Subgroups — C₂×#22562432 #31011114

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×#16863801	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562432	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863197	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	—	2	1	no	—
7	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
8	#16863801	—	2	2	no	—
9	#16863801	C₂²	1	4	yes	complemented
10	C₂×#14523977	—	2	1	no	—
11	C₂×#14523977	C₂²	1	2	yes	—
12	#16863197	C₂²	1	16	yes	complemented
13	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	complemented
14	C₂²×D₈×C₆₇	—	2	1	no	—
15	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
16	C₂×#127×C₆₇	—	2	1	no	—
17	#493×C₆₇	—	2	2	no	—
18	#493×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₃₄	—	4	1	no	—
21	#14524043	—	2	4	no	—
22	#14524043	—	4	2	no	—
23	C₂×C₄⋊Dic₆₇	—	2	2	no	—
24	C₂×C₄⋊Dic₆₇	—	4	1	no	—
25	C₂×C₆₇⋊C₁₆	—	2	4	no	—
26	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
27	#14523977	—	2	2	no	—
28	#14523977	—	2	4	no	—
29	#14523977	C₂³	1	4	yes	—
30	C₂²×D₄×C₆₇	—	2	2	no	—
31	C₂²×D₄×C₆₇	—	4	1	no	—
32	C₂²×D₄×C₆₇	—	4	1	no	—
33	C₂×D₈×C₆₇	—	2	2	no	—
34	C₂×D₈×C₆₇	—	2	4	no	—
35	C₂×D₈×C₆₇	—	2	8	no	—
36	C₂×D₈×C₆₇	—	4	4	no	—
37	C₂×D₈×C₆₇	C₂³	1	4	yes	—
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	#127×C₆₇	—	2	4	no	—
41	#127×C₆₇	—	4	2	no	—
42	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	2	yes	—
44	C₄×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂×Dic₁₃₄	—	4	4	no	—
46	C₂×Dic₁₃₄	—	8	4	no	—
47	C₂²×Dic₆₇	—	4	2	no	—
48	C₄⋊Dic₆₇	—	2	4	no	—
49	C₄⋊Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	4	4	no	—
51	C₆₇⋊C₁₆	—	2	4	no	—
52	C₂⁴×C₆₇	—	4	2	no	—
53	C₂×D₄×C₆₇	—	2	4	no	—
54	C₂×D₄×C₆₇	—	4	4	no	—
55	C₂×D₄×C₆₇	—	4	4	no	—
56	C₂×D₄×C₆₇	—	4	4	no	—
57	C₂×D₄×C₆₇	—	4	2	no	—
58	C₂×D₄×C₆₇	—	4	4	no	—
59	C₂×D₄×C₆₇	—	8	4	no	—
60	C₂×D₄×C₆₇	—	8	4	no	—
61	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	D₈×C₆₇	—	2	8	no	—
65	D₈×C₆₇	—	4	4	no	—
66	D₈×C₆₇	—	4	8	no	—
67	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
68	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₆₇	#41	1	4	yes	—
71	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₄²×C₆₇	D₈	1	2	yes	—
73	C₄²×C₆₇	C₂×D₄	1	2	yes	—
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	D₄×C₆₇	—	4	4	no	—
80	D₄×C₆₇	—	4	4	no	—
81	D₄×C₆₇	—	8	4	no	—
82	D₄×C₆₇	—	8	8	no	—
83	D₄×C₆₇	—	8	8	no	—
84	C₂×C₄×C₆₇	D₁₆	1	4	yes	—
85	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
86	C₂×C₄×C₆₇	#124	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₆₇	—	4	4	no	—
93	C₂×Dic₆₇	—	8	4	no	—
94	Dic₁₃₄	—	8	8	no	—
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	#486	1	2	yes	—
107	C₄×C₆₇	C₂×D₁₆	1	4	yes	—
108	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
109	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
110	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
111	Dic₆₇	—	8	4	no	—
112	C₂×#2240	—	67	1	no	maximal, Sylow
