Subgroups — C₆₇×#33463 #30712714

Order 17152.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3414	C₂	1	1	yes	maximal, characteristic
2	C₂×#443×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2179	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2178	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2143	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₆₇	C₂²	1	2	yes	—
9	C₂×#131×C₆₇	—	2	2	no	—
10	#510×C₆₇	C₄	1	4	yes	complemented
11	#509×C₆₇	C₄	1	4	yes	complemented
12	#508×C₆₇	C₂²	1	4	yes	—
13	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	#443×C₆₇	—	2	2	no	—
16	#443×C₆₇	—	2	2	no	—
17	#443×C₆₇	C₂²	1	4	yes	complemented
18	#443×C₆₇	C₂²	1	4	yes	complemented
19	C₂×#108×C₆₇	C₂²	1	2	yes	—
20	C₂×#104×C₆₇	—	2	2	no	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#368×C₆₇	C₂²	1	4	yes	complemented
23	#359×C₆₇	C₂²	1	8	yes	complemented
24	#346×C₆₇	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₆₇	C₂²	1	2	yes	complemented
26	C₂×#41×C₆₇	—	2	2	no	—
27	#135×C₆₇	—	2	2	no	—
28	#135×C₆₇	—	2	2	no	—
29	#135×C₆₇	C₂×C₄	1	4	yes	—
30	#135×C₆₇	C₂³	1	4	yes	—
31	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×SD₁₆×C₆₇	—	2	2	no	—
33	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
34	#131×C₆₇	—	2	4	no	—
35	#131×C₆₇	—	2	4	no	—
36	#131×C₆₇	—	4	2	no	—
37	#131×C₆₇	—	4	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	4	2	no	—
40	C₂²×C₈×C₆₇	—	4	2	no	—
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
44	Q₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
45	SD₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
46	D₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
47	M₃₂×C₆₇	—	2	2	no	—
48	M₃₂×C₆₇	C₂×C₄	1	4	yes	—
49	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
51	#108×C₆₇	—	2	2	no	—
52	#108×C₆₇	C₂³	1	4	yes	—
53	#104×C₆₇	—	2	4	no	—
54	#104×C₆₇	—	2	4	no	—
55	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	#41×C₆₇	—	2	4	no	—
60	#41×C₆₇	—	2	4	no	—
61	#41×C₆₇	—	4	2	no	—
62	#41×C₆₇	—	4	2	no	—
63	C₂×Q₈×C₆₇	—	2	2	no	—
64	C₂×D₄×C₆₇	—	2	2	no	—
65	C₂×D₄×C₆₇	—	4	2	no	—
66	C₂²×C₄×C₆₇	—	2	1	no	—
67	C₂²×C₄×C₆₇	—	4	2	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	Q₁₆×C₆₇	—	2	2	no	—
70	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
71	SD₁₆×C₆₇	—	2	4	no	—
72	SD₁₆×C₆₇	—	2	4	no	—
73	D₈×C₆₇	—	2	2	no	—
74	D₈×C₆₇	C₂²×C₄	1	4	yes	—
75	M₁₆×C₆₇	—	2	4	no	—
76	M₁₆×C₆₇	—	4	2	no	—
77	M₁₆×C₆₇	—	4	4	no	—
78	M₁₆×C₆₇	—	4	4	no	—
79	C₂×C₈×C₆₇	—	4	2	no	—
80	C₂×C₈×C₆₇	—	4	4	no	—
81	C₂×C₈×C₆₇	—	4	4	no	—
82	C₂×C₈×C₆₇	D₈	1	2	yes	—
83	C₂×C₈×C₆₇	D₈	1	2	yes	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	#41	1	2	yes	—
89	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
90	C₄²×C₆₇	—	2	1	no	—
91	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₆₇	—	2	2	no	—
94	C₁₆×C₆₇	C₂²×C₄	1	4	yes	—
95	C₂³×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	#135	1	1	yes	characteristic
97	Q₈×C₆₇	—	2	4	no	—
98	Q₈×C₆₇	—	4	2	no	—
99	D₄×C₆₇	—	2	4	no	—
100	D₄×C₆₇	—	4	4	no	—
101	D₄×C₆₇	—	4	4	no	—
102	D₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₆₇	—	4	4	no	—
114	C₈×C₆₇	C₄×D₄	1	2	yes	—
115	C₈×C₆₇	C₂×D₈	1	2	yes	—
116	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
117	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
118	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
120	C₂²×C₆₇	—	4	1	no	—
121	C₂²×C₆₇	—	4	4	no	—
122	C₂²×C₆₇	—	8	4	no	—
123	C₂²×C₆₇	C₄×D₈	1	2	yes	—
124	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	4	1	no	—
127	C₄×C₆₇	—	4	4	no	—
128	C₄×C₆₇	C₄×D₈	1	2	yes	—
129	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
131	#33463	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₆₇	—	2	1	no	—
133	C₂×C₆₇	—	2	2	no	—
134	C₂×C₆₇	—	8	4	no	—
135	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
136	#3414	C₂×C₆₇	1	1	yes	characteristic
137	C₂×#443	C₂×C₆₇	1	2	yes	complemented
138	#2179	C₂×C₆₇	1	4	yes	complemented
139	#2178	C₂×C₆₇	1	4	yes	complemented
140	#2161	C₂×C₆₇	1	1	yes	characteristic, complemented
141	#2143	C₂×C₆₇	1	2	yes	complemented
142	C₄×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
143	C₆₇	#33463	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×#135	C₂²×C₆₇	1	2	yes	—
145	C₂×#131	—	2	2	no	—
146	#510	C₄×C₆₇	1	4	yes	complemented
147	#509	C₄×C₆₇	1	4	yes	complemented
148	#508	C₂²×C₆₇	1	4	yes	—
149	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
150	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
