Subgroups — #16950406

Order 4352.

410 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762626	C₂	1	2	yes	maximal
3	#14528956	C₂	1	1	yes	maximal, characteristic
4	#14526893	C₂	1	2	yes	maximal
5	C₁₇×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₃₄	—	2	1	no	—
8	C₄×Dic₆₈	—	2	1	no	—
9	C₄×Dic₆₈	—	2	1	no	—
10	C₄×Dic₆₈	C₄	1	2	yes	—
11	C₄×Dic₆₈	C₄	1	2	yes	—
12	C₂×#10587490	—	2	1	no	—
13	C₂×#10587490	C₂²	1	2	yes	—
14	C₂×#10587489	C₂²	1	1	yes	characteristic
15	#11762833	—	2	1	no	—
16	C₂×#10587487	—	2	1	no	—
17	#11762626	—	2	4	no	—
18	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×C₄⋊C₁₆	—	2	1	no	—
21	C₁₇×C₄⋊C₁₆	—	2	1	no	—
22	C₂²×Dic₃₄	—	2	1	no	—
23	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₇	—	2	1	no	—
27	C₂×#63225	—	4	1	no	—
28	C₂×C₄×Dic₁₇	—	4	1	no	—
29	C₂×#63223	—	2	2	no	—
30	C₂×Dic₆₈	—	2	1	no	—
31	C₂×Dic₆₈	—	2	1	no	—
32	C₂×Dic₆₈	—	2	1	no	—
33	C₂×Dic₆₈	C₂×C₄	1	2	yes	—
34	C₂×Dic₆₈	D₄	1	2	yes	—
35	C₂×Dic₆₈	Q₈	1	2	yes	—
36	C₄×Dic₃₄	—	2	2	no	—
37	C₄×Dic₃₄	—	2	2	no	—
38	C₄×Dic₃₄	—	4	1	no	—
39	C₄×Dic₃₄	—	4	1	no	—
40	C₁₇×M₃₂	—	2	2	no	—
41	C₁₇×M₃₂	—	2	2	no	—
42	C₂×C₁₆×C₁₇	—	2	2	no	—
43	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	#10587490	—	2	2	no	—
48	#10587490	—	2	2	no	—
49	#10587490	—	4	2	no	—
50	#10587489	—	2	1	no	—
51	#10587489	C₂×C₄	1	2	yes	—
52	#10587487	—	2	2	no	—
53	#10587487	—	2	2	no	—
54	C₄⋊#3622	—	4	1	no	—
55	C₄⋊#3622	—	4	1	no	—
56	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₇	—	4	1	no	—
60	C₂²×Dic₁₇	—	4	1	no	—
61	C₂×Dic₃₄	—	2	2	no	—
62	C₂×Dic₃₄	—	2	2	no	—
63	C₂×Dic₃₄	—	2	2	no	—
64	C₂×Dic₃₄	—	4	1	no	—
65	C₂×Dic₃₄	—	4	1	no	—
66	C₂×Dic₃₄	—	4	1	no	—
67	C₂×C₈×C₁₇	—	2	1	no	—
68	C₂×C₈×C₁₇	—	2	1	no	—
69	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
76	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
77	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₁₇	D₈	1	1	yes	characteristic
80	C₄²×C₁₇	D₈	1	1	yes	characteristic
81	C₄⋊Dic₁₇	—	2	2	no	—
82	C₄⋊Dic₁₇	—	4	1	no	—
83	#63225	—	4	2	no	—
84	#63225	—	4	2	no	—
85	C₄×Dic₁₇	—	4	2	no	—
86	C₄×Dic₁₇	—	4	2	no	—
87	#63223	—	4	2	no	—
88	#63223	—	4	2	no	—
89	C₂×#3622	—	4	2	no	—
90	Dic₆₈	—	2	2	no	—
91	Dic₆₈	—	2	2	no	—
92	Dic₆₈	—	2	2	no	—
93	Dic₆₈	C₄⋊C₄	1	4	yes	—
94	C₁₆×C₁₇	—	4	2	no	—
95	#12819	—	17	1	no	maximal, Sylow
96	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₇	—	2	2	no	—
98	C₂×C₄×C₁₇	—	4	1	no	—
99	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
100	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
109	C₂×Dic₁₇	—	4	2	no	—
110	C₂×Dic₁₇	—	4	2	no	—
111	C₂×Dic₁₇	—	4	2	no	—
112	C₂×Dic₁₇	—	4	2	no	—
113	C₂×Dic₁₇	—	8	2	no	—
114	Dic₃₄	—	2	4	no	—
115	Dic₃₄	—	4	2	no	—
116	Dic₃₄	—	4	2	no	—
117	Dic₃₄	—	4	2	no	—
118	C₈×C₁₇	—	2	1	no	—
119	C₈×C₁₇	—	2	1	no	—
