Subgroups — C₆₇×#33706 #30712957

Order 17152.

598 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	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2243	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2242	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2141	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2140	C₂	1	2	yes	maximal
7	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×SD₁₆×C₆₇	—	2	1	no	—
10	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
11	#533×C₆₇	—	2	1	no	—
12	C₂×#124×C₆₇	—	2	1	no	—
13	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
14	#500×C₆₇	C₂²	1	2	yes	complemented
15	#499×C₆₇	—	2	1	no	—
16	#499×C₆₇	C₂²	1	2	yes	complemented
17	#498×C₆₇	—	2	1	no	—
18	#495×C₆₇	—	2	1	no	—
19	#495×C₆₇	—	2	1	no	—
20	#495×C₆₇	C₂²	1	2	yes	—
21	#495×C₆₇	C₂²	1	2	yes	—
22	#427×C₆₇	—	2	1	no	—
23	#427×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#358×C₆₇	C₂²	1	8	yes	complemented
26	#357×C₆₇	C₂²	1	8	yes	—
27	#346×C₆₇	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₆₇	C₂²	1	2	yes	complemented
29	C₂²×Q₈×C₆₇	—	2	1	no	—
30	C₂²×D₄×C₆₇	—	2	1	no	—
31	C₂×Q₁₆×C₆₇	—	2	1	no	—
32	C₂×Q₁₆×C₆₇	—	2	4	no	—
33	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
34	C₂×SD₁₆×C₆₇	—	2	2	no	—
35	C₂×SD₁₆×C₆₇	—	2	2	no	—
36	C₂×SD₁₆×C₆₇	—	4	4	no	—
37	C₂×D₈×C₆₇	—	2	1	no	—
38	C₂×D₈×C₆₇	—	2	4	no	—
39	C₂×D₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	#125×C₆₇	—	2	2	no	—
44	#124×C₆₇	—	2	2	no	—
45	#124×C₆₇	—	2	2	no	—
46	#124×C₆₇	—	4	2	no	—
47	C₂²⋊Q₈×C₆₇	—	4	2	no	—
48	C₄×Q₈×C₆₇	—	4	2	no	—
49	#117×C₆₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	M₃₂×C₆₇	—	2	4	no	—
53	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
54	#107×C₆₇	—	2	2	no	—
55	#107×C₆₇	C₂³	1	4	yes	—
56	#106×C₆₇	—	2	2	no	—
57	#106×C₆₇	—	2	2	no	—
58	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂⁴×C₆₇	—	4	1	no	—
63	C₂×Q₈×C₆₇	—	2	2	no	—
64	C₂×Q₈×C₆₇	—	4	2	no	—
65	C₂×Q₈×C₆₇	—	4	1	no	—
66	C₂×Q₈×C₆₇	—	4	2	no	—
67	C₂×Q₈×C₆₇	—	4	2	no	—
68	C₂×D₄×C₆₇	—	2	2	no	—
69	C₂×D₄×C₆₇	—	4	2	no	—
70	C₂×D₄×C₆₇	—	4	1	no	—
71	C₂×D₄×C₆₇	—	4	2	no	—
72	C₂²×C₄×C₆₇	—	4	1	no	—
73	C₂²×C₄×C₆₇	D₈	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	Q₁₆×C₆₇	—	2	4	no	—
76	Q₁₆×C₆₇	—	4	2	no	—
77	SD₁₆×C₆₇	—	4	4	no	—
78	SD₁₆×C₆₇	—	4	4	no	—
79	D₈×C₆₇	—	2	4	no	—
80	D₈×C₆₇	—	4	2	no	—
81	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
82	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	#41	1	4	yes	—
86	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	4	4	no	—
89	C₄⋊C₄×C₆₇	—	4	2	no	—
90	C₂²⋊C₄×C₆₇	—	4	2	no	—
91	C₂²⋊C₄×C₆₇	—	4	4	no	—
92	C₂²⋊C₄×C₆₇	—	4	4	no	—
93	C₄²×C₆₇	—	4	2	no	—
94	C₄²×C₆₇	D₈	1	2	yes	—
95	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₁₆×C₆₇	—	2	4	no	—
98	C₂³×C₆₇	—	4	2	no	—
99	C₂³×C₆₇	—	8	4	no	—
100	C₂³×C₆₇	—	8	2	no	—
101	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
102	Q₈×C₆₇	—	4	2	no	—
103	Q₈×C₆₇	—	4	2	no	—
104	Q₈×C₆₇	—	8	4	no	—
105	Q₈×C₆₇	—	8	2	no	—
106	D₄×C₆₇	—	4	2	no	—
107	D₄×C₆₇	—	4	2	no	—
108	D₄×C₆₇	—	8	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	2	no	—
112	C₂×C₄×C₆₇	—	8	2	no	—
113	C₂×C₄×C₆₇	#124	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₆₇	#124	1	2	yes	—
120	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
121	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
122	C₈×C₆₇	C₂×#41	1	2	yes	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	8	2	no	—
125	C₂²×C₆₇	—	8	4	no	—
126	C₂²×C₆₇	—	16	4	no	—
127	C₂²×C₆₇	#486	1	2	yes	—
128	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	8	2	no	—
132	C₄×C₆₇	—	8	4	no	—
133	C₄×C₆₇	#486	1	2	yes	—
134	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
135	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
136	#33706	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	—	16	2	no	—
139	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
140	C₂×C₆₇	#3414	1	2	yes	—
141	#3160	C₂×C₆₇	1	1	yes	characteristic, complemented
142	C₂×#495	C₂×C₆₇	1	1	yes	characteristic
143	#2243	C₂×C₆₇	1	4	yes	complemented
144	#2242	C₂×C₆₇	1	4	yes	complemented
145	#2141	C₂×C₆₇	1	2	yes	complemented
146	#2140	C₂×C₆₇	1	2	yes	—
