Subgroups — #499×Dic₆₇ #30864576

Order 17152.

774 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	#125×Dic₆₇	C₂	1	2	yes	maximal
2	#22561175	C₂	1	2	yes	maximal
3	#22560943	C₂	1	2	yes	maximal
4	#22560938	C₂	1	2	yes	maximal
5	#22560373	C₂	1	1	yes	maximal, characteristic
6	#107×Dic₆₇	C₂	1	2	yes	maximal
7	#106×Dic₆₇	C₂	1	2	yes	maximal
8	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#14523977	C₂²	1	2	yes	—
11	C₂×#14523975	C₂²	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
17	#16863295	C₂²	1	4	yes	—
18	#16863282	—	2	4	no	—
19	#16863262	C₂²	1	2	yes	—
20	C₄⋊C₄×Dic₆₇	—	2	4	no	—
21	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
22	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
23	#16863158	C₂²	1	4	yes	—
24	#16863158	C₂²	1	4	yes	—
25	C₂×#125×C₆₇	C₂²	1	2	yes	—
26	#499×C₆₇	C₄	1	8	yes	complemented
27	C₂×#107×C₆₇	C₂²	1	2	yes	—
28	C₂×#106×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₆₇	—	2	4	no	—
33	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
34	C₂×C₄×Dic₆₇	—	2	4	no	—
35	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
36	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
37	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
39	#14523977	—	2	4	no	—
40	#14523975	—	2	4	no	—
41	C₄×C₆₇⋊C₈	—	2	2	no	—
42	C₈×Dic₆₇	—	2	2	no	—
43	C₈×Dic₆₇	—	2	2	no	—
44	C₈×Dic₆₇	Q₈	1	4	yes	—
45	C₈×Dic₆₇	Q₈	1	4	yes	—
46	C₆₇⋊#95	—	2	4	no	—
47	C₆₇⋊#95	—	2	4	no	—
48	C₆₇⋊#95	—	2	4	no	—
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	#125×C₆₇	—	2	4	no	—
52	#125×C₆₇	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	#107×C₆₇	C₂×C₄	1	8	yes	—
57	#106×C₆₇	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₂²×Dic₆₇	—	2	4	no	—
60	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
62	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₆₇⋊C₈	—	2	1	no	—
64	C₂×C₆₇⋊C₈	—	2	1	no	—
65	C₂×C₆₇⋊C₈	—	2	1	no	—
66	C₂×C₆₇⋊C₈	—	2	1	no	—
67	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
68	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₄⋊Dic₆₇	—	4	8	no	—
72	C₄×Dic₆₇	—	2	1	no	—
73	C₄×Dic₆₇	—	2	1	no	—
74	C₄×Dic₆₇	—	2	4	no	—
75	C₄×Dic₆₇	—	2	4	no	—
76	C₄×Dic₆₇	—	4	8	no	—
77	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
78	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₆₇	#41	1	2	yes	—
87	C₂×C₈×C₆₇	#41	1	2	yes	—
88	C₄⋊C₄×C₆₇	—	2	16	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
91	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₆₇	#41	1	2	yes	—
93	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	8	no	—
97	C₂×C₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	—	4	4	no	—
99	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
106	C₂×Dic₆₇	—	2	2	no	—
107	C₂×Dic₆₇	—	2	2	no	—
108	C₂×Dic₆₇	—	2	2	no	—
109	C₂×Dic₆₇	—	4	4	no	—
110	C₂×Dic₆₇	—	4	4	no	—
111	C₂×Dic₆₇	—	4	4	no	—
112	C₂×Dic₆₇	C₄⋊Q₈	1	2	yes	—
113	C₂×Dic₆₇	C₄⋊Q₈	1	2	yes	—
114	C₂×Dic₆₇	#135	1	4	yes	—
115	C₂×Dic₆₇	#135	1	4	yes	—
116	C₈×C₆₇	C₄×Q₈	1	4	yes	—
117	C₈×C₆₇	C₄×Q₈	1	4	yes	—
118	C₆₇⋊C₈	—	2	2	no	—
119	C₆₇⋊C₈	—	2	2	no	—
120	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₆₇	#532	1	1	yes	characteristic
123	C₂²×C₆₇	C₂×#135	1	2	yes	—
124	C₂²×C₆₇	C₂×#135	1	2	yes	—
125	C₄×C₆₇	—	2	4	no	—
126	C₄×C₆₇	—	4	8	no	—
127	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
128	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
129	C₄×C₆₇	#533	1	1	yes	characteristic
130	C₄×C₆₇	#533	1	1	yes	characteristic
131	Dic₆₇	—	2	2	no	—
132	Dic₆₇	—	2	4	no	—
133	Dic₆₇	—	8	4	no	—
134	Dic₆₇	#499	1	4	yes	direct product factor
135	C₄×#499	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₆₇	C₄×#135	1	2	yes	—
138	C₂×C₆₇	C₂×#499	1	1	yes	characteristic
139	C₂×C₆₇	#3160	1	1	yes	characteristic
140	C₂×C₆₇	#3168	1	2	yes	—
141	C₂×#499	D₆₇	1	1	yes	characteristic
142	C₄×#125	—	67	2	no	—
143	#1991	—	67	2	no	—
144	#1942	—	67	2	no	—
145	#1941	—	67	2	no	—
146	#1846	—	67	1	no	—
147	C₄×#107	—	67	2	no	—
148	C₄×#106	—	67	2	no	—
149	C₄²×C₈	—	67	1	no	—
150	C₆₇	C₄×#499	1	1	yes	characteristic, complemented, Sylow
151	C₂×#125	C₂×D₆₇	1	2	yes	—
152	#499	Dic₆₇	1	8	yes	direct product factor
153	C₂×#107	—	67	2	no	—
154	C₂×#107	C₂×D₆₇	1	2	yes	—
155	C₂×#106	—	67	2	no	—
156	C₂×#106	C₂×D₆₇	1	2	yes	—
157	C₂×C₄×C₈	—	67	1	no	—
