Subgroups — #500×Dic₆₇ #30864567

Order 17152.

878 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×Dic₆₇	C₂	1	2	yes	maximal
2	#22561171	C₂	1	2	yes	maximal
3	#22560943	C₂	1	4	yes	maximal
4	#22560369	C₂	1	1	yes	maximal, characteristic
5	#107×Dic₆₇	C₂	1	4	yes	maximal
6	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₆₇	—	2	2	no	—
9	C₂×#14523977	C₂²	1	4	yes	—
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
12	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
13	#16863303	—	2	2	no	—
14	#16863295	—	2	2	no	—
15	#16863295	C₂²	1	4	yes	—
16	#16863266	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₆₇	—	2	2	no	—
18	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
19	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
20	#16863158	C₂²	1	8	yes	—
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	#500×C₆₇	C₄	1	8	yes	complemented
23	C₂×#107×C₆₇	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
26	Q₈×Dic₆₇	—	4	8	no	—
27	C₂×C₄⋊Dic₆₇	—	2	4	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
31	C₂×C₄×Dic₆₇	—	2	4	no	—
32	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
38	#14523977	—	2	8	no	—
39	C₄×C₆₇⋊C₈	—	2	2	no	—
40	C₈×Dic₆₇	—	2	4	no	—
41	C₈×Dic₆₇	Q₈	1	8	yes	complemented
42	C₆₇⋊#95	—	2	4	no	—
43	C₆₇⋊#95	—	2	4	no	—
44	C₂²×Q₈×C₆₇	—	2	2	no	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₇	—	2	4	no	—
47	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	#107×C₆₇	C₂×C₄	1	16	yes	—
52	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
53	C₂²×Dic₆₇	—	2	4	no	—
54	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₆₇⋊C₈	—	2	2	no	—
59	C₂×C₆₇⋊C₈	—	2	2	no	—
60	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
61	C₄⋊Dic₆₇	—	2	8	no	—
62	C₄⋊Dic₆₇	—	4	4	no	—
63	C₄⋊Dic₆₇	—	4	4	no	—
64	C₄⋊Dic₆₇	—	4	8	no	—
65	C₄×Dic₆₇	—	2	2	no	—
66	C₄×Dic₆₇	—	2	2	no	—
67	C₄×Dic₆₇	—	2	2	no	—
68	C₄×Dic₆₇	—	4	8	no	—
69	C₄×Dic₆₇	D₈	1	4	yes	—
70	C₄×Dic₆₇	Q₁₆	1	4	yes	—
71	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
72	C₂×Q₈×C₆₇	—	2	8	no	—
73	C₂×Q₈×C₆₇	—	4	4	no	—
74	C₂×Q₈×C₆₇	—	4	4	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
81	C₂×C₈×C₆₇	#41	1	4	yes	—
82	C₄⋊C₄×C₆₇	—	2	8	no	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
85	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₆₇	#41	1	2	yes	—
87	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
88	Q₈×C₆₇	—	4	16	no	—
89	C₂×C₄×C₆₇	—	2	8	no	—
90	C₂×C₄×C₆₇	—	4	4	no	—
91	C₂×C₄×C₆₇	—	4	4	no	—
92	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
96	C₂×C₄×C₆₇	#135	1	2	yes	—
97	C₂×C₄×C₆₇	#135	1	2	yes	—
98	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
102	C₂×Dic₆₇	—	2	1	no	—
103	C₂×Dic₆₇	—	2	1	no	—
104	C₂×Dic₆₇	—	2	1	no	—
105	C₂×Dic₆₇	—	2	1	no	—
106	C₂×Dic₆₇	—	2	1	no	—
107	C₂×Dic₆₇	—	2	1	no	—
108	C₂×Dic₆₇	—	4	4	no	—
109	C₂×Dic₆₇	—	4	4	no	—
110	C₂×Dic₆₇	—	4	4	no	—
111	C₂×Dic₆₇	C₄⋊Q₈	1	2	yes	—
112	C₂×Dic₆₇	C₄⋊Q₈	1	2	yes	—
113	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
