Subgroups — C₂×C₆₇×#2178 #30712712

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂×C₈×C₆₇	D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₄×D₈
prime radical	C₂×#508×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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×#2178	C₂	direct (N × H)
4. C₂×#368×C₆₇	C₂	semidirect
5. C₂×#359×C₆₇	C₂	semidirect
6. C₂×C₄×C₁₆×C₆₇	C₂	semidirect
9. #508×C₆₇	C₄	semidirect
11. C₂×Q₃₂×C₆₇	C₄	semidirect
12. C₂×SD₃₂×C₆₇	C₄	semidirect
13. C₂×D₁₆×C₆₇	C₄	semidirect
17. #443×C₆₇	C₂²	semidirect
21. #368×C₆₇	C₂²	semidirect
22. #359×C₆₇	C₂²	semidirect
23. C₄×C₁₆×C₆₇	C₂²	semidirect
40. Q₃₂×C₆₇	C₂×C₄	semidirect
41. SD₃₂×C₆₇	C₂×C₄	semidirect
42. D₁₆×C₆₇	C₂×C₄	semidirect
129. C₂×#2178	C₆₇	direct (N × H)
133. C₂×C₆₇	#2178	direct (N × H)
136. C₂×#443	C₂×C₆₇	semidirect
137. #2178	C₂×C₆₇	direct (N × H)
138. C₂×#368	C₂×C₆₇	semidirect
139. C₂×#359	C₂×C₆₇	semidirect
140. C₂×C₄×C₁₆	C₂×C₆₇	semidirect
141. C₆₇	C₂×#2178	direct (N × H)
144. #508	C₄×C₆₇	semidirect
146. C₂×Q₃₂	C₄×C₆₇	semidirect
147. C₂×SD₃₂	C₄×C₆₇	semidirect
148. C₂×D₁₆	C₄×C₆₇	semidirect
152. #443	C₂²×C₆₇	semidirect
156. #368	C₂²×C₆₇	semidirect
157. #359	C₂²×C₆₇	semidirect
158. C₄×C₁₆	C₂²×C₆₇	semidirect
175. Q₃₂	C₂×C₄×C₆₇	semidirect
176. SD₃₂	C₂×C₄×C₆₇	semidirect
177. D₁₆	C₂×C₄×C₆₇	semidirect
267. C₂	C₆₇×#2178	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#508×C₆₇	2
C₂×#443×C₆₇	2
C₆₇×#2178	2
C₂×#368×C₆₇	2
C₂×#359×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#2178	67

Subgroups — C2×C67×#2178 #30712712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2178 #30712712