Subgroups — C₂×#22562495 #31011328

Order 17152.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863798	C₂	1	1	yes	maximal, characteristic
3	#22562495	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863224	C₂	1	2	yes	maximal
5	C₂×#16863222	C₂	1	2	yes	maximal
6	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524043	—	2	1	no	—
8	C₂×#14524035	—	2	1	no	—
9	#16863801	—	2	2	no	—
10	#16863801	C₂²	1	4	yes	—
11	#16863798	—	2	2	no	—
12	#16863798	C₂²	1	4	yes	—
13	C₂×#14523977	—	2	1	no	—
14	C₂×#14523977	C₂²	1	1	yes	characteristic
15	C₂×#14523977	C₂²	1	2	yes	—
16	C₂×#14523977	C₂²	1	1	yes	characteristic
17	C₂×#14523975	—	2	1	no	—
18	#16863224	C₂²	1	8	yes	—
19	#16863222	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#346×C₆₇	C₂²	1	4	yes	—
23	C₂²×Dic₁₃₄	—	4	1	no	—
24	#14524043	—	2	4	no	—
25	#14524043	—	4	2	no	—
26	#14524035	—	2	4	no	—
27	#14524035	—	4	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	1	no	—
29	C₂×C₄⋊Dic₆₇	—	2	1	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	—	4	1	no	—
32	C₂×Dic₆₇⋊C₄	—	4	1	no	—
33	C₂×Dic₆₇⋊C₄	—	4	1	no	—
34	#14523977	—	2	1	no	—
35	#14523977	—	2	1	no	—
36	#14523977	—	2	2	no	—
37	#14523977	—	2	4	no	—
38	#14523977	C₂³	1	2	yes	—
39	#14523977	C₂³	1	2	yes	—
40	#14523977	C₂³	1	4	yes	—
41	#14523975	—	2	4	no	—
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₆₇	Q₈	1	8	yes	complemented
46	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
47	C₄×C₈×C₆₇	D₄	1	2	yes	—
48	C₄×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂×Dic₁₃₄	—	4	4	no	—
50	C₂×Dic₁₃₄	—	8	4	no	—
51	C₂²×Dic₆₇	—	4	1	no	—
52	C₂²×Dic₆₇	—	4	1	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	C₄⋊Dic₆₇	—	2	2	no	—
55	C₄⋊Dic₆₇	—	2	2	no	—
56	C₄⋊Dic₆₇	—	2	4	no	—
57	C₄⋊Dic₆₇	—	4	1	no	—
58	C₄⋊Dic₆₇	—	4	1	no	—
59	C₄⋊Dic₆₇	—	4	2	no	—
60	C₄⋊Dic₆₇	—	4	4	no	—
61	Dic₆₇⋊C₄	—	4	4	no	—
62	Dic₆₇⋊C₄	—	4	4	no	—
63	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
71	C₄²×C₆₇	D₈	1	2	yes	—
72	C₄²×C₆₇	C₂×D₄	1	2	yes	—
73	C₁₆×C₆₇	C₂×Q₈	1	8	yes	complemented
74	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₆₇	—	4	2	no	—
84	C₂×Dic₆₇	—	4	2	no	—
85	C₂×Dic₆₇	—	4	4	no	—
86	C₂×Dic₆₇	—	8	2	no	—
87	C₂×Dic₆₇	—	8	4	no	—
88	C₂×Dic₆₇	—	8	2	no	—
89	Dic₁₃₄	—	8	8	no	—
90	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
91	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
92	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
93	C₂²×C₆₇	#500	1	2	yes	—
94	C₂²×C₆₇	#509	1	2	yes	—
95	C₂²×C₆₇	#510	1	2	yes	—
96	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₆₇	—	2	2	no	—
98	C₄×C₆₇	—	2	2	no	—
99	C₄×C₆₇	#500	1	2	yes	—
100	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₆₇	—	8	4	no	—
103	Dic₆₇	—	8	2	no	—
104	Dic₆₇	—	8	2	no	—
105	C₂×#2255	—	67	1	no	maximal, Sylow
106	C₂×C₆₇	#2255	1	4	yes	complemented
107	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
108	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
109	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
110	C₂×#500	—	67	1	no	—
111	C₂×#499	—	67	1	no	—
112	#2255	—	67	8	no	—
113	C₂×#367	—	67	2	no	—
114	C₂×#366	—	67	2	no	—
115	C₂×#346	D₆₇	1	1	yes	characteristic
116	C₆₇	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄⋊Q₈	—	134	1	no	—
118	C₂×#125	—	134	1	no	—
119	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
120	#500	—	67	4	no	—
121	#500	—	134	2	no	—
122	#499	—	67	4	no	—
123	#499	—	134	2	no	—
124	C₂×#107	—	67	1	no	—
125	C₂×#107	—	67	1	no	—
126	C₂×#107	—	67	2	no	—
