Subgroups — C₂×C₆₇×#2240 #30712945

Order 17152.

726 conjugacy classes of subgroups.

The subgroup lattice has 236 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×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2240	C₂	1	8	yes	maximal, direct product factor
4	C₂×#357×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₆₇	—	2	1	no	—
7	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
9	C₂×#127×C₆₇	—	2	1	no	—
10	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
11	#500×C₆₇	—	2	2	no	—
12	#500×C₆₇	C₂²	1	4	yes	complemented
13	#493×C₆₇	—	2	2	no	—
14	#493×C₆₇	C₂²	1	4	yes	—
15	C₂×#107×C₆₇	—	2	1	no	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#357×C₆₇	C₂²	1	16	yes	complemented
19	C₄×C₁₆×C₆₇	C₂²	1	4	yes	complemented
20	C₂²×Q₈×C₆₇	—	4	1	no	—
21	C₂²×D₄×C₆₇	—	2	2	no	—
22	C₂²×D₄×C₆₇	—	4	1	no	—
23	C₂²×D₄×C₆₇	—	4	1	no	—
24	C₂×D₈×C₆₇	—	2	2	no	—
25	C₂×D₈×C₆₇	—	2	4	no	—
26	C₂×D₈×C₆₇	—	2	8	no	—
27	C₂×D₈×C₆₇	—	4	4	no	—
28	C₂×D₈×C₆₇	C₂³	1	4	yes	—
29	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₆₇	—	2	4	no	—
32	C₄⋊Q₈×C₆₇	—	4	2	no	—
33	#127×C₆₇	—	2	4	no	—
34	#127×C₆₇	—	4	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₆₇	—	2	4	no	—
39	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
40	#107×C₆₇	—	2	2	no	—
41	#107×C₆₇	—	2	4	no	—
42	#107×C₆₇	C₂³	1	4	yes	—
43	C₄×C₈×C₆₇	D₄	1	2	yes	—
44	C₄×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂⁴×C₆₇	—	4	2	no	—
46	C₂×Q₈×C₆₇	—	4	4	no	—
47	C₂×Q₈×C₆₇	—	8	4	no	—
48	C₂×D₄×C₆₇	—	2	4	no	—
49	C₂×D₄×C₆₇	—	4	4	no	—
50	C₂×D₄×C₆₇	—	4	4	no	—
51	C₂×D₄×C₆₇	—	4	4	no	—
52	C₂×D₄×C₆₇	—	4	2	no	—
53	C₂×D₄×C₆₇	—	4	4	no	—
54	C₂×D₄×C₆₇	—	8	4	no	—
55	C₂×D₄×C₆₇	—	8	4	no	—
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	D₈×C₆₇	—	2	8	no	—
61	D₈×C₆₇	—	4	4	no	—
62	D₈×C₆₇	—	4	8	no	—
63	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
64	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₆₇	#41	1	4	yes	—
67	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₄⋊C₄×C₆₇	—	4	2	no	—
70	C₄⋊C₄×C₆₇	—	4	4	no	—
71	C₄²×C₆₇	D₈	1	2	yes	—
72	C₄²×C₆₇	C₂×D₄	1	2	yes	—
73	C₁₆×C₆₇	—	2	4	no	—
74	C₂³×C₆₇	—	4	4	no	—
75	C₂³×C₆₇	—	8	4	no	—
76	C₂³×C₆₇	—	8	4	no	—
77	C₂³×C₆₇	—	8	4	no	—
78	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
79	Q₈×C₆₇	—	8	8	no	—
80	D₄×C₆₇	—	4	4	no	—
81	D₄×C₆₇	—	4	4	no	—
82	D₄×C₆₇	—	8	4	no	—
83	D₄×C₆₇	—	8	8	no	—
84	D₄×C₆₇	—	8	8	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	—	8	4	no	—
87	C₂×C₄×C₆₇	D₁₆	1	4	yes	—
88	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
89	C₂×C₄×C₆₇	#124	1	2	yes	—
90	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
94	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	C₈×C₆₇	#124	1	2	yes	—
96	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
97	C₈×C₆₇	C₂×#41	1	2	yes	—
98	C₂²×C₆₇	—	8	4	no	—
99	C₂²×C₆₇	—	8	4	no	—
100	C₂²×C₆₇	—	8	4	no	—
101	C₂²×C₆₇	—	16	8	no	—
102	C₂²×C₆₇	#486	1	2	yes	—
103	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
104	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
105	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₆₇	—	8	4	no	—
107	C₄×C₆₇	#486	1	2	yes	—
108	C₄×C₆₇	C₂×D₁₆	1	4	yes	—
109	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
110	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
111	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
112	C₂×#2240	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₆₇	—	16	4	no	—
114	C₂×C₆₇	#2240	1	4	yes	direct product factor
115	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
116	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
117	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
118	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
119	C₂×#493	C₂×C₆₇	1	1	yes	characteristic
120	#2240	C₂×C₆₇	1	8	yes	direct product factor
121	C₂×#357	C₂×C₆₇	1	4	yes	complemented
122	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
123	C₆₇	C₂×#2240	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×D₈	—	2	1	no	—
125	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
