Subgroups — #30854426

Order 17152.

998 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	C₂×#16863519	C₂	1	2	yes	maximal
2	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
3	#22568634	C₂	1	2	yes	maximal, complemented
4	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560093	C₂	1	2	yes	maximal, complemented
6	#22560046	C₂	1	4	yes	maximal, complemented
7	#22560020	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	1	yes	characteristic
10	C₂×#14524049	C₂²	1	4	yes	—
11	C₄×#11761628	C₂²	1	2	yes	—
12	#16863519	C₄	1	16	yes	complemented
13	C₄×Dic₂₆₈	C₄	1	2	yes	complemented
14	C₄×#11761617	C₄	1	4	yes	complemented
15	C₄×D₅₃₆	C₄	1	2	yes	complemented
16	C₂×#14523979	C₂²	1	2	yes	—
17	#16863434	C₂²	1	1	yes	characteristic
18	C₂×#14523971	C₂²	1	4	yes	—
19	#16863393	C₂²	1	2	yes	—
20	#16863353	—	2	4	no	—
21	#16863351	C₂²	1	4	yes	—
22	C₄×#11761612	C₂²	1	2	yes	—
23	#16863348	—	2	2	no	—
24	#97×D₆₇	—	2	2	no	—
25	#16863161	C₂²	1	2	yes	—
26	#16863156	C₂²	1	4	yes	—
27	C₂×#97×C₆₇	C₂²	1	2	yes	—
28	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#376×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#11761628	C₂³	1	2	yes	—
33	#14524065	C₂×C₄	1	8	yes	—
34	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
35	C₂×#11761617	C₂×C₄	1	2	yes	—
36	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
37	#14524049	—	2	8	no	—
38	#14524049	C₂×C₄	1	16	yes	complemented
39	C₂×#11761612	C₂³	1	4	yes	—
40	C₂×#11761611	—	2	4	no	—
41	C₂×C₈×D₆₇	—	2	4	no	—
42	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
43	C₄×D₂₆₈	C₂×C₄	1	4	yes	—
44	C₄×C₂²⋊D₆₇	—	2	4	no	—
45	#14524002	C₂³	1	2	yes	—
46	#14524000	—	2	2	no	—
47	C₄²×D₆₇	—	2	2	no	—
48	#14523979	C₂×C₄	1	8	yes	complemented
49	#14523977	C₂×C₄	1	2	yes	—
50	#14523975	C₂×C₄	1	2	yes	—
51	#14523971	C₂×C₄	1	16	yes	complemented
52	#14523968	C₂×C₄	1	4	yes	—
53	#14523965	C₂×C₄	1	4	yes	—
54	#14523954	—	2	2	no	—
55	#14523953	—	2	4	no	—
56	C₄×C₆₇⋊C₈	—	2	2	no	—
57	C₈×Dic₆₇	—	2	4	no	—
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	#97×C₆₇	C₂×C₄	1	8	yes	complemented
64	#97×C₆₇	D₄	1	2	yes	—
65	#97×C₆₇	D₄	1	2	yes	—
66	C₄×C₈×C₆₇	—	2	4	no	—
67	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
69	#95×C₆₇	—	2	2	no	—
70	#11761628	—	2	4	no	—
71	#11761628	C₂²×C₄	1	8	yes	—
72	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
73	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
74	C₂×C₂²⋊D₆₇	—	2	2	no	—
75	C₂×C₄×D₆₇	—	2	2	no	—
76	C₂×C₄×D₆₇	—	2	4	no	—
77	Dic₂₆₈	C₄²	1	4	yes	—
78	#11761617	C₄²	1	8	yes	—
79	D₅₃₆	C₄²	1	4	yes	—
80	#11761612	—	2	4	no	—
81	#11761612	C₂²×C₄	1	8	yes	—
82	#11761611	—	2	16	no	—
83	C₂×C₆₇⋊C₈	—	2	4	no	—
84	C₈×D₆₇	—	2	16	no	—
85	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
86	Dic₆₇⋊C₄	—	2	4	no	—
87	C₂²⋊Dic₆₇	—	2	2	no	—
88	(C₂×D₆₇)⋊C₄	—	2	4	no	—
89	C₄×Dic₆₇	—	2	4	no	—
90	C₄×Dic₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	#41	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₆₇	#41	1	4	yes	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