127	C₂×#107	—	134	1	no	—
128	C₂×#106	—	134	1	no	—
129	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
130	#367	—	67	8	no	—
131	#366	—	67	8	no	—
132	#346	C₂×D₆₇	1	4	yes	—
133	C₂²×Q₈	—	268	1	no	—
134	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	268	2	no	—
137	C₄⋊Q₈	—	134	4	no	—
138	#125	—	268	2	no	—
139	#125	—	134	4	no	—
140	C₂×C₄⋊C₄	—	268	1	no	—
141	C₂×C₄⋊C₄	—	268	1	no	—
142	C₂×C₄⋊C₄	—	268	1	no	—
143	C₂×C₄⋊C₄	—	134	1	no	—
144	C₂×C₄⋊C₄	—	134	1	no	—
145	C₂×C₄⋊C₄	—	134	2	no	—
146	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
147	C₂×C₁₆	Dic₁₃₄	1	8	yes	complemented
148	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
149	#107	—	67	2	no	—
150	#107	—	67	4	no	—
151	#107	—	67	2	no	—
152	#107	—	134	1	no	—
153	#107	—	134	1	no	—
154	#107	—	134	2	no	—
155	#107	—	134	4	no	—
156	#106	—	134	4	no	—
157	C₄×C₈	D₂₆₈	1	2	yes	—
158	C₄×C₈	C₂²×D₆₇	1	2	yes	—
159	C₂×Q₈	—	268	4	no	—
160	C₂×Q₈	—	536	4	no	—
161	C₂²×C₄	—	268	1	no	—
162	C₂²×C₄	—	268	1	no	—
163	C₂²×C₄	—	268	2	no	—
164	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
165	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
167	C₂×C₈	Dic₂₆₈	1	4	yes	—
168	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
170	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
171	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
172	C₄⋊C₄	—	268	4	no	—
173	C₄⋊C₄	—	268	4	no	—
174	C₄⋊C₄	—	268	4	no	—
175	C₄⋊C₄	—	268	1	no	—
176	C₄⋊C₄	—	268	1	no	—
177	C₄⋊C₄	—	268	2	no	—
178	C₄⋊C₄	—	134	2	no	—
179	C₄⋊C₄	—	134	2	no	—
180	C₄⋊C₄	—	134	4	no	—
181	C₄²	D₅₃₆	1	2	yes	—
182	C₄²	C₂×D₂₆₈	1	2	yes	—
183	C₁₆	C₂×Dic₁₃₄	1	8	yes	complemented
184	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
185	Q₈	—	536	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	268	2	no	—
189	C₂×C₄	—	268	2	no	—
190	C₂×C₄	—	268	4	no	—
191	C₂×C₄	—	536	4	no	—
192	C₂×C₄	—	536	2	no	—
193	C₂×C₄	—	536	2	no	—
194	C₂×C₄	#14524043	1	2	yes	—
195	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
196	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
197	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
198	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
199	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
200	C₈	#14524043	1	2	yes	—
201	C₈	C₂×Dic₂₆₈	1	4	yes	—
202	C₈	C₂²×Dic₁₃₄	1	2	yes	—
203	C₂²	#16863801	1	2	yes	—
204	C₂²	#16863834	1	2	yes	—
205	C₂²	#16863839	1	2	yes	—
206	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	536	4	no	—
210	C₄	—	536	2	no	—
211	C₄	—	536	2	no	—
212	C₄	#16863801	1	2	yes	—
213	C₄	C₂×#14524043	1	1	yes	characteristic
214	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
215	C₂	#22562495	1	4	yes	direct product factor
216	C₂	C₂×#16863801	1	1	yes	characteristic
217	C₂	C₂×#16863834	1	1	yes	characteristic
218	C₂	C₂×#16863839	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#346×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×#346	D₆₇
hypocenter	C₆₇	C₂×#2255
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #22562495	C₂	direct (N × H)
45. C₂×C₁₆×C₆₇	Q₈	semidirect
73. C₁₆×C₆₇	C₂×Q₈	semidirect
106. C₂×C₆₇	#2255	semidirect
116. C₆₇	C₂×#2255	semidirect
147. C₂×C₁₆	Dic₁₃₄	semidirect
183. C₁₆	C₂×Dic₁₃₄	semidirect
215. C₂	#22562495	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×#16863801	2
C₂×#16863798	2
#22562495	2
C₂×#16863224	2
C₂×#16863222	2
C₂×#346×C₆₇	2
C₂×#2255	67

Subgroups — C2×#22562495 #31011328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562495 #31011328