114	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
115	C₂×Dic₆₇	C₂×Q₁₆	1	2	yes	—
116	C₂×Dic₆₇	C₂×Q₁₆	1	2	yes	—
117	C₈×C₆₇	C₄×Q₈	1	8	yes	complemented
118	C₆₇⋊C₈	—	2	4	no	—
119	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂²×C₆₇	#532	1	1	yes	characteristic
122	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
123	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
124	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
126	C₄×C₆₇	—	4	8	no	—
127	C₄×C₆₇	C₄×D₈	1	4	yes	—
128	C₄×C₆₇	C₄×Q₁₆	1	4	yes	—
129	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
130	C₄×C₆₇	#533	1	2	yes	—
131	Dic₆₇	—	2	2	no	—
132	Dic₆₇	—	2	2	no	—
133	Dic₆₇	—	2	2	no	—
134	Dic₆₇	—	8	4	no	—
135	Dic₆₇	#500	1	4	yes	direct product factor
136	C₄×#500	—	67	1	no	maximal, Sylow
137	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
138	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
139	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
140	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
141	C₂×C₆₇	#3160	1	1	yes	characteristic
142	C₂×C₆₇	#3169	1	1	yes	characteristic
143	C₂×C₆₇	#3170	1	1	yes	characteristic
144	C₂×#500	D₆₇	1	1	yes	characteristic
145	C₄×C₄⋊Q₈	—	67	2	no	—
146	#1990	—	67	2	no	—
147	#1942	—	67	4	no	—
148	#1845	—	67	1	no	—
149	C₄×#107	—	67	4	no	—
150	C₄²×C₈	—	67	1	no	—
151	C₆₇	C₄×#500	1	1	yes	characteristic, complemented, Sylow
152	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
153	C₂×C₄×Q₈	—	134	2	no	—
154	#500	Dic₆₇	1	8	yes	direct product factor
155	C₂×#107	—	67	4	no	—
156	C₂×#107	C₂×D₆₇	1	4	yes	—
157	C₂×C₄×C₈	—	67	2	no	—
158	C₂×C₄×C₈	—	67	2	no	—
159	C₂×C₄×C₈	—	67	1	no	—
160	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
161	#391	—	134	2	no	—
162	#389	—	67	4	no	—
163	#389	—	134	2	no	—
164	#384	—	67	2	no	—
165	C₄×C₄⋊C₄	—	67	4	no	—
166	C₄×C₄⋊C₄	—	134	2	no	—
167	C₄³	—	67	1	no	—
168	#340	—	67	8	no	—
169	C₂²×Q₈	—	2	2	no	—
170	C₂²×C₈	—	67	2	no	—
171	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
172	C₄⋊Q₈	—	2	4	no	—
173	C₄⋊Q₈	C₂×Dic₆₇	1	8	yes	—
174	C₄×Q₈	—	268	8	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	67	4	no	—
177	C₂×C₄⋊C₄	—	134	4	no	—
178	C₂×C₄⋊C₄	—	134	2	no	—
179	C₂×C₄⋊C₄	—	134	2	no	—
180	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
181	C₂×C₄²	—	67	1	no	—
182	C₂×C₄²	—	67	1	no	—
183	C₂×C₄²	—	67	1	no	—
184	C₂×C₄²	—	67	1	no	—
185	C₂×C₄²	—	67	1	no	—
186	C₂×C₄²	—	67	1	no	—
187	C₂×C₄²	—	134	4	no	—
188	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
189	#107	—	134	8	no	—
190	#107	C₂×Dic₆₇	1	16	yes	—
191	C₄×C₈	—	67	8	no	—
192	C₄×C₈	—	134	4	no	—
193	C₄×C₈	—	134	2	no	—
194	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
195	#95	—	134	4	no	—
196	#95	—	134	4	no	—
197	C₂×Q₈	—	2	8	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×Q₈	—	4	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	67	1	no	—
202	C₂²×C₄	—	67	1	no	—
203	C₂²×C₄	—	67	1	no	—
204	C₂²×C₄	—	67	1	no	—
205	C₂²×C₄	—	134	4	no	—
206	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
