Subgroups — C₂×#22562111 #31009311

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22562111	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₆₇⋊C₁₆	C₂	1	2	yes	maximal
4	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₆×Dic₆₇	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₃₄	C₄	1	4	yes	—
11	C₈×Dic₁₃₄	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
13	C₂×Dic₆₇⋊C₈	C₄	1	2	yes	—
14	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
15	Dic₆₇⋊C₁₆	C₂²	1	8	yes	—
16	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	—
17	C₁₆×Dic₆₇	C₂²	1	8	yes	—
18	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
21	C₂²×Dic₁₃₄	C₈	1	1	yes	characteristic
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₃₄	C₈	1	4	yes	—
24	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
26	C₂×Dic₆₇⋊C₄	C₈	1	2	yes	—
27	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₆₇⋊C₁₆	—	2	2	no	—
29	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
30	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
31	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
32	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
33	Dic₆₇⋊C₈	C₂×C₄	1	8	yes	—
34	C₈×Dic₆₇	C₂×C₄	1	4	yes	—
35	C₈×Dic₆₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₆₇	—	2	4	no	—
40	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂×Dic₁₃₄	C₁₆	1	8	yes	complemented
44	C₂×Dic₁₃₄	C₂×C₈	1	4	yes	—
45	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
46	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
47	C₂×C₆₇⋊C₈	—	2	4	no	—
48	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
49	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
50	Dic₆₇⋊C₄	C₂×C₈	1	8	yes	—
51	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
52	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
53	C₆₇⋊C₁₆	—	2	2	no	—
54	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
55	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₆₇	#41	1	2	yes	—
63	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×C₈	1	2	yes	—
65	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
66	C₁₆×C₆₇	—	2	4	no	—
67	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	C₂×C₁₆	1	4	yes	—
70	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₆₇	#131	1	2	yes	—
75	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×Dic₆₇	—	2	4	no	—
77	C₂×Dic₆₇	C₂×C₁₆	1	8	yes	—
78	C₂×Dic₆₇	C₂²×C₈	1	4	yes	—
79	C₂×Dic₆₇	C₂²×C₈	1	4	yes	—
80	Dic₁₃₄	C₂×C₁₆	1	16	yes	complemented
81	C₈×C₆₇	—	2	2	no	—
82	C₈×C₆₇	C₄×Q₈	1	2	yes	—
83	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
85	C₆₇⋊C₈	—	2	4	no	—
86	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
87	C₂²×C₆₇	C₂²×C₁₆	1	2	yes	—
88	C₂²×C₆₇	#504	1	2	yes	—
89	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₆₇	—	2	2	no	—
91	C₄×C₆₇	C₈×Q₈	1	2	yes	—
92	C₄×C₆₇	C₂²×C₁₆	1	4	yes	—
93	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
94	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
95	Dic₆₇	—	2	4	no	—
96	Dic₆₇	C₂²×C₁₆	1	8	yes	—
97	C₂×Q₈×C₁₆	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	Q₈×C₁₆	1	4	yes	complemented
99	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
100	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
101	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
102	C₂×C₈×Q₈	—	67	1	no	—
103	Q₈×C₁₆	—	67	8	no	—
104	C₂×C₄⋊C₁₆	—	67	2	no	—
105	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
106	C₂×C₄×C₁₆	—	67	2	no	—
107	C₂×C₄×C₁₆	—	67	1	no	—