158	C₂×C₄×C₈	—	67	1	no	—
159	C₂×C₄×C₈	—	67	1	no	—
160	C₂×C₄×C₈	—	67	1	no	—
161	C₂×C₄×C₈	—	67	1	no	—
162	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
163	#389	—	67	4	no	—
164	#387	—	134	4	no	—
165	#382	—	67	2	no	—
166	C₄×C₄⋊C₄	—	67	4	no	—
167	C₄×C₄⋊C₄	—	134	4	no	—
168	C₄³	—	67	1	no	—
169	#340	—	67	4	no	—
170	#340	—	67	4	no	—
171	C₂²×C₈	—	67	1	no	—
172	C₂²×C₈	—	67	1	no	—
173	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
175	#125	—	2	4	no	—
176	#125	C₂×Dic₆₇	1	8	yes	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	—	67	4	no	—
179	C₂×C₄⋊C₄	—	134	4	no	—
180	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
181	C₂×C₄²	—	67	1	no	—
182	C₂×C₄²	—	67	2	no	—
183	C₂×C₄²	—	67	2	no	—
184	C₂×C₄²	—	67	1	no	—
185	C₂×C₄²	—	134	4	no	—
186	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
187	#107	—	134	4	no	—
188	#107	C₂×Dic₆₇	1	8	yes	—
189	#106	—	134	4	no	—
190	#106	C₂×Dic₆₇	1	8	yes	—
191	C₄×C₈	—	67	4	no	—
192	C₄×C₈	—	67	4	no	—
193	C₄×C₈	—	134	2	no	—
194	C₄×C₈	—	134	2	no	—
195	C₄×C₈	—	134	2	no	—
196	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
197	#95	—	134	4	no	—
198	#95	—	134	4	no	—
199	#95	—	134	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	67	1	no	—
202	C₂²×C₄	—	67	1	no	—
203	C₂²×C₄	—	67	2	no	—
204	C₂²×C₄	—	134	4	no	—
205	C₂²×C₄	D₄×D₆₇	1	2	yes	—
206	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
207	C₂×C₈	—	67	2	no	—
208	C₂×C₈	—	67	2	no	—
209	C₂×C₈	—	134	1	no	—
210	C₂×C₈	—	134	1	no	—
211	C₂×C₈	—	134	1	no	—
212	C₂×C₈	—	134	1	no	—
213	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
214	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
215	C₂×C₈	Q₈×D₆₇	1	2	yes	—
216	C₂×C₈	Q₈×D₆₇	1	2	yes	—
217	C₂×C₈	#11761629	1	2	yes	—
218	C₂×C₈	#11761629	1	2	yes	—
219	C₄⋊C₄	—	2	16	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	268	8	no	—
222	C₄⋊C₄	—	134	4	no	—
223	C₄⋊C₄	—	134	4	no	—
224	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
225	C₄²	—	268	8	no	—
226	C₄²	—	67	2	no	—
227	C₄²	—	67	2	no	—
228	C₄²	—	134	4	no	—
229	C₄²	—	134	4	no	—
230	C₄²	—	134	1	no	—
231	C₄²	—	134	1	no	—
232	C₄²	C₂²×Dic₆₇	1	2	yes	—
233	C₄²	#11761627	1	2	yes	—
234	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	268	4	no	—
241	C₂×C₄	—	268	4	no	—
242	C₂×C₄	—	268	4	no	—
243	C₂×C₄	—	67	4	no	—
244	C₂×C₄	—	67	2	no	—
245	C₂×C₄	—	67	4	no	—
246	C₂×C₄	—	67	2	no	—
247	C₂×C₄	—	134	2	no	—
248	C₂×C₄	—	134	2	no	—
249	C₂×C₄	—	134	2	no	—
250	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
251	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
252	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
253	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
254	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
255	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
256	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
257	C₈	—	134	2	no	—
258	C₈	—	134	2	no	—
259	C₈	Q₈×Dic₆₇	1	4	yes	—
260	C₈	Q₈×Dic₆₇	1	4	yes	—
261	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
262	C₂²	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
263	C₂²	#16863947	1	1	yes	characteristic
264	C₂²	#135×D₆₇	1	2	yes	—
265	C₂²	#135×D₆₇	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	536	4	no	—
269	C₄	—	67	4	no	—
270	C₄	—	134	4	no	—
271	C₄	—	134	2	no	—
272	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
273	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
274	C₄	#16863951	1	1	yes	characteristic
275	C₄	#16863951	1	1	yes	characteristic
276	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
277	C₂	#135×Dic₆₇	1	2	yes	—
278	C₂	#499×D₆₇	1	1	yes	characteristic
279	C₂	#22570722	1	1	yes	characteristic
280	C₂	#22570781	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#499×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#499	D₆₇
hypocenter	C₆₇	C₄×#499
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
26. #499×C₆₇	C₄	semidirect
134. Dic₆₇	#499	direct (N × H)
150. C₆₇	C₄×#499	semidirect
152. #499	Dic₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
#125×Dic₆₇	2
#22561175	2
#22560943	2
#22560938	2
#22560373	2
#107×Dic₆₇	2
#106×Dic₆₇	2
C₄×C₈×Dic₆₇	2
C₂×#499×C₆₇	2
C₄×#499	67

Subgroups — #499×Dic67 #30864576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #499×Dic₆₇ #30864576