Subgroups — C₂×#22570733 #31725447

Order 17152.

1902 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863953	C₂	1	2	yes	maximal
2	#22570733	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16863805	C₂	1	8	yes	maximal, complemented
4	C₂×#16863434	C₂	1	4	yes	maximal
5	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863953	—	2	8	no	—
7	#16863953	C₂²	1	16	yes	—
8	C₂×#14524043	C₂²	1	4	yes	—
9	C₂×#14524035	C₂²	1	4	yes	—
10	C₂×#14524022	—	2	4	no	—
11	C₂×C₄×Dic₁₃₄	—	2	4	no	—
12	C₂×#14524002	C₂²	1	4	yes	—
13	#16863805	C₂²	1	64	yes	complemented
14	#16863434	C₂²	1	32	yes	—
15	C₂×#14523977	C₂²	1	8	yes	—
16	C₂×#14523975	C₂²	1	8	yes	—
17	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#97×C₆₇	C₂²	1	4	yes	—
20	C₂×#97×C₆₇	C₂²	1	8	yes	—
21	C₂²×Dic₁₃₄	—	2	4	no	—
22	#14524043	—	2	8	no	—
23	#14524043	C₂³	1	16	yes	—
24	#14524035	—	2	8	no	—
25	#14524035	C₂³	1	16	yes	—
26	#14524022	—	2	32	no	—
27	C₄×Dic₁₃₄	—	2	32	no	—
28	#14524002	—	2	8	no	—
29	#14524002	C₂³	1	16	yes	—
30	C₂×C₄⋊Dic₆₇	—	2	4	no	—
31	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
32	C₂×Dic₆₇⋊C₄	—	2	8	no	—
33	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
34	C₂×C₄×Dic₆₇	—	2	4	no	—
35	#14523977	C₂³	1	32	yes	—
36	#14523975	C₂³	1	32	yes	—
37	C₂³×C₄×C₆₇	D₄	1	2	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	Q₈	1	16	yes	complemented
40	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
42	C₂×C₄²×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	D₄	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
46	#97×C₆₇	C₂³	1	16	yes	—
47	C₂×Dic₁₃₄	—	2	16	no	—
48	C₂×Dic₁₃₄	—	4	16	no	—
49	C₂²×Dic₆₇	—	2	8	no	—
50	C₄⋊Dic₆₇	—	2	16	no	—
51	C₄⋊Dic₆₇	—	2	8	no	—
52	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
53	Dic₆₇⋊C₄	—	2	32	no	—
54	C₂²⋊Dic₆₇	—	2	16	no	—
55	C₄×Dic₆₇	—	2	16	no	—
56	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	—	4	4	no	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
65	C₂×C₈×C₆₇	C₂×Q₈	1	16	yes	—
66	C₂×C₈×C₆₇	C₂×Q₈	1	32	yes	complemented
67	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
68	C₄²×C₆₇	C₂×D₄	1	8	yes	—
69	C₄²×C₆₇	C₂×D₄	1	4	yes	—
70	C₄²×C₆₇	C₂⁴	1	4	yes	—
71	C₂³×C₆₇	—	2	4	no	—
72	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
73	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
74	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	—	2	8	no	—
76	C₂×C₄×C₆₇	—	4	8	no	—
77	C₂×C₄×C₆₇	C₄⋊Q₈	1	8	yes	—
78	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
79	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
80	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
81	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
83	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
84	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
85	C₂×Dic₆₇	—	2	16	no	—
86	C₂×Dic₆₇	—	4	16	no	—
87	Dic₁₃₄	—	4	32	no	—
88	C₈×C₆₇	C₂²×Q₈	1	16	yes	—
89	C₂²×C₆₇	—	2	8	no	—
90	C₂²×C₆₇	—	2	4	no	—
91	C₂²×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
92	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
93	C₂²×C₆₇	#575	1	4	yes	—
94	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
95	C₄×C₆₇	—	2	8	no	—
96	C₄×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
97	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
98	C₄×C₆₇	C₂³×Q₈	1	2	yes	—
99	Dic₆₇	—	4	16	no	—
100	C₂×#3162	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	4	no	—
102	C₂×C₆₇	#3162	1	4	yes	complemented
103	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₆₇	C₂×#575	1	2	yes	—
105	C₂×#533	—	67	2	no	—
106	#3162	—	67	16	no	—
107	C₂×#501	—	67	8	no	—
108	C₂×#427	—	67	4	no	—
109	C₂²×#97	D₆₇	1	1	yes	characteristic
110	C₆₇	C₂×#3162	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₈	C₂×D₆₇	1	2	yes	—
