Subgroups — #16950366

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762611	C₂	1	2	yes	maximal
3	#14528944	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526876	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₃₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₆₈	—	2	1	no	—
8	C₄×D₁₃₆	—	2	2	no	—
9	C₄×D₁₃₆	C₄	1	4	yes	complemented
10	C₂×#10587489	—	2	1	no	—
11	C₂×#10587489	C₂²	1	2	yes	—
12	C₂×#10587489	C₂²	1	1	yes	characteristic
13	C₂×#10587492	—	2	1	no	—
14	#11762679	—	2	1	no	—
15	#11762611	—	2	4	no	—
16	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×C₄⋊C₁₆	—	2	2	no	—
19	C₂²×D₆₈	—	2	1	no	—
20	C₂²×C₄×D₁₇	—	4	1	no	—
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
24	C₂×#63227	—	4	1	no	—
25	C₂×C₄⋊Dic₁₇	—	2	1	no	—
26	C₂×C₄⋊Dic₁₇	—	2	2	no	—
27	C₂×C₄⋊Dic₁₇	—	4	1	no	—
28	C₂×C₄⋊Dic₁₇	—	4	1	no	—
29	C₂×D₁₃₆	—	2	1	no	—
30	C₂×D₁₃₆	—	2	1	no	—
31	C₂×D₁₃₆	—	2	1	no	—
32	C₂×D₁₃₆	C₂×C₄	1	2	yes	—
33	C₂×D₁₃₆	D₄	1	2	yes	—
34	C₂×D₁₃₆	Q₈	1	2	yes	—
35	C₄×D₆₈	—	2	4	no	—
36	C₄×D₆₈	—	4	2	no	—
37	C₂×C₁₆×C₁₇	—	2	2	no	—
38	C₂×C₁₆×C₁₇	—	2	2	no	—
39	C₂×C₁₆×C₁₇	—	2	2	no	—
40	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₇	D₄	1	2	yes	—
42	#10587492	—	2	4	no	—
43	#10587489	—	2	1	no	—
44	#10587489	—	2	2	no	—
45	#10587489	—	2	2	no	—
46	#10587489	—	4	2	no	—
47	#10587489	C₂×C₄	1	2	yes	—
48	C₂³×D₁₇	—	4	1	no	—
49	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₁₇	—	4	1	no	—
53	C₂²×Dic₁₇	—	4	2	no	—
54	C₂×D₆₈	—	2	2	no	—
55	C₂×D₆₈	—	2	2	no	—
56	C₂×D₆₈	—	2	2	no	—
57	C₂×D₆₈	—	4	1	no	—
58	C₂×D₆₈	—	4	1	no	—
59	C₂×D₆₈	—	4	1	no	—
60	C₂×C₄×D₁₇	—	4	4	no	—
61	C₂×C₄×D₁₇	—	8	2	no	—
62	C₂×C₄×D₁₇	—	8	2	no	—
63	C₂×C₈×C₁₇	—	2	1	no	—
64	C₂×C₈×C₁₇	—	2	1	no	—
65	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
71	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
72	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₇	SD₁₆	1	2	yes	—
74	#63227	—	4	4	no	—
75	C₄⋊Dic₁₇	—	2	2	no	—
76	C₄⋊Dic₁₇	—	4	1	no	—
77	C₄⋊Dic₁₇	—	4	2	no	—
78	C₄⋊Dic₁₇	—	4	2	no	—
79	C₄⋊Dic₁₇	—	8	2	no	—
80	C₄⋊Dic₁₇	—	8	2	no	—
81	D₁₃₆	—	2	2	no	—
82	D₁₃₆	—	2	2	no	—
83	D₁₃₆	—	2	2	no	—
84	D₁₃₆	C₄⋊C₄	1	4	yes	complemented
85	C₁₆×C₁₇	—	4	2	no	—
86	#12811	—	17	1	no	maximal, Sylow
87	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
88	C₂²×D₁₇	—	4	2	no	—
89	C₂²×D₁₇	—	4	2	no	—
90	C₂²×D₁₇	—	4	2	no	—
91	C₂²×D₁₇	—	8	1	no	—
92	C₂²×D₁₇	—	8	1	no	—
93	C₂²×D₁₇	—	8	1	no	—
94	C₂²×D₁₇	—	8	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	#102	1	2	yes	—
98	C₂×C₄×C₁₇	#102	1	2	yes	—
99	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
100	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
101	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
107	C₂×Dic₁₇	—	4	2	no	—
108	C₂×Dic₁₇	—	8	1	no	—
109	C₂×Dic₁₇	—	8	1	no	—
110	C₂×Dic₁₇	—	8	2	no	—
111	C₂×Dic₁₇	—	8	2	no	—
112	C₂×Dic₁₇	—	8	2	no	—
113	D₆₈	—	2	4	no	—
114	D₆₈	—	4	2	no	—
115	D₆₈	—	4	2	no	—
116	D₆₈	—	4	2	no	—
117	C₄×D₁₇	—	8	8	no	—
118	C₈×C₁₇	—	2	2	no	—