151	#443	—	2	2	no	—
152	#443	—	2	2	no	—
153	#443	C₂²×C₆₇	1	4	yes	complemented
154	#443	C₂²×C₆₇	1	4	yes	complemented
155	C₂×#108	C₂²×C₆₇	1	2	yes	—
156	C₂×#104	—	2	2	no	—
157	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
158	#368	C₂²×C₆₇	1	4	yes	complemented
159	#359	C₂²×C₆₇	1	8	yes	complemented
160	#346	C₂²×C₆₇	1	2	yes	complemented
161	C₄×C₁₆	C₂²×C₆₇	1	2	yes	complemented
162	C₂×#41	—	2	2	no	—
163	#135	—	2	2	no	—
164	#135	—	2	2	no	—
165	#135	C₂×C₄×C₆₇	1	4	yes	—
166	#135	C₂³×C₆₇	1	4	yes	—
167	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
170	#131	—	2	4	no	—
171	#131	—	2	4	no	—
172	#131	—	4	2	no	—
173	#131	—	4	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	4	2	no	—
176	C₂²×C₈	—	4	2	no	—
177	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
180	Q₃₂	C₂×C₄×C₆₇	1	4	yes	complemented
181	SD₃₂	C₂×C₄×C₆₇	1	8	yes	complemented
182	D₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
183	M₃₂	—	2	2	no	—
184	M₃₂	C₂×C₄×C₆₇	1	4	yes	—
185	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
186	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
187	#108	—	2	2	no	—
188	#108	C₂³×C₆₇	1	4	yes	—
189	#104	—	2	4	no	—
190	#104	—	2	4	no	—
191	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
192	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
195	#41	—	2	4	no	—
196	#41	—	2	4	no	—
197	#41	—	4	2	no	—
198	#41	—	4	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
205	Q₁₆	—	2	2	no	—
206	Q₁₆	C₂²×C₄×C₆₇	1	4	yes	—
207	SD₁₆	—	2	4	no	—
208	SD₁₆	—	2	4	no	—
209	D₈	—	2	2	no	—
210	D₈	C₂²×C₄×C₆₇	1	4	yes	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	4	no	—
214	M₁₆	—	4	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	—	4	4	no	—
218	C₂×C₈	D₈×C₆₇	1	2	yes	—
219	C₂×C₈	D₈×C₆₇	1	2	yes	—
220	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
221	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
224	C₂×C₈	#41×C₆₇	1	2	yes	—
225	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
226	C₄²	—	2	1	no	—
227	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
228	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
229	C₁₆	—	2	2	no	—
230	C₁₆	C₂²×C₄×C₆₇	1	4	yes	—
231	C₂³	—	4	2	no	—
232	C₂³	#135×C₆₇	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	4	2	no	—
235	D₄	—	2	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
246	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
247	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	C₄×D₄×C₆₇	1	2	yes	—
251	C₈	C₂×D₈×C₆₇	1	1	yes	characteristic
252	C₈	C₂×D₈×C₆₇	1	2	yes	—
253	C₈	C₂×D₈×C₆₇	1	1	yes	characteristic
254	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
255	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	8	4	no	—
259	C₂²	C₄×D₈×C₆₇	1	2	yes	—
260	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	4	no	—
264	C₄	C₄×D₈×C₆₇	1	2	yes	—
265	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
266	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₇	C₂×D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂×C₄×D₈
prime radical	C₆₇×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₇ ◁ C₂²×C₈×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#443×C₆₇	C₂	semidirect
3. C₆₇×#2179	C₂	semidirect
4. C₆₇×#2178	C₂	semidirect
5. C₆₇×#2161	C₂	semidirect
6. C₆₇×#2143	C₂	semidirect
7. C₄×M₃₂×C₆₇	C₂	semidirect
10. #510×C₆₇	C₄	semidirect
11. #509×C₆₇	C₄	semidirect
17. #443×C₆₇	C₂²	semidirect
18. #443×C₆₇	C₂²	semidirect
22. #368×C₆₇	C₂²	semidirect
23. #359×C₆₇	C₂²	semidirect
24. #346×C₆₇	C₂²	semidirect
25. C₄×C₁₆×C₆₇	C₂²	semidirect
44. Q₃₂×C₆₇	C₂×C₄	semidirect
45. SD₃₂×C₆₇	C₂×C₄	semidirect
46. D₁₆×C₆₇	C₂×C₄	semidirect
131. #33463	C₆₇	direct (N × H)
137. C₂×#443	C₂×C₆₇	semidirect
138. #2179	C₂×C₆₇	semidirect
139. #2178	C₂×C₆₇	semidirect
140. #2161	C₂×C₆₇	semidirect
141. #2143	C₂×C₆₇	semidirect
142. C₄×M₃₂	C₂×C₆₇	semidirect
143. C₆₇	#33463	direct (N × H)
146. #510	C₄×C₆₇	semidirect
147. #509	C₄×C₆₇	semidirect
153. #443	C₂²×C₆₇	semidirect
154. #443	C₂²×C₆₇	semidirect
158. #368	C₂²×C₆₇	semidirect
159. #359	C₂²×C₆₇	semidirect
160. #346	C₂²×C₆₇	semidirect
161. C₄×C₁₆	C₂²×C₆₇	semidirect
180. Q₃₂	C₂×C₄×C₆₇	semidirect
181. SD₃₂	C₂×C₄×C₆₇	semidirect
182. D₁₆	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3414	2
C₂×#443×C₆₇	2
C₆₇×#2179	2
C₆₇×#2178	2
C₆₇×#2161	2
C₆₇×#2143	2
C₄×M₃₂×C₆₇	2
#33463	67

Subgroups — C67×#33463 #30712714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33463 #30712714