120	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₈×C₁₇	#117	1	1	yes	characteristic
122	C₈×C₁₇	#121	1	1	yes	characteristic
123	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
124	#3622	—	8	2	no	—
125	C₂×C₄×Q₁₆	—	17	1	no	—
126	#2150	D₁₇	1	1	yes	characteristic
127	C₂×#362	—	17	2	no	—
128	#1848	—	17	1	no	—
129	#1387	—	17	2	no	—
130	C₂²×C₁₇	—	2	1	no	—
131	C₂²×C₁₇	—	2	1	no	—
132	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
133	C₂²×C₁₇	#461	1	1	yes	characteristic
134	C₂²×C₁₇	#476	1	1	yes	characteristic
135	C₄×C₁₇	—	2	1	no	—
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	#362	1	2	yes	complemented
139	C₄×C₁₇	#380	1	1	yes	characteristic
140	C₄×C₁₇	#417	1	1	yes	characteristic
141	C₄×C₁₇	#459	1	1	yes	characteristic
142	C₄×C₁₇	#474	1	1	yes	characteristic
143	Dic₁₇	—	4	4	no	—
144	Dic₁₇	—	8	2	no	—
145	C₂²×Q₁₆	—	17	1	no	—
146	C₂×C₄×Q₈	—	34	1	no	—
147	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
148	C₄×Q₁₆	—	17	2	no	—
149	C₄×Q₁₆	—	17	2	no	—
150	C₄×Q₁₆	—	34	1	no	—
151	C₄×Q₁₆	—	34	1	no	—
152	C₂×#108	—	17	2	no	—
153	C₂×#108	—	34	1	no	—
154	C₂×#107	—	17	1	no	—
155	#423	—	34	1	no	—
156	C₂×#103	—	34	1	no	—
157	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
158	C₄⋊C₁₆	—	2	1	no	—
159	C₄⋊C₁₆	—	2	1	no	—
160	#362	—	34	4	no	—
161	C₂×C₁₇	—	2	1	no	—
162	C₂×C₁₇	—	2	1	no	—
163	C₂×C₁₇	#1802	1	1	yes	characteristic
164	C₂×C₁₇	C₂×#362	1	1	yes	characteristic
165	C₂×C₁₇	#2148	1	1	yes	characteristic
166	C₂²×Q₈	—	34	1	no	—
167	C₂×Q₁₆	—	17	2	no	—
168	C₂×Q₁₆	—	17	2	no	—
169	C₂×Q₁₆	—	17	2	no	—
170	C₂×Q₁₆	—	34	1	no	—
171	C₂×Q₁₆	—	34	1	no	—
172	C₂×Q₁₆	—	34	1	no	—
173	C₂×M₁₆	—	34	2	no	—
174	C₂²×C₈	D₆₈	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
176	C₄×Q₈	—	34	2	no	—
177	C₄×Q₈	—	34	2	no	—
178	C₄×Q₈	—	68	1	no	—
179	C₄×Q₈	—	68	1	no	—
180	C₂×C₄⋊C₄	—	34	1	no	—
181	C₂×C₄⋊C₄	—	68	1	no	—
182	C₂×C₄²	—	68	1	no	—
183	C₂×C₄²	D₆₈	1	1	yes	characteristic
184	M₃₂	—	2	2	no	—
185	M₃₂	—	2	2	no	—
186	C₂×C₁₆	—	2	2	no	—
187	#108	—	34	2	no	—
188	#108	—	34	2	no	—
189	#108	—	68	2	no	—
190	#107	—	17	2	no	—
191	#107	—	34	1	no	—
192	C₄⋊C₈	—	68	1	no	—
193	C₄⋊C₈	—	68	1	no	—
194	#103	—	34	2	no	—
195	#103	—	34	2	no	—
196	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
197	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
198	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
200	C₁₇	#12819	1	1	yes	characteristic, complemented, Sylow
201	C₂×Q₈	—	34	2	no	—
202	C₂×Q₈	—	34	2	no	—
203	C₂×Q₈	—	34	2	no	—
204	C₂×Q₈	—	68	1	no	—
205	C₂×Q₈	—	68	1	no	—
206	C₂×Q₈	—	68	1	no	—
207	C₂²×C₄	—	68	1	no	—
208	C₂²×C₄	—	68	1	no	—
209	C₂²×C₄	#63219	1	1	yes	characteristic
210	C₂²×C₄	#63219	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
212	Q₁₆	—	17	4	no	—
