Subgroups — #30821547

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	2	yes	maximal
2	#22558512	C₂	1	4	yes	maximal, complemented
3	#22558506	C₂	1	4	yes	maximal, complemented
4	#22558501	C₂	1	2	yes	maximal, complemented
5	#22558492	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863453	C₄	1	4	yes	—
8	#16863453	C₄	1	4	yes	—
9	#16863449	C₂²	1	4	yes	—
10	#16863449	C₂²	1	4	yes	—
11	#16863420	C₄	1	4	yes	—
12	#16863360	C₄	1	4	yes	—
13	C₄×#11761612	C₂²	1	4	yes	—
14	#16863118	C₂²	1	8	yes	complemented
15	#16863117	C₂²	1	4	yes	complemented
16	C₈×C₆₇⋊C₈	C₂²	1	4	yes	complemented
17	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₆₇	C₄	1	2	yes	—
20	C₄×M₁₆×C₆₇	C₄	1	2	yes	—
21	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
22	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
23	C₂×#97×C₆₇	C₄	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#11761612	C₂×C₄	1	8	yes	—
27	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	—
28	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	—
29	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	—
30	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	—
31	#14523954	C₂×C₄	1	8	yes	—
32	C₄×C₆₇⋊C₈	C₂³	1	8	yes	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	8	no	—
36	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	#97×C₆₇	C₂×C₄	1	4	yes	—
50	#97×C₆₇	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	C₄×C₈×C₆₇	C₂³	1	2	yes	—
55	#11761612	—	4	8	no	—
56	C₂×C₆₇⋊C₈	C₂²×C₄	1	16	yes	—
57	C₂⁴×C₆₇	C₄²	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	4	4	no	—
60	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
64	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
65	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	M₁₆×C₆₇	—	2	8	no	—
73	M₁₆×C₆₇	—	2	8	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	M₁₆	1	8	yes	complemented
76	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
78	C₄²×C₆₇	C₄²	1	2	yes	—
79	C₄²×C₆₇	C₄²	1	2	yes	—
80	C₄²×C₆₇	C₄²	1	2	yes	—
81	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
88	C₂³×C₆₇	—	2	2	no	—
89	C₂³×C₆₇	—	4	2	no	—
90	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
91	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
92	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₆₇	—	2	4	no	—
114	C₈×C₆₇	C₂×M₁₆	1	8	yes	—
115	C₆₇⋊C₈	—	2	16	no	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	—	2	1	no	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	—	2	2	no	—
121	C₂²×C₆₇	—	4	4	no	—
122	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
123	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	—	2	1	no	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	—	2	2	no	—
129	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
130	C₄×C₆₇	C₂²×M₁₆	1	2	yes	—
131	C₄×C₆₇	#568	1	4	yes	—
132	#17334	—	67	1	no	maximal, Sylow
133	C₂×C₆₇	—	2	2	no	—
134	C₂×C₆₇	—	4	1	no	—
135	C₂×C₆₇	—	4	1	no	—
136	C₂×C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic
137	C₂×C₆₇	#2873	1	2	yes	—
138	#2920	—	67	2	no	—
139	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
140	#1455	—	67	4	no	—
141	#1453	—	67	4	no	—
142	#1452	—	67	2	no	—
143	#1450	—	67	2	no	—
144	C₆₇	#17334	1	1	yes	characteristic, complemented, Sylow
145	C₂²×M₁₆	Dic₆₇	1	2	yes	—
146	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
147	#432	—	67	4	no	—
148	#432	—	67	4	no	—
149	#431	—	67	4	no	—
150	#431	—	67	4	no	—
151	#423	—	67	4	no	—
152	#407	—	67	4	no	—
153	C₄×M₁₆	—	67	4	no	—
154	C₄×M₁₆	Dic₆₇	1	2	yes	—
155	C₄×M₁₆	Dic₆₇	1	2	yes	—
156	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
157	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
158	C₂×#97	Dic₆₇	1	2	yes	—
159	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
161	#322	—	67	4	no	—
162	#322	—	67	8	no	—
163	C₈²	—	67	4	no	—
164	C₂³×C₄	C₄×D₆₇	1	2	yes	—
165	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
166	C₂×M₁₆	—	2	8	no	—
167	C₂×M₁₆	—	67	8	no	—
168	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
169	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
170	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
171	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
172	C₂×C₄²	C₄×D₆₇	1	2	yes	—
173	C₂×C₄²	C₄×D₆₇	1	2	yes	—
174	C₂×C₄²	C₄×D₆₇	1	2	yes	—
175	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
176	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
181	C₄⋊C₈	—	67	8	no	—
182	C₄⋊C₈	—	67	8	no	—
183	C₂²⋊C₈	—	67	8	no	—
184	C₂²⋊C₈	—	67	8	no	—
185	#97	—	67	8	no	—
186	#97	C₂×Dic₆₇	1	4	yes	—
187	#97	C₂×Dic₆₇	1	4	yes	—
188	C₄×C₈	—	67	8	no	—
189	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
190	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
191	C₄×C₈	C₂²×D₆₇	1	2	yes	—
192	C₄×C₈	C₂²×D₆₇	1	2	yes	—
193	C₂⁴	C₄×Dic₆₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
197	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
200	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
201	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
203	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
204	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
208	M₁₆	—	2	8	no	—
209	M₁₆	—	2	8	no	—
210	M₁₆	—	268	8	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	67	16	no	—
213	C₂×C₈	#11761612	1	8	yes	complemented
214	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
215	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
216	C₄²	C₄×Dic₆₇	1	2	yes	—
217	C₄²	C₄×Dic₆₇	1	2	yes	—
218	C₄²	C₄×Dic₆₇	1	2	yes	—
219	C₄²	C₂×C₄×D₆₇	1	2	yes	—
220	C₄²	C₂×C₄×D₆₇	1	2	yes	—
221	C₄²	C₂×C₄×D₆₇	1	2	yes	—
222	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
223	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
224	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
225	C₄²	C₂³×D₆₇	1	1	yes	characteristic
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
229	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
230	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
247	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
248	C₂×C₄	C₂×#11761612	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
250	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
251	C₈	—	2	4	no	—
252	C₈	—	134	16	no	—
253	C₈	C₂×#11761612	1	8	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	4	no	—
260	C₂²	C₄×#11761612	1	2	yes	—
261	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	C₄×#11761612	1	2	yes	—
268	C₄	C₂²×#11761612	1	2	yes	—
269	C₄	#16864162	1	4	yes	—
270	C₂	—	2	2	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	1	no	—
273	C₂	C₂×C₄×#11761612	1	1	yes	characteristic
274	C₂	#22568627	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₄²
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
prime radical	C₂⁴×C₆₇	C₄²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#17334
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22558512	C₂	semidirect
3. #22558506	C₂	semidirect
4. #22558501	C₂	semidirect
5. #22558492	C₂	semidirect
14. #16863118	C₂²	semidirect
15. #16863117	C₂²	semidirect
16. C₈×C₆₇⋊C₈	C₂²	semidirect
75. C₂×C₈×C₆₇	M₁₆	semidirect
144. C₆₇	#17334	semidirect
213. C₂×C₈	#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
#22558512	2
#22558506	2
#22558501	2
#22558492	2
C₂×C₄×M₁₆×C₆₇	2
#17334	67