Subgroups — #30864017

Order 17152.

730 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570555	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568922	C₂	1	1	yes	maximal, characteristic
4	#22561190	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561041	C₂	1	1	yes	maximal, characteristic
6	#22560920	C₂	1	4	yes	maximal, complemented
7	#22560350	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560046	C₂	1	4	yes	maximal, complemented
9	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524049	C₂²	1	2	yes	—
11	#16863953	C₂²	1	1	yes	characteristic
12	#16863946	C₂²	1	1	yes	characteristic
13	#16863765	C₄	1	2	yes	complemented
14	#16863761	C₄	1	2	yes	complemented
15	#16863756	C₄	1	4	yes	complemented
16	#16863526	—	2	2	no	—
17	C₈×Dic₁₃₄	—	2	2	no	—
18	#16863475	—	2	2	no	—
19	C₈×D₂₆₈	—	2	2	no	—
20	#16863422	C₂²	1	2	yes	—
21	#16863420	C₂²	1	1	yes	characteristic
22	#16863420	C₂²	1	1	yes	characteristic
23	C₂×#14523971	C₂²	1	4	yes	—
24	#16863393	C₂²	1	4	yes	—
25	#16863362	C₂²	1	2	yes	—
26	#16863351	C₂²	1	2	yes	—
27	#16863161	C₂²	1	4	yes	—
28	#16863156	C₂²	1	4	yes	—
29	C₂×#97×C₆₇	C₂²	1	2	yes	—
30	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#376×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761628	C₂³	1	2	yes	—
35	#14524049	—	2	8	no	—
36	C₂×#11761612	C₂³	1	2	yes	—
37	C₂×#11761612	C₂³	1	2	yes	—
38	C₂×#11761611	—	2	2	no	—
39	C₂×C₈×D₆₇	—	2	2	no	—
40	#14524043	C₂×C₄	1	1	yes	characteristic
41	#14524043	C₂×C₄	1	2	yes	—
42	#14524040	C₂×C₄	1	1	yes	characteristic
43	#14524035	C₂×C₄	1	2	yes	—
44	#14524031	C₂×C₄	1	2	yes	—
45	#14524022	—	2	2	no	—
46	#14524017	—	2	2	no	—
47	C₄×Dic₁₃₄	—	2	2	no	—
48	C₄×D₂₆₈	—	2	2	no	—
49	#14524002	C₂³	1	2	yes	—
50	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
52	Dic₆₇⋊C₈	—	2	4	no	—
53	#14523971	—	2	8	no	—
54	#14523968	C₂×C₄	1	8	yes	—
55	#14523965	C₂×C₄	1	8	yes	—
56	(C₂×D₆₇)⋊C₈	—	2	4	no	—
57	#14523953	—	2	2	no	—
58	C₈×Dic₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#97×C₆₇	—	2	4	no	—
65	#97×C₆₇	C₂×C₄	1	2	yes	—
66	#97×C₆₇	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₆₇	—	2	4	no	—
68	C₄×C₈×C₆₇	D₄	1	2	yes	—
69	C₄×C₈×C₆₇	D₄	1	2	yes	—
70	#95×C₆₇	—	2	2	no	—
71	#11761628	—	2	4	no	—
72	#11761628	—	2	4	no	—
73	C₂×Dic₁₃₄	—	2	2	no	—
74	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
75	C₂×D₂₆₈	—	2	2	no	—
76	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
77	C₂×C₂²⋊D₆₇	—	2	2	no	—
78	C₂×C₄×D₆₇	—	2	2	no	—
79	#11761612	—	2	4	no	—
80	#11761612	—	2	4	no	—
81	#11761611	—	4	4	no	—
82	C₂×C₆₇⋊C₈	—	2	2	no	—
83	C₂×C₆₇⋊C₈	—	2	2	no	—
84	C₈×D₆₇	—	4	4	no	—
85	C₄⋊Dic₆₇	—	2	2	no	—
86	C₄⋊Dic₆₇	—	2	2	no	—
87	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
88	Dic₆₇⋊C₄	—	2	4	no	—
89	C₂²⋊Dic₆₇	—	2	2	no	—
90	(C₂×D₆₇)⋊C₄	—	2	4	no	—
91	C₄×Dic₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	#41	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	#41	1	4	yes	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₂×D₄	1	2	yes	—