209	C₂×C₈	—	67	4	no	—
210	C₂×C₈	—	134	2	no	—
211	C₂×C₈	—	134	2	no	—
212	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
213	C₂×C₈	Q₈×D₆₇	1	4	yes	—
214	C₂×C₈	#11761629	1	4	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	268	8	no	—
218	C₄⋊C₄	—	268	4	no	—
219	C₄⋊C₄	—	268	4	no	—
220	C₄⋊C₄	—	134	8	no	—
221	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
222	C₄²	—	268	8	no	—
223	C₄²	—	67	4	no	—
224	C₄²	—	67	4	no	—
225	C₄²	—	67	4	no	—
226	C₄²	—	134	2	no	—
227	C₄²	—	134	2	no	—
228	C₄²	—	134	2	no	—
229	C₄²	C₂²×Dic₆₇	1	2	yes	—
230	C₄²	#11761627	1	2	yes	—
231	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
232	Q₈	—	4	16	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	268	4	no	—
237	C₂×C₄	—	268	4	no	—
238	C₂×C₄	—	268	4	no	—
239	C₂×C₄	—	67	2	no	—
240	C₂×C₄	—	67	2	no	—
241	C₂×C₄	—	67	2	no	—
242	C₂×C₄	—	67	2	no	—
243	C₂×C₄	—	67	2	no	—
244	C₂×C₄	—	67	2	no	—
245	C₂×C₄	—	134	1	no	—
246	C₂×C₄	—	134	1	no	—
247	C₂×C₄	—	134	1	no	—
248	C₂×C₄	—	134	1	no	—
249	C₂×C₄	—	134	1	no	—
250	C₂×C₄	—	134	1	no	—
251	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
252	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
253	C₂×C₄	D₈×D₆₇	1	2	yes	—
254	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
255	C₂×C₄	#14524062	1	2	yes	—
256	C₂×C₄	#14524066	1	2	yes	—
257	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
259	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
260	C₂×C₄	C₂×#11761629	1	2	yes	—
261	C₈	—	134	4	no	—
262	C₈	Q₈×Dic₆₇	1	8	yes	complemented
263	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
264	C₂²	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
265	C₂²	#16863947	1	1	yes	characteristic
266	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
267	C₂²	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
268	C₂²	C₂×#14524062	1	1	yes	characteristic
269	C₂²	C₂×#14524066	1	1	yes	characteristic
270	C₄	—	4	8	no	—
271	C₄	—	536	4	no	—
272	C₄	—	67	4	no	—
273	C₄	—	134	2	no	—
274	C₄	—	134	2	no	—
275	C₄	—	134	2	no	—
276	C₄	D₈×Dic₆₇	1	4	yes	—
277	C₄	Q₁₆×Dic₆₇	1	4	yes	—
278	C₄	C₂×Q₈×Dic₆₇	1	2	yes	—
279	C₄	#16863951	1	2	yes	—
280	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
281	C₂	C₂×D₈×Dic₆₇	1	1	yes	characteristic
282	C₂	C₂×Q₁₆×Dic₆₇	1	1	yes	characteristic
283	C₂	#500×D₆₇	1	1	yes	characteristic
284	C₂	#22570720	1	1	yes	characteristic
285	C₂	#22570789	1	1	yes	characteristic
286	C₂	#22570796	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	C₄×#500
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
22. #500×C₆₇	C₄	semidirect
41. C₈×Dic₆₇	Q₈	semidirect
117. C₈×C₆₇	C₄×Q₈	semidirect
135. Dic₆₇	#500	direct (N × H)
151. C₆₇	C₄×#500	semidirect
154. #500	Dic₆₇	direct (N × H)
262. C₈	Q₈×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×Dic₆₇	2
#22561171	2
#22560943	2
#22560369	2
#107×Dic₆₇	2
C₄×C₈×Dic₆₇	2
C₂×#500×C₆₇	2
C₄×#500	67

Subgroups — #500×Dic67 #30864567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #500×Dic₆₇ #30864567