2	no	—
186	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
187	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
188	C₂×C₄²	D₆₈	1	1	yes	characteristic
189	C₂×C₄²	D₆₈	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	34	2	no	—
196	C₄⋊C₈	—	34	2	no	—
197	#95	—	34	2	no	—
198	#95	—	34	2	no	—
199	C₁₇	#8593	1	1	yes	characteristic, complemented, Sylow
200	C₂×Q₈	—	34	2	no	—
201	C₂×Q₈	—	34	2	no	—
202	C₂×Q₈	—	68	2	no	—
203	C₂×Q₈	—	68	2	no	—
204	C₂²×C₄	—	34	2	no	—
205	C₂²×C₄	—	34	2	no	—
206	C₂²×C₄	#63227	1	1	yes	characteristic
207	C₂²×C₄	#63227	1	1	yes	characteristic
208	C₂²×C₄	#63227	1	1	yes	characteristic
209	C₂²×C₄	#63227	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	68	2	no	—
217	C₂×C₈	—	68	2	no	—
218	C₂×C₈	—	68	2	no	—
219	C₄⋊C₄	—	17	4	no	—
220	C₄⋊C₄	—	34	2	no	—
221	C₄⋊C₄	—	34	2	no	—
222	C₄⋊C₄	—	34	2	no	—
223	C₄⋊C₄	—	68	2	no	—
224	C₄⋊C₄	—	68	4	no	—
225	C₄⋊C₄	—	68	2	no	—
226	C₄⋊C₄	—	68	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	68	4	no	—
234	C₄²	D₄×D₁₇	1	1	yes	characteristic
235	C₄²	#63254	1	1	yes	characteristic
236	C₄²	Q₈×D₁₇	1	1	yes	characteristic
237	C₄²	#63256	1	1	yes	characteristic
238	C₂³	C₂×#63227	1	1	yes	characteristic
239	Q₈	—	68	4	no	—
240	Q₈	—	68	4	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	34	4	no	—
255	C₂×C₄	—	68	2	no	—
256	C₂×C₄	—	68	2	no	—
257	C₂×C₄	—	68	2	no	—
258	C₂×C₄	—	68	2	no	—
259	C₂×C₄	—	68	2	no	—
260	C₂×C₄	#10587495	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
262	C₂×C₄	#10587578	1	1	yes	characteristic
263	C₂×C₄	#10587581	1	1	yes	characteristic
264	C₂×C₄	#10587583	1	1	yes	characteristic
265	C₂×C₄	#10587590	1	2	yes	—
266	C₂×C₄	#10587594	1	2	yes	—
267	C₂×C₄	#10587624	1	1	yes	characteristic
268	C₂×C₄	#10587632	1	1	yes	characteristic
269	C₂×C₄	#10587636	1	2	yes	—
270	C₂×C₄	#10587637	1	2	yes	—
271	C₈	—	8	2	no	—
272	C₈	—	136	2	no	—
273	C₂²	#11762666	1	1	yes	characteristic
274	C₂²	C₂×#10587495	1	1	yes	characteristic
275	C₂²	#11762791	1	1	yes	characteristic
276	C₂²	#11762803	1	1	yes	characteristic
277	C₂²	#11762806	1	1	yes	characteristic
278	C₂²	#11762815	1	1	yes	characteristic
279	C₂²	#11762817	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	68	4	no	—
287	C₄	—	136	2	no	—
288	C₄	#11763025	1	2	yes	—
289	C₄	#11763028	1	2	yes	—
290	C₄	#11763110	1	2	yes	—
291	C₄	#11763113	1	2	yes	—
292	C₂	#14527448	1	1	yes	characteristic
293	C₂	#14527490	1	1	yes	characteristic
294	C₂	#14527492	1	1	yes	characteristic
295	C₂	#14527500	1	1	yes	characteristic
296	C₂	#14528777	1	1	yes	characteristic
297	C₂	#14528795	1	1	yes	characteristic
298	C₂	#14528796	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	#8593
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
199. C₁₇	#8593	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587555	2
#14528901	2
C₂×#11762554	2
#14526668	2
C₁₇×#1831	2
#8593	17