104	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
105	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
108	C₂²×D₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
118	C₂²⋊D₆₇	—	2	8	no	—
119	C₂×Dic₆₇	—	2	2	no	—
120	C₂×Dic₆₇	—	2	4	no	—
121	D₂₆₈	—	2	2	no	—
122	D₂₆₈	C₂×C₄²	1	4	yes	—
123	C₄×D₆₇	—	2	8	no	—
124	C₄×D₆₇	—	8	4	no	—
125	Dic₁₃₄	—	2	2	no	—
126	Dic₁₃₄	C₂×C₄²	1	4	yes	—
127	C₈×C₆₇	—	2	2	no	—
128	C₈×C₆₇	C₂×C₄²	1	2	yes	—
129	C₈×C₆₇	C₂×C₄²	1	2	yes	—
130	C₈×C₆₇	C₄×D₄	1	8	yes	—
131	C₆₇⋊C₈	—	2	8	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₆₇	C₄×#41	1	2	yes	—
135	C₂×D₆₇	—	2	4	no	—
136	C₂×D₆₇	—	4	4	no	—
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	4	4	no	—
140	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
141	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
142	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
143	Dic₆₇	—	2	4	no	—
144	Dic₆₇	—	4	4	no	—
145	#20513	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
148	C₂×C₆₇	#2957	1	2	yes	—
149	D₆₇	—	4	4	no	—
150	C₂×#444	—	67	2	no	—
151	C₄×#135	—	67	1	no	—
152	#2875	—	67	2	no	—
153	C₄×#108	—	67	1	no	—
154	#1773	—	67	2	no	—
155	#1764	—	67	4	no	—
156	#1759	—	67	2	no	—
157	#1726	D₆₇	1	1	yes	characteristic, complemented
158	C₆₇	#20513	1	1	yes	characteristic, complemented, Sylow
159	C₂×#135	—	67	1	no	—
160	C₂×#131	—	67	4	no	—
161	C₄×#41	—	67	2	no	—
162	#444	—	67	16	no	—
163	C₄×Q₁₆	—	67	2	no	—
164	C₄×SD₁₆	—	67	4	no	—
165	C₄×D₈	—	67	2	no	—
166	C₂×#108	—	67	2	no	—
167	#427	—	67	1	no	—
168	C₂×#104	—	67	4	no	—
169	#416	—	67	2	no	—
170	#405	—	67	4	no	—
171	#405	—	134	4	no	—
172	C₄×M₁₆	—	67	2	no	—
173	C₄×M₁₆	—	134	2	no	—
174	C₂×#97	—	134	2	no	—
175	C₂×#97	C₂×D₆₇	1	2	yes	—
176	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
178	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
179	#376	C₂×D₆₇	1	1	yes	characteristic
180	#341	—	67	2	no	—
181	#339	—	67	4	no	—
182	C₂×#41	—	67	2	no	—
183	#135	—	67	8	no	—
184	C₂×Q₁₆	—	67	1	no	—
185	C₂×SD₁₆	—	67	2	no	—
186	C₂×D₈	—	67	1	no	—
187	#131	—	67	16	no	—
188	#131	—	134	8	no	—
189	C₂×M₁₆	—	67	4	no	—
190	C₂×M₁₆	—	134	4	no	—
191	C₂²×C₈	—	134	4	no	—
192	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
195	C₄×Q₈	—	67	4	no	—
196	C₄×D₄	—	67	4	no	—
197	C₄×D₄	—	134	4	no	—
198	#117	—	67	2	no	—
199	#117	—	134	2	no	—
200	C₂×C₄²	—	134	2	no	—
201	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
202	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
203	#108	—	67	8	no	—
204	#107	—	67	2	no	—
205	#106	—	67	2	no	—
206	#104	—	67	16	no	—
207	#103	—	67	4	no	—
208	#102	—	67	4	no	—
209	#97	—	134	4	no	—
210	#97	—	134	2	no	—
211	#97	C₄×D₆₇	1	8	yes	complemented
212	#97	D₂₆₈	1	2	yes	—
213	#97	D₂₆₈	1	2	yes	—
214	C₄×C₈	—	2	4	no	—
215	C₄×C₈	—	134	4	no	—
216	C₄×C₈	—	134	2	no	—
217	C₄×C₈	C₄×D₆₇	1	2	yes	—
218	C₄×C₈	C₄×D₆₇	1	2	yes	—
219	#95	—	2	2	no	—
