Subgroups — #30874306

Order 17152.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569093	C₂	1	2	yes	maximal
2	#22568804	C₂	1	2	yes	maximal, complemented
3	#22560990	C₂	1	2	yes	maximal, complemented
4	#22560437	C₂	1	4	yes	maximal, complemented
5	#22560366	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560268	C₂	1	2	yes	maximal, complemented
7	#22560088	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863879	C₂²	1	4	yes	—
10	#16863508	—	2	2	no	—
11	#16863503	—	2	2	no	—
12	SD₁₆×Dic₆₇	—	2	4	no	—
13	C₂×#14523979	C₂²	1	1	yes	characteristic
14	#16863438	—	2	4	no	—
15	#16863438	C₂²	1	4	yes	—
16	#16863434	—	2	2	no	—
17	#16863434	C₂²	1	1	yes	characteristic
18	C₂×#14523975	—	2	2	no	—
19	C₂×#14523975	C₂²	1	2	yes	—
20	C₂×#14523975	C₂²	1	2	yes	—
21	#16863397	C₂²	1	4	yes	—
22	#16863351	C₂²	1	2	yes	—
23	#16863266	C₂²	1	2	yes	—
24	#16863164	C₂²	1	4	yes	—
25	#16863156	C₂²	1	2	yes	—
26	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#131×C₆₇	C₂²	1	2	yes	—
28	#443×C₆₇	C₄	1	8	yes	complemented
29	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#104×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#11761612	C₂³	1	2	yes	—
33	Q₈×Dic₆₇	—	2	4	no	—
34	D₄×Dic₆₇	—	2	4	no	—
35	D₄×Dic₆₇	—	4	4	no	—
36	#14524002	—	2	4	no	—
37	#14524002	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₆₇	—	2	4	no	—
39	C₂×C₄⋊Dic₆₇	—	4	2	no	—
40	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
41	#14523979	—	2	2	no	—
42	#14523977	—	2	4	no	—
43	#14523977	D₄	1	2	yes	—
44	#14523975	—	2	4	no	—
45	#14523975	—	2	2	no	—
46	#14523975	—	2	2	no	—
47	#14523975	—	2	2	no	—
48	#14523975	—	2	2	no	—
49	#14523975	D₄	1	2	yes	—
50	#14523967	—	2	4	no	—
51	#14523964	—	2	4	no	—
52	#14523953	—	2	2	no	—
53	C₈×Dic₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	C₂³	1	2	yes	—
55	#135×C₆₇	C₂×C₄	1	4	yes	—
56	#135×C₆₇	Q₈	1	4	yes	—
57	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
58	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
59	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
60	#131×C₆₇	—	2	4	no	—
61	#131×C₆₇	C₂×C₄	1	8	yes	complemented
62	C₂×M₁₆×C₆₇	—	2	2	no	—
63	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	—	2	2	no	—
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	#108×C₆₇	C₂×C₄	1	4	yes	complemented
69	#104×C₆₇	C₂×C₄	1	8	yes	complemented
70	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
71	C₂²×Dic₆₇	—	2	4	no	—
72	C₂²×Dic₆₇	—	4	4	no	—
73	#11761612	—	2	2	no	—
74	#11761612	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₄⋊Dic₆₇	—	2	4	no	—
77	C₄⋊Dic₆₇	—	2	4	no	—
78	C₄⋊Dic₆₇	—	4	4	no	—
79	C₄⋊Dic₆₇	—	8	4	no	—
80	C₄⋊Dic₆₇	C₂×D₄	1	4	yes	—
81	C₂²⋊Dic₆₇	—	2	2	no	—
82	C₂²⋊Dic₆₇	—	4	4	no	—
83	C₄×Dic₆₇	—	2	2	no	—
84	C₄×Dic₆₇	—	4	4	no	—
85	#41×C₆₇	—	2	2	no	—
86	#41×C₆₇	—	2	2	no	—
87	#41×C₆₇	C₂²×C₄	1	4	yes	—
88	#41×C₆₇	C₂×Q₈	1	4	yes	—
89	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₇	—	2	1	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	Q₁₆×C₆₇	C₄⋊C₄	1	4	yes	complemented
96	SD₁₆×C₆₇	C₄⋊C₄	1	8	yes	complemented
97	D₈×C₆₇	C₄⋊C₄	1	4	yes	complemented
98	M₁₆×C₆₇	—	2	2	no	—
99	M₁₆×C₆₇	—	2	8	no	—
100	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	8	no	—
103	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
110	C₂×C₈×C₆₇	#41	1	2	yes	—
111	C₄²×C₆₇	—	2	1	no	—
112	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂³×C₆₇	—	2	2	no	—
114	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
