Subgroups — C₂×#22562067 #31009105

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863490	C₂	1	2	yes	maximal
3	#22562067	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863199	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
8	C₂×#14524008	—	2	2	no	—
9	C₂×Dic₅₃₆	C₄	1	8	yes	complemented
10	C₂×Dic₅₃₆	C₂²	1	4	yes	—
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16863490	—	2	4	no	—
13	#16863490	C₂²	1	8	yes	—
14	C₂×#14523966	—	2	2	no	—
15	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	#16863199	C₂²	1	8	yes	—
17	C₁₆×Dic₆₇	C₂²	1	4	yes	—
18	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#107×C₆₇	C₂²	1	2	yes	—
20	#366×C₆₇	C₂²	1	4	yes	—
21	C₂²×Dic₁₃₄	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	—	2	4	no	—
24	C₂×Dic₂₆₈	C₂×C₄	1	8	yes	—
25	C₂×Dic₂₆₈	C₂³	1	4	yes	—
26	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
27	#14524008	—	2	8	no	—
28	#14524008	—	4	4	no	—
29	C₂×Dic₆₇⋊C₄	—	4	2	no	—
30	C₂×C₄×Dic₆₇	—	4	2	no	—
31	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
32	Dic₅₃₆	C₂×C₄	1	16	yes	complemented
33	C₂×C₆₇⋊C₁₆	—	2	2	no	—
34	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
35	#14523966	—	2	8	no	—
36	C₈×Dic₆₇	D₄	1	2	yes	—
37	C₈×Dic₆₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
42	#107×C₆₇	—	2	2	no	—
43	#107×C₆₇	C₂³	1	4	yes	—
44	C₂×Dic₁₃₄	—	2	8	no	—
45	C₂×Dic₁₃₄	—	2	4	no	—
46	C₂×Dic₁₃₄	—	4	2	no	—
47	C₂×Dic₁₃₄	—	4	4	no	—
48	C₂²×Dic₆₇	—	4	2	no	—
49	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
50	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
51	Dic₂₆₈	—	2	8	no	—
52	Dic₂₆₈	—	2	4	no	—
53	Dic₂₆₈	C₂²×C₄	1	8	yes	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
56	Dic₆₇⋊C₄	—	4	8	no	—
57	C₄×Dic₆₇	—	4	8	no	—
58	C₄×Dic₆₇	D₈	1	2	yes	—
59	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
60	C₆₇⋊C₁₆	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	4	2	no	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	#41	1	2	yes	—
66	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₄⋊C₄×C₆₇	—	4	2	no	—
69	C₁₆×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	—	4	4	no	—
72	C₂×C₄×C₆₇	—	8	4	no	—
73	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#135	1	2	yes	—
76	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×Dic₆₇	—	2	2	no	—
78	C₂×Dic₆₇	—	4	4	no	—
79	C₂×Dic₆₇	—	4	8	no	—
80	C₂×Dic₆₇	Q₃₂	1	4	yes	complemented
81	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
82	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
83	Dic₁₃₄	—	2	8	no	—
84	Dic₁₃₄	—	4	4	no	—
85	Dic₁₃₄	—	4	8	no	—
86	C₈×C₆₇	C₄×D₄	1	2	yes	—
87	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
88	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
89	C₆₇⋊C₈	—	2	2	no	—
90	C₂²×C₆₇	C₄×D₈	1	2	yes	—
91	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
92	C₂²×C₆₇	#508	1	2	yes	—
93	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₆₇	—	8	4	no	—
95	C₄×C₆₇	C₄×D₈	1	2	yes	—
96	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
97	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
98	Dic₆₇	—	2	2	no	—
99	Dic₆₇	—	4	8	no	—
100	Dic₆₇	C₂×Q₃₂	1	4	yes	complemented
101	C₂×C₄×Q₃₂	—	67	1	no	maximal, Sylow
102	C₂×C₆₇	C₄×Q₃₂	1	4	yes	complemented
103	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
104	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
105	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
106	C₂²×Q₃₂	—	67	1	no	—
107	C₂×C₄×Q₁₆	—	67	2	no	—
108	C₄×Q₃₂	—	67	8	no	—
109	C₂×#366	D₆₇	1	1	yes	characteristic
110	C₂×#358	—	67	2	no	—
111	C₂×C₄×C₁₆	—	67	1	no	—
