Subgroups — #30745302

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863453	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11761612	C₂	1	1	yes	maximal, characteristic
3	#22558043	C₂	1	4	yes	maximal, complemented
4	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₄	1	2	yes	—
6	#16863453	—	2	2	no	—
7	#16863453	C₄	1	4	yes	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
10	C₄×#11761612	—	2	2	no	—
11	C₄×#11761612	—	2	2	no	—
12	C₂×#14523954	—	2	1	no	—
13	C₂×#14523954	C₄	1	2	yes	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₆₇	—	2	1	no	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#423×C₆₇	—	2	2	no	—
18	#423×C₆₇	C₄	1	4	yes	—
19	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×#11761612	—	2	8	no	—
22	C₂×#11761612	—	2	4	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
28	#14523954	—	2	4	no	—
29	#14523954	—	2	4	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	4	no	—
36	C₂×M₁₆×C₆₇	—	4	4	no	—
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	D₄	1	2	yes	—
42	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	D₄	1	2	yes	—
44	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₆₇	—	2	4	no	—
47	C₄⋊C₈×C₆₇	—	2	2	no	—
48	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
49	#11761612	—	2	16	no	—
50	C₂×C₆₇⋊C₈	—	2	4	no	—
51	C₂×C₆₇⋊C₈	—	2	4	no	—
52	C₂×C₆₇⋊C₈	—	2	4	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	M₁₆	1	8	yes	complemented
56	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	4	2	no	—
61	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
73	M₁₆×C₆₇	—	4	8	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	4	4	no	—
76	C₄²×C₆₇	C₄²	1	2	yes	—
77	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₆₇	—	2	2	no	—
87	C₂³×C₆₇	—	2	2	no	—
88	C₂³×C₆₇	—	2	2	no	—
89	C₂³×C₆₇	#95	1	2	yes	—
90	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	#95	1	2	yes	—
103	C₂×C₄×C₆₇	#95	1	2	yes	—
104	C₂×C₄×C₆₇	#95	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
112	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₆₇	#137	1	2	yes	—
114	C₈×C₆₇	—	4	4	no	—
115	C₆₇⋊C₈	—	2	8	no	—
116	C₆₇⋊C₈	—	4	4	no	—
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	—	2	2	no	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	—	4	4	no	—
121	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
122	C₂²×C₆₇	#423	1	2	yes	—
123	C₂²×C₆₇	#428	1	2	yes	—
124	C₂²×C₆₇	#430	1	2	yes	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
131	C₄×C₆₇	#407	1	2	yes	—
132	C₄×C₆₇	#417	1	2	yes	—
133	C₄×C₆₇	#418	1	1	yes	characteristic
134	C₄×C₆₇	#419	1	1	yes	characteristic
135	#7876	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	—	4	2	no	—
137	C₂×C₆₇	#1569	1	4	yes	—
138	C₂×C₆₇	#1728	1	1	yes	characteristic
139	C₂×C₆₇	#1736	1	1	yes	characteristic
140	C₂×C₆₇	#1738	1	1	yes	characteristic
141	C₂×#432	—	67	1	no	—
142	C₂×#423	D₆₇	1	1	yes	characteristic
143	C₂×C₄×M₁₆	—	67	1	no	—
144	#1277	—	67	4	no	—
145	C₆₇	#7876	1	1	yes	characteristic, complemented, Sylow
146	C₂²×M₁₆	—	2	1	no	—
147	C₂²×M₁₆	—	67	2	no	—
148	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
149	#432	—	67	4	no	—
150	#432	—	134	2	no	—
151	#423	—	2	2	no	—
152	#423	Dic₆₇	1	4	yes	—
153	C₂×C₄⋊C₈	—	2	1	no	—
154	C₂×C₄⋊C₈	—	67	2	no	—
155	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
156	C₂×C₂²⋊C₈	—	134	1	no	—
157	C₄×M₁₆	—	134	2	no	—
158	C₄×M₁₆	—	134	2	no	—
159	C₂×#97	—	67	2	no	—
160	C₂×#97	—	134	1	no	—
161	C₂×C₄×C₈	—	67	2	no	—
162	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
163	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
164	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	4	4	no	—
167	C₂×M₁₆	—	134	8	no	—
168	C₂×M₁₆	—	134	4	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	67	4	no	—
171	C₂²×C₈	—	134	2	no	—
172	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
174	C₂×C₄²	C₄×D₆₇	1	2	yes	—
175	C₂×C₄²	D₂₆₈	1	2	yes	—
176	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
177	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
179	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
180	C₄⋊C₈	—	2	4	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	67	4	no	—
183	C₄⋊C₈	—	134	2	no	—
184	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
185	C₂²⋊C₈	—	134	4	no	—
186	#97	—	134	4	no	—
187	#97	—	134	4	no	—
188	C₄×C₈	—	134	2	no	—
189	C₄×C₈	—	134	2	no	—
190	C₂⁴	Dic₆₇⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
196	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
197	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
198	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
199	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
203	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
204	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
207	M₁₆	—	4	8	no	—
208	M₁₆	—	134	16	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	4	no	—
211	C₂×C₈	—	268	2	no	—
212	C₂×C₈	—	268	2	no	—
213	C₂×C₈	—	67	8	no	—
214	C₂×C₈	—	134	4	no	—
215	C₂×C₈	—	134	4	no	—
216	C₂×C₈	—	134	4	no	—
217	C₄²	C₄×Dic₆₇	1	2	yes	—
218	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
219	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
220	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
221	C₄²	Dic₆₇⋊C₄	1	2	yes	—
222	C₄²	C₄⋊Dic₆₇	1	2	yes	—
223	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
224	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
226	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₆₇⋊#95	1	2	yes	—
231	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	C₆₇⋊#95	1	2	yes	—
244	C₂×C₄	C₆₇⋊#95	1	2	yes	—
245	C₂×C₄	C₆₇⋊#95	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
247	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
250	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
252	C₂×C₄	M₁₆×D₆₇	1	4	yes	—
253	C₂×C₄	#14524069	1	2	yes	—
254	C₂×C₄	#14524074	1	2	yes	—
255	C₈	—	4	4	no	—
256	C₈	—	268	4	no	—
257	C₈	—	134	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
263	C₂²	#16863419	1	2	yes	—
264	C₂²	#16863436	1	2	yes	—
265	C₂²	#16863444	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	M₁₆×Dic₆₇	1	2	yes	—
272	C₄	#16863359	1	2	yes	—
273	C₄	#16863395	1	2	yes	—
274	C₄	#16863401	1	1	yes	characteristic
275	C₄	#16863405	1	1	yes	characteristic
276	C₂	—	4	2	no	—
277	C₂	#22558974	1	4	yes	—
278	C₂	#22559902	1	1	yes	characteristic
279	C₂	#22559928	1	1	yes	characteristic
280	C₂	#22559935	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#423×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#423	D₆₇
hypocenter	C₆₇	#7876
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #22558043	C₂	semidirect
55. C₂×C₆₇⋊C₈	M₁₆	semidirect
145. C₆₇	#7876	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863453	2
C₂×C₄×#11761612	2
#22558043	2
C₂×#423×C₆₇	2
#7876	67