112	#533	—	67	16	no	—
113	#533	—	134	8	no	—
114	C₂×C₄⋊Q₈	—	67	4	no	—
115	C₂×#125	—	67	4	no	—
116	C₂×C₂²⋊Q₈	—	134	4	no	—
117	C₂×C₄×Q₈	—	134	4	no	—
118	C₂×#117	—	67	4	no	—
119	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
120	#501	—	67	64	no	—
121	#427	—	67	32	no	—
122	C₂×#107	—	67	8	no	—
123	C₂×#106	—	67	8	no	—
124	C₂×#97	C₂×D₆₇	1	4	yes	—
125	C₂×#97	C₂×D₆₇	1	8	yes	—
126	C₂²×Q₈	—	134	4	no	—
127	C₂³×C₄	D₂₆₈	1	2	yes	—
128	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
129	C₂²×C₈	Dic₁₃₄	1	16	yes	complemented
130	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
131	C₂²×C₈	C₂²×D₆₇	1	8	yes	—
132	C₄⋊Q₈	—	67	16	no	—
133	C₄⋊Q₈	—	134	8	no	—
134	#125	—	67	16	no	—
135	#125	—	134	8	no	—
136	C₂²⋊Q₈	—	134	32	no	—
137	C₄×Q₈	—	134	32	no	—
138	#117	—	67	16	no	—
139	#117	—	134	8	no	—
140	C₂×C₄⋊C₄	—	67	8	no	—
141	C₂×C₄⋊C₄	—	134	8	no	—
142	C₂×C₄⋊C₄	—	134	4	no	—
143	C₂×C₂²⋊C₄	—	134	4	no	—
144	C₂×C₄²	—	134	4	no	—
145	C₂×C₄²	D₂₆₈	1	4	yes	—
146	C₂×C₄²	D₂₆₈	1	2	yes	—
147	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
148	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
149	#107	—	67	32	no	—
150	#106	—	67	32	no	—
151	#97	C₂²×D₆₇	1	16	yes	—
152	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
153	C₂×Q₈	—	268	16	no	—
154	C₂×Q₈	—	134	16	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	134	8	no	—
157	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
158	C₂²×C₄	C₂×D₂₆₈	1	8	yes	—
159	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
160	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
161	C₂²×C₄	C₂×Dic₁₃₄	1	8	yes	—
162	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
163	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₃₄	1	16	yes	—
165	C₂×C₈	C₂×Dic₁₃₄	1	32	yes	complemented
166	C₂×C₈	C₂³×D₆₇	1	8	yes	—
167	C₄⋊C₄	—	67	16	no	—
168	C₄⋊C₄	—	134	8	no	—
169	C₄⋊C₄	—	134	32	no	—
170	C₄⋊C₄	—	134	16	no	—
171	C₂²⋊C₄	—	134	16	no	—
172	C₄²	—	134	16	no	—
173	C₄²	C₂×D₂₆₈	1	8	yes	—
174	C₄²	C₂×D₂₆₈	1	4	yes	—
175	C₄²	C₂³×D₆₇	1	4	yes	—
176	C₂³	—	2	4	no	—
177	C₂³	#14524043	1	4	yes	—
178	C₂³	C₂²×D₂₆₈	1	2	yes	—
179	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
180	Q₈	—	268	32	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	268	16	no	—
184	C₂×C₄	—	134	16	no	—
185	C₂×C₄	#14524043	1	8	yes	—
186	C₂×C₄	#14524043	1	4	yes	—
187	C₂×C₄	C₂²×D₂₆₈	1	8	yes	—
188	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
189	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₁₃₄	1	4	yes	—
191	C₂×C₄	C₂²×Dic₁₃₄	1	8	yes	—
192	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
193	C₈	C₂²×Dic₁₃₄	1	16	yes	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	C₂×#14524043	1	4	yes	—
197	C₂²	C₂×#14524043	1	2	yes	—
198	C₂²	#16864196	1	4	yes	—
199	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
200	C₄	—	2	8	no	—
201	C₄	—	268	16	no	—
202	C₄	C₂×#14524043	1	2	yes	—
203	C₄	C₂×#14524043	1	4	yes	—
204	C₄	C₂³×Dic₁₃₄	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	#22570733	1	4	yes	direct product factor
207	C₂	C₂²×#14524043	1	1	yes	characteristic
208	C₂	C₂×#16864196	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	C₂×#3162
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #22570733	C₂	direct (N × H)
3. C₂×#16863805	C₂	semidirect
13. #16863805	C₂²	semidirect
39. C₂²×C₈×C₆₇	Q₈	semidirect
66. C₂×C₈×C₆₇	C₂×Q₈	semidirect
102. C₂×C₆₇	#3162	semidirect
110. C₆₇	C₂×#3162	semidirect
129. C₂²×C₈	Dic₁₃₄	semidirect
165. C₂×C₈	C₂×Dic₁₃₄	semidirect
206. C₂	#22570733	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×#16863953	2
#22570733	2
C₂×#16863805	2
C₂×#16863434	2
C₂²×#97×C₆₇	2
C₂×#3162	67

Subgroups — C2×#22570733 #31725447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22570733 #31725447