112	C₆₇	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₁₆	—	67	2	no	—
114	C₂×C₄×Q₈	—	134	2	no	—
115	C₂×Q₃₂	—	67	8	no	—
116	C₂×Q₃₂	—	67	4	no	—
117	C₂²×C₁₆	—	67	1	no	—
118	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
119	C₄×Q₁₆	—	67	8	no	—
120	C₄×Q₁₆	—	134	4	no	—
121	C₂×#107	C₂×D₆₇	1	2	yes	—
122	C₂×#103	—	134	2	no	—
123	C₂×C₄×C₈	—	67	1	no	—
124	#366	C₂×D₆₇	1	4	yes	—
125	#358	—	67	8	no	—
126	C₄×C₁₆	—	67	4	no	—
127	C₂²×Q₈	—	134	2	no	—
128	C₂×Q₁₆	—	67	8	no	—
129	C₂×Q₁₆	—	67	4	no	—
130	C₂×Q₁₆	—	134	2	no	—
131	C₂×Q₁₆	—	134	4	no	—
132	C₂²×C₈	—	67	1	no	—
133	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
134	C₄×Q₈	—	268	4	no	—
135	C₄×Q₈	—	134	8	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	268	2	no	—
138	C₂×C₄²	—	268	2	no	—
139	C₂×C₄²	—	67	1	no	—
140	Q₃₂	—	67	16	no	—
141	C₂×C₁₆	—	67	2	no	—
142	C₂×C₁₆	—	134	2	no	—
143	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
144	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
145	#107	—	2	2	no	—
146	#107	C₂²×D₆₇	1	4	yes	—
147	#103	—	134	8	no	—
148	C₄×C₈	—	67	2	no	—
149	C₄×C₈	—	67	2	no	—
150	C₂×Q₈	—	268	2	no	—
151	C₂×Q₈	—	268	4	no	—
152	C₂×Q₈	—	134	8	no	—
153	C₂×Q₈	—	134	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	268	2	no	—
156	C₂²×C₄	—	67	1	no	—
157	C₂²×C₄	—	67	1	no	—
158	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
159	Q₁₆	—	67	8	no	—
160	Q₁₆	—	134	8	no	—
161	Q₁₆	—	134	4	no	—
162	C₂×C₈	—	67	2	no	—
163	C₂×C₈	—	134	2	no	—
164	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
165	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
166	C₂×C₈	#11761629	1	2	yes	—
167	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	268	8	no	—
171	C₄²	—	268	8	no	—
172	C₄²	—	67	2	no	—
173	C₄²	—	67	2	no	—
174	C₁₆	—	134	2	no	—
175	C₁₆	C₂×C₄×D₆₇	1	4	yes	—
176	C₂³	D₈×D₆₇	1	1	yes	characteristic
177	Q₈	—	268	4	no	—
178	Q₈	—	268	8	no	—
179	Q₈	—	134	8	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	4	no	—
182	C₂×C₄	—	268	4	no	—
183	C₂×C₄	—	268	8	no	—
184	C₂×C₄	—	67	4	no	—
185	C₂×C₄	—	67	2	no	—
186	C₂×C₄	—	67	2	no	—
187	C₂×C₄	—	134	2	no	—
188	C₂×C₄	#14524005	1	2	yes	—
189	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
190	C₂×C₄	#14524066	1	2	yes	—
191	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
192	C₈	—	134	2	no	—
193	C₈	#14524005	1	2	yes	—
194	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
195	C₈	C₂×#11761629	1	1	yes	characteristic
196	C₂²	#16863478	1	2	yes	—
197	C₂²	Q₃₂×D₆₇	1	2	yes	—
198	C₂²	#16863825	1	2	yes	—
199	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	268	8	no	—
202	C₄	—	67	4	no	—
203	C₄	—	134	2	no	—
204	C₄	#16863478	1	2	yes	—
205	C₄	C₂×#14524005	1	1	yes	characteristic
206	C₄	C₂×#14524066	1	1	yes	characteristic
207	C₂	#22562067	1	4	yes	direct product factor
208	C₂	C₂×#16863478	1	1	yes	characteristic
209	C₂	C₂×Q₃₂×D₆₇	1	1	yes	characteristic
210	C₂	C₂×#16863825	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#366×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#366	D₆₇
hypocenter	C₆₇	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
3. #22562067	C₂	direct (N × H)
9. C₂×Dic₅₃₆	C₄	semidirect
32. Dic₅₃₆	C₂×C₄	semidirect
80. C₂×Dic₆₇	Q₃₂	semidirect
100. Dic₆₇	C₂×Q₃₂	semidirect
102. C₂×C₆₇	C₄×Q₃₂	semidirect
112. C₆₇	C₂×C₄×Q₃₂	semidirect
207. C₂	#22562067	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₃₆	2
C₂×#16863490	2
#22562067	2
C₂×#16863199	2
C₂×C₁₆×Dic₆₇	2
C₂×#366×C₆₇	2
C₂×C₄×Q₃₂	67

Subgroups — C2×#22562067 #31009105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562067 #31009105