119	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₁₇	#117	1	1	yes	characteristic
121	C₈×C₁₇	#121	1	1	yes	characteristic
122	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×D₈	—	17	1	no	—
124	C₂×C₄⋊C₁₆	D₁₇	1	1	yes	characteristic, complemented
125	C₂×#357	—	17	2	no	—
126	#1845	—	17	1	no	—
127	#1380	—	17	2	no	—
128	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
129	C₂²×C₁₇	#459	1	1	yes	characteristic
130	C₂²×C₁₇	#474	1	1	yes	characteristic
131	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
132	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
133	C₂²×C₁₇	#509	1	1	yes	characteristic
134	C₂²×C₁₇	#510	1	1	yes	characteristic
135	C₂×D₁₇	—	4	4	no	—
136	C₂×D₁₇	—	8	2	no	—
137	C₂×D₁₇	—	8	2	no	—
138	C₂×D₁₇	—	8	2	no	—
139	C₂×D₁₇	—	8	2	no	—
140	C₂×D₁₇	—	8	2	no	—
141	C₂×D₁₇	—	8	2	no	—
142	C₄×C₁₇	—	2	2	no	—
143	C₄×C₁₇	#357	1	4	yes	complemented
144	C₄×C₁₇	#380	1	1	yes	characteristic
145	C₄×C₁₇	#417	1	1	yes	characteristic
146	C₄×C₁₇	#461	1	1	yes	characteristic
147	C₄×C₁₇	#476	1	1	yes	characteristic
148	Dic₁₇	—	8	2	no	—
149	Dic₁₇	—	16	2	no	—
150	C₂²×D₈	—	17	1	no	—
151	C₂×C₄×D₄	—	34	1	no	—
152	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
153	C₄×D₈	—	17	4	no	—
154	C₄×D₈	—	34	2	no	—
155	C₂×#107	—	17	1	no	—
156	C₂×#107	—	17	2	no	—
157	C₂×#107	—	34	1	no	—
158	C₂×#102	—	34	1	no	—
159	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
160	#384	—	34	1	no	—
161	C₄⋊C₁₆	—	2	2	no	—
162	#357	—	34	4	no	—
163	C₂×C₁₇	#1802	1	1	yes	characteristic
164	C₂×C₁₇	C₂×#357	1	1	yes	characteristic
165	C₂×C₁₇	#2139	1	1	yes	characteristic
166	C₂×C₁₇	#2206	1	1	yes	characteristic
167	C₂×C₁₇	#2208	1	1	yes	characteristic
168	C₂×C₁₇	#2224	1	1	yes	characteristic
169	C₂×C₁₇	#2226	1	1	yes	characteristic
170	D₁₇	—	8	4	no	—
171	C₂²×D₄	—	34	1	no	—
172	C₂³×C₄	—	68	1	no	—
173	C₂×D₈	—	17	2	no	—
174	C₂×D₈	—	17	2	no	—
175	C₂×D₈	—	17	2	no	—
176	C₂×D₈	—	34	1	no	—
177	C₂×D₈	—	34	1	no	—
178	C₂×D₈	—	34	1	no	—
179	C₂²×C₈	D₆₈	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
181	C₄×D₄	—	34	4	no	—
182	C₄×D₄	—	68	2	no	—
183	C₂×C₄⋊C₄	—	34	1	no	—
184	C₂×C₄⋊C₄	—	34	2	no	—
185	C₂×C₄⋊C₄	—	68	1	no	—
186	C₂×C₄⋊C₄	—	68	1	no	—
187	C₂×C₂²⋊C₄	—	68	1	no	—
188	C₂×C₄²	D₆₈	1	1	yes	characteristic
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	—	2	2	no	—
191	C₂×C₁₆	—	2	2	no	—
192	#107	—	17	2	no	—
193	#107	—	34	1	no	—
194	#107	—	34	2	no	—
195	#107	—	34	2	no	—
196	#107	—	68	2	no	—
197	#102	—	34	4	no	—
198	C₄×C₈	C₄×D₁₇	1	2	yes	—
199	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
200	C₁₇	#12811	1	1	yes	characteristic, complemented, Sylow
201	C₂⁴	—	68	1	no	—
202	C₂×D₄	—	34	2	no	—
203	C₂×D₄	—	34	2	no	—
204	C₂×D₄	—	34	2	no	—
205	C₂×D₄	—	68	1	no	—
206	C₂×D₄	—	68	1	no	—
207	C₂×D₄	—	68	1	no	—
208	C₂²×C₄	—	68	1	no	—
209	C₂²×C₄	—	68	4	no	—
210	C₂²×C₄	—	68	2	no	—
211	C₂²×C₄	—	136	2	no	—
212	C₂²×C₄	—	136	2	no	—
213	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
214	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
216	D₈	—	17	4	no	—
217	D₈	—	34	2	no	—