115	Q₈×C₆₇	—	2	2	no	—
116	Q₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
117	D₄×C₆₇	—	2	4	no	—
118	D₄×C₆₇	—	2	4	no	—
119	D₄×C₆₇	—	2	2	no	—
120	D₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
132	C₂×Dic₆₇	—	2	4	no	—
133	C₂×Dic₆₇	—	4	4	no	—
134	C₂×Dic₆₇	—	8	4	no	—
135	C₂×Dic₆₇	—	16	4	no	—
136	C₈×C₆₇	—	2	4	no	—
137	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
138	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
139	C₈×C₆₇	C₄×D₄	1	2	yes	—
140	C₈×C₆₇	C₄×D₄	1	2	yes	—
141	C₆₇⋊C₈	—	4	2	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	2	4	no	—
144	C₂²×C₆₇	—	4	4	no	—
145	C₂²×C₆₇	#520	1	1	yes	characteristic
146	C₂²×C₆₇	D₄²	1	1	yes	characteristic
147	C₂²×C₆₇	#554	1	1	yes	characteristic
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	—	2	4	no	—
150	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₆₇	#547	1	1	yes	characteristic
153	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
154	Dic₆₇	—	4	4	no	—
155	Dic₆₇	—	8	4	no	—
156	#22689	—	67	1	no	maximal, Sylow
157	C₂×C₆₇	—	2	1	no	—
158	C₂×C₆₇	—	2	1	no	—
159	C₂×C₆₇	—	4	4	no	—
160	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₆₇	#3291	1	2	yes	—
162	C₂×#443	D₆₇	1	1	yes	characteristic
163	#2952	—	67	2	no	—
164	#2912	—	67	2	no	—
165	#1951	—	67	2	no	—
166	#1859	—	67	4	no	—
167	#1844	—	67	1	no	—
168	#1815	—	67	2	no	—
169	#1772	—	67	1	no	—
170	C₆₇	#22689	1	1	yes	characteristic, complemented, Sylow
171	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
172	C₂×#131	C₂×D₆₇	1	2	yes	—
173	#520	—	67	4	no	—
174	#443	Dic₆₇	1	8	yes	complemented
175	#442	—	134	2	no	—
176	#441	—	134	2	no	—
177	C₄×SD₁₆	—	134	4	no	—
178	C₂×#108	—	67	1	no	—
179	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
180	#428	—	67	4	no	—
181	#428	—	134	4	no	—
182	#427	—	67	1	no	—
183	#427	—	134	2	no	—
184	C₂×#106	—	67	2	no	—
185	C₂×#106	—	67	2	no	—
186	C₂×#106	—	134	2	no	—
187	C₂×#104	C₂×D₆₇	1	2	yes	—
188	#417	—	67	4	no	—
189	#405	—	67	2	no	—
190	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
191	#384	—	67	2	no	—
192	#343	—	67	4	no	—
193	#339	—	67	2	no	—
194	C₂×#41	C₂²×D₆₇	1	2	yes	—
195	#135	Dic₁₃₄	1	4	yes	—
196	#135	C₂×Dic₆₇	1	4	yes	—
197	C₂×Q₁₆	D₂₆₈	1	1	yes	characteristic
198	C₂×SD₁₆	D₂₆₈	1	2	yes	—
199	C₂×D₈	D₂₆₈	1	1	yes	characteristic
200	#131	—	2	4	no	—
201	#131	C₂×Dic₆₇	1	8	yes	complemented
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	67	2	no	—
204	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
208	C₄×Q₈	—	134	4	no	—
209	C₄×D₄	—	268	4	no	—
210	C₄×D₄	—	134	4	no	—
211	#117	—	67	2	no	—
212	#117	—	134	4	no	—
213	C₂×C₄⋊C₄	—	268	2	no	—
214	C₂×C₄⋊C₄	—	67	4	no	—
215	C₂×C₄⋊C₄	—	134	4	no	—
216	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
217	#108	—	134	2	no	—
218	#108	C₂×Dic₆₇	1	4	yes	complemented
219	#107	—	67	2	no	—
220	#107	—	134	4	no	—
221	#106	—	67	2	no	—
222	#106	—	134	4	no	—
223	#106	—	134	2	no	—
224	#106	—	134	2	no	—
225	#106	—	134	2	no	—
226	#106	—	134	2	no	—
227	#104	C₂×Dic₆₇	1	8	yes	complemented
228	#103	—	134	4	no	—
229	#102	—	134	4	no	—
230	#97	—	134	2	no	—
231	C₄×C₈	—	134	2	no	—
232	C₄×C₈	C₂×Dic₆₇	1	4	yes	complemented
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	C₂²×Dic₆₇	1	4	yes	—
236	#41	C₂×Dic₁₃₄	1	4	yes	—
237	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	C₂×D₂₆₈	1	2	yes	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	268	4	no	—
243	C₂²×C₄	—	134	4	no	—