147	C₄×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
148	C₆₇	#33706	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
150	C₂²×SD₁₆	—	2	1	no	—
151	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
152	#533	—	2	1	no	—
153	C₂×#124	—	2	1	no	—
154	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
155	#500	C₂²×C₆₇	1	2	yes	complemented
156	#499	—	2	1	no	—
157	#499	C₂²×C₆₇	1	2	yes	complemented
158	#498	—	2	1	no	—
159	#495	—	2	1	no	—
160	#495	—	2	1	no	—
161	#495	C₂²×C₆₇	1	2	yes	—
162	#495	C₂²×C₆₇	1	2	yes	—
163	#427	—	2	1	no	—
164	#427	C₂²×C₆₇	1	2	yes	—
165	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
166	#358	C₂²×C₆₇	1	8	yes	complemented
167	#357	C₂²×C₆₇	1	8	yes	—
168	#346	C₂²×C₆₇	1	2	yes	complemented
169	C₄×C₁₆	C₂²×C₆₇	1	2	yes	complemented
170	C₂²×Q₈	—	2	1	no	—
171	C₂²×D₄	—	2	1	no	—
172	C₂×Q₁₆	—	2	1	no	—
173	C₂×Q₁₆	—	2	4	no	—
174	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×SD₁₆	—	2	2	no	—
177	C₂×SD₁₆	—	4	4	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	—	2	4	no	—
180	C₂×D₈	C₂³×C₆₇	1	2	yes	—
181	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	2	no	—
184	#125	—	2	2	no	—
185	#124	—	2	2	no	—
186	#124	—	2	2	no	—
187	#124	—	4	2	no	—
188	C₂²⋊Q₈	—	4	2	no	—
189	C₄×Q₈	—	4	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	4	2	no	—
192	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
193	M₃₂	—	2	4	no	—
194	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
195	#107	—	2	2	no	—
196	#107	C₂³×C₆₇	1	4	yes	—
197	#106	—	2	2	no	—
198	#106	—	2	2	no	—
199	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
200	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
203	C₂⁴	—	4	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	D₈×C₆₇	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
216	Q₁₆	—	2	4	no	—
217	Q₁₆	—	4	2	no	—
218	SD₁₆	—	4	4	no	—
219	SD₁₆	—	4	4	no	—
220	D₈	—	2	4	no	—
221	D₈	—	4	2	no	—
222	C₂×C₈	SD₁₆×C₆₇	1	4	yes	—
223	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
226	C₂×C₈	#41×C₆₇	1	4	yes	—
227	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	4	2	no	—
235	C₄²	D₈×C₆₇	1	2	yes	—
236	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
238	C₁₆	—	2	4	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	4	no	—
241	C₂³	—	8	2	no	—
242	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	8	4	no	—
246	Q₈	—	8	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	8	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	#124×C₆₇	1	2	yes	—
255	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
256	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
257	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
259	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
260	C₈	#124×C₆₇	1	2	yes	—
261	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
262	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
263	C₈	C₂×#41×C₆₇	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	16	4	no	—
268	C₂²	#486×C₆₇	1	2	yes	—
269	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	8	2	no	—
273	C₄	—	8	4	no	—
274	C₄	#486×C₆₇	1	2	yes	—
275	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
276	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×#486×C₆₇	1	1	yes	characteristic
280	C₂	C₆₇×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3160	C₂	semidirect
3. C₆₇×#2243	C₂	semidirect
4. C₆₇×#2242	C₂	semidirect
5. C₆₇×#2141	C₂	semidirect
7. C₄×M₃₂×C₆₇	C₂	semidirect
14. #500×C₆₇	C₂²	semidirect
16. #499×C₆₇	C₂²	semidirect
25. #358×C₆₇	C₂²	semidirect
27. #346×C₆₇	C₂²	semidirect
28. C₄×C₁₆×C₆₇	C₂²	semidirect
136. #33706	C₆₇	direct (N × H)
141. #3160	C₂×C₆₇	semidirect
143. #2243	C₂×C₆₇	semidirect
144. #2242	C₂×C₆₇	semidirect
145. #2141	C₂×C₆₇	semidirect
147. C₄×M₃₂	C₂×C₆₇	semidirect
148. C₆₇	#33706	direct (N × H)
155. #500	C₂²×C₆₇	semidirect
157. #499	C₂²×C₆₇	semidirect
166. #358	C₂²×C₆₇	semidirect
168. #346	C₂²×C₆₇	semidirect
169. C₄×C₁₆	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3160	2
C₂×#495×C₆₇	2
C₆₇×#2243	2
C₆₇×#2242	2
C₆₇×#2141	2
C₆₇×#2140	2
C₄×M₃₂×C₆₇	2
#33706	67

Subgroups — C67×#33706 #30712957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33706 #30712957