218	D₈	—	34	2	no	—
219	D₈	—	34	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	#63227	1	2	yes	—
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	1	no	—
231	C₄⋊C₄	—	68	2	no	—
232	C₄⋊C₄	—	68	2	no	—
233	C₄⋊C₄	—	136	2	no	—
234	C₄⋊C₄	—	136	2	no	—
235	C₂²⋊C₄	—	68	4	no	—
236	C₄²	#63219	1	2	yes	—
237	C₄²	#63227	1	2	yes	—
238	C₁₆	—	4	2	no	—
239	C₂³	—	68	2	no	—
240	C₂³	—	68	2	no	—
241	C₂³	—	68	2	no	—
242	C₂³	—	136	1	no	—
243	C₂³	—	136	1	no	—
244	C₂³	—	136	1	no	—
245	C₂³	—	136	1	no	—
246	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
247	D₄	—	34	4	no	—
248	D₄	—	68	2	no	—
249	D₄	—	68	2	no	—
250	D₄	—	68	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	68	2	no	—
254	C₂×C₄	—	136	2	no	—
255	C₂×C₄	—	136	2	no	—
256	C₂×C₄	—	136	2	no	—
257	C₂×C₄	—	136	1	no	—
258	C₂×C₄	—	136	1	no	—
259	C₂×C₄	—	136	8	no	—
260	C₂×C₄	D₂₇₂	1	2	yes	—
261	C₂×C₄	#10587471	1	2	yes	—
262	C₂×C₄	#10587492	1	2	yes	—
263	C₂×C₄	#10587492	1	2	yes	—
264	C₂×C₄	#10587581	1	1	yes	characteristic
265	C₂×C₄	#10587583	1	1	yes	characteristic
266	C₂×C₄	C₂×#63219	1	1	yes	characteristic
267	C₂×C₄	#10587594	1	1	yes	characteristic
268	C₂×C₄	#10587595	1	1	yes	characteristic
269	C₂×C₄	C₂×#63227	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
272	C₈	#10587578	1	1	yes	characteristic
273	C₈	#10587624	1	1	yes	characteristic
274	C₈	#10587632	1	1	yes	characteristic
275	C₂²	—	68	4	no	—
276	C₂²	—	136	2	no	—
277	C₂²	—	136	2	no	—
278	C₂²	—	136	2	no	—
279	C₂²	—	136	2	no	—
280	C₂²	—	136	2	no	—
281	C₂²	—	136	2	no	—
282	C₂²	C₂×#10587492	1	1	yes	characteristic
283	C₂²	#11763005	1	1	yes	characteristic
284	C₂²	#11763090	1	1	yes	characteristic
285	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
286	C₂²	C₂×#10587471	1	1	yes	characteristic
287	C₂²	#11763247	1	1	yes	characteristic
288	C₂²	#11763252	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	272	2	no	—
291	C₄	—	136	2	no	—
292	C₄	#11762611	1	4	yes	complemented
293	C₄	#11762666	1	1	yes	characteristic
294	C₄	#11762811	1	1	yes	characteristic
295	C₄	#11763015	1	1	yes	characteristic
296	C₄	#11763100	1	1	yes	characteristic
297	C₂	—	136	4	no	—
298	C₂	#14528785	1	1	yes	characteristic
299	C₂	C₂×#11762611	1	1	yes	characteristic
300	C₂	#14530497	1	1	yes	characteristic
301	C₂	#14530820	1	1	yes	characteristic
302	C₂	#14530830	1	1	yes	characteristic
303	C₂	#14530922	1	1	yes	characteristic
304	C₂	#14530932	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×C₄⋊C₁₆	C₂
socle	C₂³×C₁₇	C₂×D₈
prime radical	C₂²×D₁₃₆	C₂²
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₇
hypocenter	C₁₇	#12811
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #14528944	C₂	semidirect
4. #14526876	C₂	semidirect
5. C₂×C₁₇×C₄⋊C₁₆	C₂	semidirect
9. C₄×D₁₃₆	C₄	semidirect
84. D₁₃₆	C₄⋊C₄	semidirect
124. C₂×C₄⋊C₁₆	D₁₇	semidirect
143. C₄×C₁₇	#357	semidirect
200. C₁₇	#12811	semidirect
292. C₄	#11762611	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₃₆	2
C₂×#11762611	2
#14528944	2
#14526876	2
C₂×C₁₇×C₄⋊C₁₆	2
#12811	17