107	C₄²×C₆₇	#41	1	4	yes	—
108	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₂²×D₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	4	2	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₄×D₄	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	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
119	C₂²⋊D₆₇	—	4	4	no	—
120	C₂×Dic₆₇	—	2	4	no	—
121	C₂×Dic₆₇	—	2	2	no	—
122	D₂₆₈	—	2	4	no	—
123	D₂₆₈	—	4	4	no	—
124	C₄×D₆₇	—	4	2	no	—
125	C₄×D₆₇	—	4	2	no	—
126	Dic₁₃₄	—	2	4	no	—
127	Dic₁₃₄	—	4	4	no	—
128	C₈×C₆₇	—	2	4	no	—
129	C₈×C₆₇	—	2	2	no	—
130	C₈×C₆₇	#124	1	2	yes	—
131	C₈×C₆₇	#124	1	2	yes	—
132	C₆₇⋊C₈	—	4	2	no	—
133	C₆₇⋊C₈	—	4	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₆₇	#529	1	2	yes	—
137	C₂×D₆₇	—	4	4	no	—
138	C₂×D₆₇	—	4	2	no	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	4	4	no	—
142	C₄×C₆₇	C₄×#41	1	2	yes	—
143	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
144	C₄×C₆₇	#532	1	1	yes	characteristic
145	Dic₆₇	—	4	4	no	—
146	Dic₆₇	—	4	2	no	—
147	#21661	—	67	1	no	maximal, Sylow
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	C₄×#124	1	1	yes	characteristic
150	C₂×C₆₇	#2955	1	2	yes	—
151	D₆₇	—	8	2	no	—
152	#3135	—	67	1	no	—
153	#2960	—	67	1	no	—
154	#2931	—	67	1	no	—
155	#1995	—	67	1	no	—
156	#1963	—	67	1	no	—
157	#1938	—	67	4	no	—
158	#1838	—	67	1	no	—
159	#1764	—	67	4	no	—
160	#1726	D₆₇	1	1	yes	characteristic, complemented
161	C₆₇	#21661	1	1	yes	characteristic, complemented, Sylow
162	C₂×#131	—	67	2	no	—
163	#533	—	67	1	no	—
164	#532	—	67	1	no	—
165	#491	—	67	2	no	—
166	#490	—	67	2	no	—
167	#489	—	67	4	no	—
168	#446	—	134	2	no	—
169	C₈×Q₈	—	134	2	no	—
170	#436	—	134	2	no	—
171	C₈×D₄	—	134	2	no	—
172	#424	—	67	2	no	—
173	#423	—	67	1	no	—
174	#423	—	67	1	no	—
175	C₂×#104	—	67	4	no	—
176	#416	—	67	4	no	—
177	#408	—	67	2	no	—
178	#405	—	67	2	no	—
179	C₂×#97	C₂×D₆₇	1	2	yes	—
180	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
182	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
183	#376	C₂×D₆₇	1	1	yes	characteristic
184	#341	—	67	4	no	—
185	#339	—	67	4	no	—
186	C₂×#41	—	67	2	no	—
187	#131	—	134	8	no	—
188	C₂×M₁₆	—	67	2	no	—
189	C₂×M₁₆	—	67	2	no	—
190	C₂×M₁₆	—	134	2	no	—
191	C₂²×C₈	—	134	2	no	—
192	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
193	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
195	C₄⋊Q₈	—	67	2	no	—
196	C₄⋊Q₈	—	67	1	no	—
197	#127	—	67	1	no	—
198	#125	—	67	2	no	—
199	#124	—	67	2	no	—
200	C₂²⋊Q₈	—	134	2	no	—
201	#121	—	134	2	no	—
202	C₄×Q₈	—	134	2	no	—
203	C₄×D₄	—	134	2	no	—
204	#117	—	67	2	no	—
205	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
206	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
207	C₄⋊C₈	—	134	4	no	—
208	C₄⋊C₈	—	134	2	no	—
209	C₄⋊C₈	—	134	2	no	—
210	#104	—	134	8	no	—
211	#103	—	67	8	no	—
212	#102	—	67	8	no	—
213	C₂²⋊C₈	—	134	4	no	—
214	#97	—	2	4	no	—
215	#97	—	134	2	no	—
216	#97	C₄×D₆₇	1	2	yes	—
217	#97	C₄×D₆₇	1	2	yes	—
