Subgroups — C₂×C₆₇×#2243 #30712949

Order 17152.

634 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2243	C₂	1	8	yes	maximal, direct product factor
4	C₂×#358×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#357×C₆₇	C₂	1	2	yes	maximal
6	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×SD₁₆×C₆₇	—	2	1	no	—
9	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
11	C₂×#124×C₆₇	—	2	1	no	—
12	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
13	#500×C₆₇	—	2	2	no	—
14	#500×C₆₇	C₂²	1	4	yes	complemented
15	#495×C₆₇	—	2	2	no	—
16	#495×C₆₇	C₂²	1	4	yes	—
17	C₂×#107×C₆₇	—	2	1	no	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#358×C₆₇	C₂²	1	8	yes	complemented
21	#357×C₆₇	C₂²	1	8	yes	—
22	#346×C₆₇	C₂²	1	4	yes	complemented
23	C₂²×Q₈×C₆₇	—	2	1	no	—
24	C₂²×Q₈×C₆₇	—	4	1	no	—
25	C₂²×D₄×C₆₇	—	2	1	no	—
26	C₂×Q₁₆×C₆₇	—	2	1	no	—
27	C₂×Q₁₆×C₆₇	—	2	4	no	—
28	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
29	C₂×SD₁₆×C₆₇	—	2	4	no	—
30	C₂×SD₁₆×C₆₇	—	4	4	no	—
31	C₂×D₈×C₆₇	—	2	1	no	—
32	C₂×D₈×C₆₇	—	2	4	no	—
33	C₂×D₈×C₆₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊Q₈×C₆₇	—	2	4	no	—
37	C₄⋊Q₈×C₆₇	—	4	2	no	—
38	#124×C₆₇	—	2	4	no	—
39	#124×C₆₇	—	4	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
42	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
43	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
44	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₆₇	—	2	4	no	—
46	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
47	#107×C₆₇	—	2	2	no	—
48	#107×C₆₇	—	2	4	no	—
49	#107×C₆₇	C₂³	1	4	yes	—
50	C₄×C₈×C₆₇	D₄	1	2	yes	—
51	C₄×C₈×C₆₇	C₂³	1	2	yes	—
52	C₂⁴×C₆₇	—	4	1	no	—
53	C₂×Q₈×C₆₇	—	2	2	no	—
54	C₂×Q₈×C₆₇	—	4	1	no	—
55	C₂×Q₈×C₆₇	—	4	4	no	—
56	C₂×Q₈×C₆₇	—	4	2	no	—
57	C₂×Q₈×C₆₇	—	4	2	no	—
58	C₂×Q₈×C₆₇	—	8	4	no	—
59	C₂×D₄×C₆₇	—	2	2	no	—
60	C₂×D₄×C₆₇	—	4	2	no	—
61	C₂×D₄×C₆₇	—	4	1	no	—
62	C₂×D₄×C₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	4	1	no	—
64	C₂²×C₄×C₆₇	—	4	2	no	—
65	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	Q₁₆×C₆₇	—	2	4	no	—
69	Q₁₆×C₆₇	—	4	2	no	—
70	SD₁₆×C₆₇	—	4	8	no	—
71	D₈×C₆₇	—	2	4	no	—
72	D₈×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
74	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₆₇	#41	1	4	yes	—
77	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₆₇	—	2	4	no	—
79	C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₄⋊C₄×C₆₇	—	4	4	no	—
81	C₂²⋊C₄×C₆₇	—	4	4	no	—
82	C₂²⋊C₄×C₆₇	—	4	4	no	—
83	C₄²×C₆₇	D₈	1	2	yes	—
84	C₄²×C₆₇	C₂×D₄	1	2	yes	—
85	C₁₆×C₆₇	—	2	4	no	—
86	C₂³×C₆₇	—	4	2	no	—
87	C₂³×C₆₇	—	8	2	no	—
88	C₂³×C₆₇	—	8	2	no	—
89	C₂³×C₆₇	—	8	2	no	—
90	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₆₇	—	4	2	no	—
92	Q₈×C₆₇	—	4	2	no	—
93	Q₈×C₆₇	—	8	8	no	—
94	Q₈×C₆₇	—	8	2	no	—
95	D₄×C₆₇	—	4	2	no	—
96	D₄×C₆₇	—	4	2	no	—
97	D₄×C₆₇	—	8	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	8	2	no	—
103	C₂×C₄×C₆₇	—	8	4	no	—
104	C₂×C₄×C₆₇	#124	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₆₇	#124	1	2	yes	—
111	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
112	C₈×C₆₇	C₂×#41	1	2	yes	—
113	C₂²×C₆₇	—	8	2	no	—
114	C₂²×C₆₇	—	8	2	no	—
115	C₂²×C₆₇	—	8	2	no	—
116	C₂²×C₆₇	—	16	4	no	—
117	C₂²×C₆₇	#486	1	2	yes	—
118	C₂²×C₆₇	#509	1	2	yes	—
119	C₂²×C₆₇	#510	1	2	yes	—
120	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
121	C₄×C₆₇	—	2	2	no	—
122	C₄×C₆₇	—	2	2	no	—
123	C₄×C₆₇	—	8	2	no	—
124	C₄×C₆₇	—	8	4	no	—
125	C₄×C₆₇	#486	1	2	yes	—
126	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
127	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
128	C₂×#2243	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₆₇	—	16	2	no	—
130	C₂×C₆₇	#2243	1	4	yes	direct product factor
131	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
132	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
133	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
134	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
135	C₂×#495	C₂×C₆₇	1	1	yes	characteristic
136	#2243	C₂×C₆₇	1	8	yes	direct product factor
137	C₂×#358	C₂×C₆₇	1	2	yes	complemented
138	C₂×#357	C₂×C₆₇	1	2	yes	—