244	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
245	Q₁₆	C₄⋊Dic₆₇	1	4	yes	complemented
246	SD₁₆	C₄⋊Dic₆₇	1	8	yes	complemented
247	D₈	C₄⋊Dic₆₇	1	4	yes	complemented
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	8	no	—
250	M₁₆	—	134	2	no	—
251	M₁₆	—	134	2	no	—
252	M₁₆	C₂²×Dic₆₇	1	4	yes	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	8	no	—
255	C₂×C₈	—	134	2	no	—
256	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
257	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
258	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
259	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
261	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
262	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
263	C₂×C₈	#11761627	1	2	yes	—
264	C₄⋊C₄	—	268	4	no	—
265	C₄⋊C₄	—	536	4	no	—
266	C₄⋊C₄	—	67	4	no	—
267	C₄⋊C₄	—	134	4	no	—
268	C₄⋊C₄	—	134	4	no	—
269	C₂²⋊C₄	—	268	4	no	—
270	C₂²⋊C₄	—	134	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	268	4	no	—
273	C₄²	—	134	2	no	—
274	C₄²	C₂²×Dic₆₇	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	#14524089	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
279	D₄	—	2	4	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	2	no	—
282	D₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	268	4	no	—
289	C₂×C₄	—	536	4	no	—
290	C₂×C₄	—	1072	4	no	—
291	C₂×C₄	—	134	4	no	—
292	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
295	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
296	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
297	C₂×C₄	#14524090	1	1	yes	characteristic
298	C₈	—	2	4	no	—
299	C₈	—	268	2	no	—
300	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
301	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
302	C₈	D₄×Dic₆₇	1	2	yes	—
303	C₈	D₄×Dic₆₇	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	#16863879	1	1	yes	characteristic
308	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
309	C₂²	#16864103	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	2	4	no	—
312	C₄	—	268	4	no	—
313	C₄	—	536	4	no	—
314	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
315	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
316	C₄	#16864050	1	1	yes	characteristic
317	C₄	D₄×Dic₁₃₄	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	4	no	—
321	C₂	D₄×C₄⋊Dic₆₇	1	1	yes	characteristic
322	C₂	#22572003	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#443×C₆₇	C₂
socle	C₂²×C₆₇	D₄²
prime radical	C₂×#135×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#443	D₆₇
hypocenter	C₆₇	#22689
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
2. #22568804	C₂	semidirect
3. #22560990	C₂	semidirect
4. #22560437	C₂	semidirect
5. #22560366	C₂	semidirect
6. #22560268	C₂	semidirect
7. #22560088	C₂	semidirect
28. #443×C₆₇	C₄	semidirect
61. #131×C₆₇	C₂×C₄	semidirect
68. #108×C₆₇	C₂×C₄	semidirect
69. #104×C₆₇	C₂×C₄	semidirect
70. C₄×C₈×C₆₇	C₂×C₄	semidirect
95. Q₁₆×C₆₇	C₄⋊C₄	semidirect
96. SD₁₆×C₆₇	C₄⋊C₄	semidirect
97. D₈×C₆₇	C₄⋊C₄	semidirect
170. C₆₇	#22689	semidirect
174. #443	Dic₆₇	semidirect
201. #131	C₂×Dic₆₇	semidirect
218. #108	C₂×Dic₆₇	semidirect
227. #104	C₂×Dic₆₇	semidirect
232. C₄×C₈	C₂×Dic₆₇	semidirect
245. Q₁₆	C₄⋊Dic₆₇	semidirect
246. SD₁₆	C₄⋊Dic₆₇	semidirect
247. D₈	C₄⋊Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569093	2
#22568804	2
#22560990	2
#22560437	2
#22560366	2
#22560268	2
#22560088	2
C₂×#443×C₆₇	2
#22689	67