213	Q₁₆	—	34	2	no	—
214	Q₁₆	—	34	2	no	—
215	Q₁₆	—	34	2	no	—
216	M₁₆	—	68	2	no	—
217	M₁₆	—	68	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	68	2	no	—
221	C₂×C₈	#63227	1	1	yes	characteristic
222	C₂×C₈	#63227	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
224	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
225	C₂×C₈	#63254	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
227	C₂×C₈	#63256	1	1	yes	characteristic
228	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
229	C₄⋊C₄	—	34	2	no	—
230	C₄⋊C₄	—	68	2	no	—
231	C₄⋊C₄	—	68	2	no	—
232	C₄⋊C₄	—	68	1	no	—
233	C₄²	—	68	2	no	—
234	C₄²	—	68	2	no	—
235	C₄²	D₁₃₆	1	1	yes	characteristic
236	C₄²	D₁₃₆	1	1	yes	characteristic
237	C₄²	#63227	1	1	yes	characteristic
238	C₄²	#63227	1	1	yes	characteristic
239	C₁₆	—	4	2	no	—
240	C₂³	C₂×#63219	1	1	yes	characteristic
241	Q₈	—	34	4	no	—
242	Q₈	—	68	2	no	—
243	Q₈	—	68	2	no	—
244	Q₈	—	68	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	68	2	no	—
248	C₂×C₄	—	68	2	no	—
249	C₂×C₄	—	68	2	no	—
250	C₂×C₄	—	68	2	no	—
251	C₂×C₄	—	136	2	no	—
252	C₂×C₄	#10587492	1	1	yes	characteristic
253	C₂×C₄	#10587492	1	1	yes	characteristic
254	C₂×C₄	#10587492	1	1	yes	characteristic
255	C₂×C₄	#10587492	1	1	yes	characteristic
256	C₂×C₄	#10587581	1	1	yes	characteristic
257	C₂×C₄	#10587583	1	1	yes	characteristic
258	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
259	C₂×C₄	#10587594	1	1	yes	characteristic
260	C₂×C₄	#10587595	1	1	yes	characteristic
261	C₂×C₄	C₂×#63227	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	136	2	no	—
265	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
266	C₈	#10587578	1	1	yes	characteristic
267	C₈	#10587624	1	1	yes	characteristic
268	C₈	#10587632	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×#10587492	1	1	yes	characteristic
272	C₂²	#11763015	1	1	yes	characteristic
273	C₂²	#11763100	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	68	4	no	—
278	C₄	—	136	2	no	—
279	C₄	#11762626	1	2	yes	complemented
280	C₄	#11762666	1	1	yes	characteristic
281	C₄	#11762811	1	1	yes	characteristic
282	C₄	#11763005	1	1	yes	characteristic
283	C₄	#11763090	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	#14528785	1	1	yes	characteristic
287	C₂	C₂×#11762626	1	1	yes	characteristic
288	C₂	#14530534	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587492
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#2150	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#2150	D₁₇
hypocenter	C₁₇	#12819
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
138. C₄×C₁₇	#362	semidirect
200. C₁₇	#12819	semidirect
279. C₄	#11762626	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₆₈	2
C₂×#11762626	2
#14528956	2
#14526893	2
C₁₇×#2150	2
#12819	17