108	C₆₇	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄×Q₈	—	67	1	no	—
110	C₂²×C₁₆	—	67	1	no	—
111	C₂²×C₁₆	—	67	1	no	—
112	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
113	C₈×Q₈	—	67	4	no	—
114	C₈×Q₈	—	67	4	no	—
115	C₂×C₄⋊C₈	—	67	2	no	—
116	C₂×C₄⋊C₈	—	67	1	no	—
117	C₂×C₄×C₈	—	67	2	no	—
118	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
119	C₄⋊C₁₆	—	67	8	no	—
120	C₄⋊C₁₆	C₂×D₆₇	1	4	yes	—
121	C₄×C₁₆	—	67	8	no	—
122	C₄×C₁₆	—	67	4	no	—
123	C₂²×Q₈	—	67	1	no	—
124	C₂²×C₈	—	67	2	no	—
125	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
127	C₄×Q₈	—	67	4	no	—
128	C₄×Q₈	—	67	4	no	—
129	C₂×C₄⋊C₄	—	67	2	no	—
130	C₂×C₄⋊C₄	—	67	1	no	—
131	C₂×C₄²	—	67	2	no	—
132	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
133	C₂×C₁₆	—	2	4	no	—
134	C₂×C₁₆	—	67	4	no	—
135	C₂×C₁₆	—	67	2	no	—
136	C₂×C₁₆	—	67	2	no	—
137	C₂×C₁₆	—	134	2	no	—
138	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
139	C₄⋊C₈	—	67	8	no	—
140	C₄⋊C₈	—	67	4	no	—
141	C₄×C₈	—	67	4	no	—
142	C₄×C₈	—	67	4	no	—
143	C₄×C₈	C₄×D₆₇	1	2	yes	—
144	C₄×C₈	C₂²×D₆₇	1	2	yes	—
145	C₂×Q₈	—	67	8	no	—
146	C₂×Q₈	—	67	4	no	—
147	C₂²×C₄	—	67	2	no	—
148	C₂²×C₄	—	67	2	no	—
149	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
150	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	67	4	no	—
154	C₂×C₈	—	134	4	no	—
155	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
156	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
157	C₂×C₈	Q₈×D₆₇	1	2	yes	—
158	C₂×C₈	#11761627	1	2	yes	—
159	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
160	C₄⋊C₄	—	67	8	no	—
161	C₄⋊C₄	—	67	4	no	—
162	C₄²	—	67	4	no	—
163	C₄²	—	67	4	no	—
164	C₄²	C₈×D₆₇	1	2	yes	—
165	C₄²	C₂×C₄×D₆₇	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₁₆	—	67	4	no	—
168	C₁₆	—	134	2	no	—
169	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
170	Q₈	—	67	16	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	67	8	no	—
173	C₂×C₄	—	67	4	no	—
174	C₂×C₄	—	67	4	no	—
175	C₂×C₄	—	134	4	no	—
176	C₂×C₄	C₁₆×D₆₇	1	4	yes	—
177	C₂×C₄	#14524008	1	2	yes	—
178	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
179	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
181	C₂×C₄	#14524051	1	2	yes	—
182	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	134	4	no	—
185	C₈	#14524008	1	2	yes	—
186	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
187	C₈	C₂×#11761627	1	1	yes	characteristic
188	C₂²	#16863520	1	2	yes	—
189	C₂²	C₂×C₁₆×D₆₇	1	2	yes	—
190	C₂²	#16863814	1	2	yes	—
191	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	67	8	no	—
194	C₄	—	134	4	no	—
195	C₄	#16863520	1	2	yes	—
196	C₄	C₂×C₁₆×D₆₇	1	4	yes	—
197	C₄	C₂×#14524008	1	1	yes	characteristic
198	C₄	C₂×#14524051	1	1	yes	characteristic
199	C₂	#22562111	1	4	yes	direct product factor
200	C₂	C₂×#16863520	1	1	yes	characteristic
201	C₂	C₂²×C₁₆×D₆₇	1	1	yes	characteristic
202	C₂	C₂×#16863814	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #22562111	C₂	direct (N × H)
29. C₂×C₆₇⋊C₁₆	Q₈	semidirect
43. C₂×Dic₁₃₄	C₁₆	semidirect
54. C₆₇⋊C₁₆	C₂×Q₈	semidirect
80. Dic₁₃₄	C₂×C₁₆	semidirect
98. C₂×C₆₇	Q₈×C₁₆	semidirect
108. C₆₇	C₂×Q₈×C₁₆	semidirect
199. C₂	#22562111	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×C₈×Dic₁₃₄	2
#22562111	2
C₂×Dic₆₇⋊C₁₆	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂×C₁₆×Dic₆₇	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×Q₈×C₁₆	67

Subgroups — C2×#22562111 #31009311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562111 #31009311