220	#41	—	67	8	no	—
221	#41	—	134	4	no	—
222	C₂×Q₈	—	67	2	no	—
223	C₂×D₄	—	67	2	no	—
224	C₂×D₄	—	134	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	134	4	no	—
227	C₂²×C₄	—	134	2	no	—
228	C₂²×C₄	#11761628	1	2	yes	—
229	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
230	Q₁₆	—	67	4	no	—
231	SD₁₆	—	67	8	no	—
232	D₈	—	67	4	no	—
233	M₁₆	—	67	8	no	—
234	M₁₆	—	134	4	no	—
235	M₁₆	—	134	16	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	134	4	no	—
238	C₂×C₈	—	134	16	no	—
239	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
243	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
244	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
245	C₂×C₈	#11761628	1	4	yes	—
246	C₄⋊C₄	—	67	4	no	—
247	C₄⋊C₄	—	134	4	no	—
248	C₂²⋊C₄	—	134	2	no	—
249	C₂²⋊C₄	—	134	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	134	4	no	—
252	C₄²	—	134	2	no	—
253	C₄²	C₂×C₄×D₆₇	1	4	yes	—
254	C₄²	C₂×C₄×D₆₇	1	2	yes	—
255	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
256	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
257	C₂³	—	134	2	no	—
258	C₂³	C₂×#11761628	1	1	yes	characteristic
259	Q₈	—	67	4	no	—
260	Q₈	—	134	2	no	—
261	D₄	—	67	4	no	—
262	D₄	—	134	8	no	—
263	D₄	—	134	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	536	4	no	—
268	C₂×C₄	—	134	4	no	—
269	C₂×C₄	—	134	2	no	—
270	C₂×C₄	—	134	8	no	—
271	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
272	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
274	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
276	C₂×C₄	C₂×#11761628	1	2	yes	—
277	C₈	—	2	2	no	—
278	C₈	—	134	8	no	—
279	C₈	C₄²×D₆₇	1	2	yes	—
280	C₈	C₄²×D₆₇	1	2	yes	—
281	C₈	C₄×D₂₆₈	1	8	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	268	4	no	—
284	C₂²	—	134	4	no	—
285	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
286	C₂²	C₄×#11761628	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	4	no	—
290	C₄	—	268	4	no	—
291	C₄	—	134	4	no	—
292	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
293	C₄	C₂×C₄×D₂₆₈	1	2	yes	—
294	C₄	C₄×#11761628	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	268	4	no	—
297	C₂	C₄²×D₂₆₈	1	1	yes	characteristic
298	C₂	#22569127	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#20513
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
3. #22568634	C₂	semidirect
4. C₄×#14523979	C₂	semidirect
5. #22560093	C₂	semidirect
6. #22560046	C₂	semidirect
7. #22560020	C₂	semidirect
8. C₆₇×#1726	C₂	semidirect
12. #16863519	C₄	semidirect
13. C₄×Dic₂₆₈	C₄	semidirect
14. C₄×#11761617	C₄	semidirect
15. C₄×D₅₃₆	C₄	semidirect
38. #14524049	C₂×C₄	semidirect
48. #14523979	C₂×C₄	semidirect
51. #14523971	C₂×C₄	semidirect
63. #97×C₆₇	C₂×C₄	semidirect
157. #1726	D₆₇	semidirect
158. C₆₇	#20513	semidirect
211. #97	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863519	2
C₄×#14524065	2
#22568634	2
C₄×#14523979	2
#22560093	2
#22560046	2
#22560020	2
C₆₇×#1726	2
#20513	67