126	C₂×C₄⋊Q₈	—	2	1	no	—
127	C₂×#127	—	2	1	no	—
128	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
129	#500	—	2	2	no	—
130	#500	C₂²×C₆₇	1	4	yes	complemented
131	#493	—	2	2	no	—
132	#493	C₂²×C₆₇	1	4	yes	—
133	C₂×#107	—	2	1	no	—
134	C₂×#107	C₂²×C₆₇	1	2	yes	—
135	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
136	#357	C₂²×C₆₇	1	16	yes	complemented
137	C₄×C₁₆	C₂²×C₆₇	1	4	yes	complemented
138	C₂²×Q₈	—	4	1	no	—
139	C₂²×D₄	—	2	2	no	—
140	C₂²×D₄	—	4	1	no	—
141	C₂²×D₄	—	4	1	no	—
142	C₂×D₈	—	2	4	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	8	no	—
145	C₂×D₈	—	4	4	no	—
146	C₂×D₈	C₂³×C₆₇	1	4	yes	—
147	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	4	no	—
150	C₄⋊Q₈	—	4	2	no	—
151	#127	—	2	4	no	—
152	#127	—	4	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	1	no	—
155	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
156	C₂×C₁₆	—	2	4	no	—
157	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
158	#107	—	2	4	no	—
159	#107	—	2	2	no	—
160	#107	C₂³×C₆₇	1	4	yes	—
161	C₄×C₈	D₄×C₆₇	1	2	yes	—
162	C₄×C₈	C₂³×C₆₇	1	2	yes	—
163	C₂⁴	—	4	2	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	8	4	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	—	4	4	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	—	8	4	no	—
173	C₂×D₄	—	8	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
176	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
178	D₈	—	2	8	no	—
179	D₈	—	4	4	no	—
180	D₈	—	4	8	no	—
181	C₂×C₈	SD₁₆×C₆₇	1	4	yes	—
182	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
184	C₂×C₈	#41×C₆₇	1	4	yes	—
185	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄²	D₈×C₆₇	1	2	yes	—
190	C₄²	C₂×D₄×C₆₇	1	2	yes	—
191	C₁₆	—	2	4	no	—
192	C₂³	—	4	4	no	—
193	C₂³	—	8	4	no	—
194	C₂³	—	8	4	no	—
195	C₂³	—	8	4	no	—
196	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
197	Q₈	—	8	8	no	—
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	8	4	no	—
201	D₄	—	8	8	no	—
202	D₄	—	8	8	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	D₁₆×C₆₇	1	4	yes	—
206	C₂×C₄	SD₃₂×C₆₇	1	4	yes	—
207	C₂×C₄	#124×C₆₇	1	2	yes	—
208	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
209	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
210	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
211	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
212	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
213	C₈	#124×C₆₇	1	2	yes	—
214	C₈	C₂×SD₁₆×C₆₇	1	4	yes	—
215	C₈	C₂×#41×C₆₇	1	2	yes	—
216	C₂²	—	8	4	no	—
217	C₂²	—	8	4	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	16	8	no	—
220	C₂²	#486×C₆₇	1	2	yes	—
221	C₂²	C₂×D₁₆×C₆₇	1	2	yes	—
222	C₂²	C₂×SD₃₂×C₆₇	1	2	yes	—
223	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
224	C₄	—	8	4	no	—
225	C₄	#486×C₆₇	1	2	yes	—
226	C₄	C₂×D₁₆×C₆₇	1	4	yes	—
227	C₄	C₂×SD₃₂×C₆₇	1	4	yes	—
228	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
229	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
230	C₂	—	16	4	no	—
231	C₂	C₆₇×#2240	1	4	yes	direct product factor
232	C₂	C₂×#486×C₆₇	1	1	yes	characteristic
233	C₂	C₂²×D₁₆×C₆₇	1	1	yes	characteristic
234	C₂	C₂²×SD₃₂×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#493×C₆₇	C₂
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×#2240	C₂	direct (N × H)
4. C₂×#357×C₆₇	C₂	semidirect
5. C₂×C₄×C₁₆×C₆₇	C₂	semidirect
12. #500×C₆₇	C₂²	semidirect
18. #357×C₆₇	C₂²	semidirect
19. C₄×C₁₆×C₆₇	C₂²	semidirect
112. C₂×#2240	C₆₇	direct (N × H)
114. C₂×C₆₇	#2240	direct (N × H)
118. C₂×#500	C₂×C₆₇	semidirect
120. #2240	C₂×C₆₇	direct (N × H)
121. C₂×#357	C₂×C₆₇	semidirect
122. C₂×C₄×C₁₆	C₂×C₆₇	semidirect
123. C₆₇	C₂×#2240	direct (N × H)
130. #500	C₂²×C₆₇	semidirect
136. #357	C₂²×C₆₇	semidirect
137. C₄×C₁₆	C₂²×C₆₇	semidirect
231. C₂	C₆₇×#2240	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×#493×C₆₇	2
C₆₇×#2240	2
C₂×#357×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#2240	67

Subgroups — C2×C67×#2240 #30712945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2240 #30712945