139	C₂×#346	C₂×C₆₇	1	1	yes	characteristic, complemented
140	C₆₇	C₂×#2243	1	1	yes	characteristic, direct product factor, Sylow
141	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
142	C₂²×SD₁₆	—	2	1	no	—
143	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
144	C₂×C₄⋊Q₈	—	2	1	no	—
145	C₂×#124	—	2	1	no	—
146	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
147	#500	—	2	2	no	—
148	#500	C₂²×C₆₇	1	4	yes	complemented
149	#495	—	2	2	no	—
150	#495	C₂²×C₆₇	1	4	yes	—
151	C₂×#107	—	2	1	no	—
152	C₂×#107	C₂²×C₆₇	1	2	yes	—
153	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
154	#358	C₂²×C₆₇	1	8	yes	complemented
155	#357	C₂²×C₆₇	1	8	yes	—
156	#346	C₂²×C₆₇	1	4	yes	complemented
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×Q₈	—	4	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
163	C₂×SD₁₆	—	2	4	no	—
164	C₂×SD₁₆	—	4	4	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	—	2	4	no	—
167	C₂×D₈	C₂³×C₆₇	1	2	yes	—
168	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	—	4	2	no	—
172	#124	—	2	4	no	—
173	#124	—	4	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	4	1	no	—
176	C₂×C₂²⋊C₄	—	4	1	no	—
177	C₂×C₂²⋊C₄	—	4	1	no	—
178	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
181	#107	—	2	4	no	—
182	#107	—	2	2	no	—
183	#107	C₂³×C₆₇	1	4	yes	—
184	C₄×C₈	D₄×C₆₇	1	2	yes	—
185	C₄×C₈	C₂³×C₆₇	1	2	yes	—
186	C₂⁴	—	4	1	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	4	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	—	8	4	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
200	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
202	Q₁₆	—	2	4	no	—
203	Q₁₆	—	4	2	no	—
204	SD₁₆	—	4	8	no	—
205	D₈	—	2	4	no	—
206	D₈	—	4	2	no	—
207	C₂×C₈	SD₁₆×C₆₇	1	4	yes	—
208	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
210	C₂×C₈	#41×C₆₇	1	4	yes	—
211	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₄²	D₈×C₆₇	1	2	yes	—
218	C₄²	C₂×D₄×C₆₇	1	2	yes	—
219	C₁₆	—	2	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	8	2	no	—
222	C₂³	—	8	2	no	—
223	C₂³	—	8	2	no	—
224	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	8	8	no	—
228	Q₈	—	8	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	8	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	#124×C₆₇	1	2	yes	—
239	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
240	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
241	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
242	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
243	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
244	C₈	#124×C₆₇	1	2	yes	—
245	C₈	C₂×SD₁₆×C₆₇	1	4	yes	—
246	C₈	C₂×#41×C₆₇	1	2	yes	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	16	4	no	—
251	C₂²	#486×C₆₇	1	2	yes	—
252	C₂²	#509×C₆₇	1	2	yes	—
253	C₂²	#510×C₆₇	1	2	yes	—
254	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	8	2	no	—
258	C₄	—	8	4	no	—
259	C₄	#486×C₆₇	1	2	yes	—
260	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
261	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
262	C₂	—	16	2	no	—
263	C₂	C₆₇×#2243	1	4	yes	direct product factor
264	C₂	C₂×#486×C₆₇	1	1	yes	characteristic
265	C₂	C₂×#509×C₆₇	1	1	yes	characteristic
266	C₂	C₂×#510×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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. C₂×#500×C₆₇	C₂	semidirect
3. C₆₇×#2243	C₂	direct (N × H)
4. C₂×#358×C₆₇	C₂	semidirect
6. C₂×#346×C₆₇	C₂	semidirect
14. #500×C₆₇	C₂²	semidirect
20. #358×C₆₇	C₂²	semidirect
22. #346×C₆₇	C₂²	semidirect
128. C₂×#2243	C₆₇	direct (N × H)
130. C₂×C₆₇	#2243	direct (N × H)
134. C₂×#500	C₂×C₆₇	semidirect
136. #2243	C₂×C₆₇	direct (N × H)
137. C₂×#358	C₂×C₆₇	semidirect
139. C₂×#346	C₂×C₆₇	semidirect
140. C₆₇	C₂×#2243	direct (N × H)
148. #500	C₂²×C₆₇	semidirect
154. #358	C₂²×C₆₇	semidirect
156. #346	C₂²×C₆₇	semidirect
263. C₂	C₆₇×#2243	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×#500×C₆₇	2
C₂×#495×C₆₇	2
C₆₇×#2243	2
C₂×#358×C₆₇	2
C₂×#357×C₆₇	2
C₂×#346×C₆₇	2
C₂×#2243	67

Subgroups — C2×C67×#2243 #30712949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2243 #30712949