218	C₄×C₈	—	2	4	no	—
219	C₄×C₈	—	134	2	no	—
220	C₄×C₈	D₂₆₈	1	2	yes	—
221	C₄×C₈	D₂₆₈	1	2	yes	—
222	#95	—	2	2	no	—
223	#41	—	134	4	no	—
224	#41	—	134	4	no	—
225	C₂×Q₈	—	67	2	no	—
226	C₂×Q₈	—	134	2	no	—
227	C₂×D₄	—	67	2	no	—
228	C₂×D₄	—	134	2	no	—
229	C₂×D₄	—	134	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	134	2	no	—
232	C₂²×C₄	#11761628	1	2	yes	—
233	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
234	M₁₆	—	268	4	no	—
235	M₁₆	—	134	4	no	—
236	M₁₆	—	134	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	268	4	no	—
241	C₂×C₈	—	134	2	no	—
242	C₂×C₈	—	134	2	no	—
243	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
244	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
245	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
246	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
247	C₂×C₈	#11761628	1	4	yes	—
248	C₄⋊C₄	—	67	4	no	—
249	C₄⋊C₄	—	134	2	no	—
250	C₄⋊C₄	—	134	4	no	—
251	C₄⋊C₄	—	134	2	no	—
252	C₂²⋊C₄	—	134	2	no	—
253	C₂²⋊C₄	—	134	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	134	2	no	—
256	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
257	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
258	C₄²	C₂×D₂₆₈	1	2	yes	—
259	C₄²	#11761628	1	4	yes	—
260	C₂³	—	134	2	no	—
261	C₂³	C₂×#11761628	1	1	yes	characteristic
262	Q₈	—	268	4	no	—
263	Q₈	—	134	4	no	—
264	D₄	—	268	4	no	—
265	D₄	—	268	4	no	—
266	D₄	—	134	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	268	2	no	—
271	C₂×C₄	—	268	2	no	—
272	C₂×C₄	—	134	4	no	—
273	C₂×C₄	—	134	2	no	—
274	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
275	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
278	C₂×C₄	C₂×#11761628	1	2	yes	—
279	C₂×C₄	C₂×#11761628	1	2	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	2	2	no	—
282	C₈	—	268	2	no	—
283	C₈	—	268	2	no	—
284	C₈	#14524031	1	2	yes	—
285	C₈	#14524031	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	268	4	no	—
288	C₂²	—	268	2	no	—
289	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
290	C₂²	#16863930	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	268	4	no	—
295	C₄	—	268	2	no	—
296	C₄	C₄×#11761628	1	2	yes	—
297	C₄	C₂×#14524031	1	1	yes	characteristic
298	C₄	#16863946	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	536	2	no	—
301	C₂	C₄×#14524031	1	1	yes	characteristic
302	C₂	#22569113	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#21661
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
2. #22569143	C₂	semidirect
4. #22561190	C₂	semidirect
6. #22560920	C₂	semidirect
7. #22560350	C₂	semidirect
8. #22560046	C₂	semidirect
9. C₆₇×#1726	C₂	semidirect
13. #16863765	C₄	semidirect
14. #16863761	C₄	semidirect
15. #16863756	C₄	semidirect
160. #1726	D₆₇	semidirect
161. C₆₇	#21661	semidirect

Maximal subgroups

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

subgroup	index
#22570555	2
#22569143	2
#22568922	2
#22561190	2
#22561041	2
#22560920	2
#22560350	2
#22560046	2
C₆₇×#1726	2
#21661	67