113	C₂×C₆₇	—	16	4	no	—
114	C₂×C₆₇	#2240	1	4	yes	complemented
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	—	67	1	no	—
119	C₂×#493	D₆₇	1	1	yes	characteristic
120	#2240	—	67	8	no	—
121	C₂×#357	—	67	4	no	—
122	C₂×C₄×C₁₆	—	67	1	no	—
123	C₆₇	C₂×#2240	1	1	yes	characteristic, complemented, Sylow
124	C₂²×D₈	—	2	1	no	—
125	C₂²×D₈	C₂×D₆₇	1	2	yes	—
126	C₂×C₄⋊Q₈	—	134	1	no	—
127	C₂×#127	—	2	1	no	—
128	C₂²×C₁₆	—	67	2	no	—
129	#500	—	67	4	no	—
130	#500	—	134	2	no	—
131	#493	—	2	2	no	—
132	#493	C₂×D₆₇	1	4	yes	—
133	C₂×#107	—	67	2	no	—
134	C₂×#107	—	134	1	no	—
135	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
136	#357	—	67	16	no	—
137	C₄×C₁₆	—	67	4	no	—
138	C₂²×Q₈	—	268	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	C₂²×D₆₇	1	4	yes	—
147	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
149	C₄⋊Q₈	—	268	2	no	—
150	C₄⋊Q₈	—	134	4	no	—
151	#127	—	2	4	no	—
152	#127	—	4	2	no	—
153	C₂×C₄⋊C₄	—	268	1	no	—
154	C₂×C₄⋊C₄	—	134	2	no	—
155	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂×C₁₆	—	67	4	no	—
157	C₂×C₁₆	—	134	4	no	—
158	#107	—	67	4	no	—
159	#107	—	134	2	no	—
160	#107	—	134	4	no	—
161	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
162	C₄×C₈	C₂²×D₆₇	1	2	yes	—
163	C₂⁴	—	4	2	no	—
164	C₂×Q₈	—	268	4	no	—
165	C₂×Q₈	—	536	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	—	268	2	no	—
175	C₂²×C₄	#11761613	1	1	yes	characteristic
176	C₂²×C₄	#11761613	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
178	D₈	—	2	8	no	—
179	D₈	—	4	4	no	—
180	D₈	—	4	8	no	—
181	C₂×C₈	#11761616	1	4	yes	—
182	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
184	C₂×C₈	#11761627	1	4	yes	—
185	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
186	C₄⋊C₄	—	268	4	no	—
187	C₄⋊C₄	—	268	2	no	—
188	C₄⋊C₄	—	134	4	no	—
189	C₄²	#11761613	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
191	C₁₆	—	134	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×#11761613	1	1	yes	characteristic
197	Q₈	—	536	8	no	—
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	8	8	no	—
201	D₄	—	8	8	no	—
202	D₄	—	8	4	no	—
203	C₂×C₄	—	268	4	no	—
204	C₂×C₄	—	536	4	no	—
205	C₂×C₄	#14523984	1	4	yes	—
206	C₂×C₄	#14523987	1	4	yes	—
207	C₂×C₄	#14524030	1	2	yes	—
208	C₂×C₄	C₂×#11761613	1	2	yes	—
209	C₂×C₄	C₂×#11761613	1	2	yes	—
210	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
211	C₂×C₄	C₂×#11761616	1	2	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₈	#14524030	1	2	yes	—
214	C₈	C₂×#11761616	1	4	yes	—
215	C₈	C₂×#11761627	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	#16863742	1	2	yes	—
221	C₂²	C₂×#14523984	1	2	yes	—
222	C₂²	C₂×#14523987	1	2	yes	—
223	C₂²	C₂²×#11761613	1	1	yes	characteristic
224	C₄	—	536	4	no	—
225	C₄	#16863742	1	2	yes	—
226	C₄	C₂×#14523984	1	4	yes	—
227	C₄	C₂×#14523987	1	4	yes	—
228	C₄	C₂×#14524030	1	1	yes	characteristic
229	C₄	C₂²×#11761616	1	1	yes	characteristic
230	C₂	—	16	4	no	—
231	C₂	#22562432	1	4	yes	direct product factor
232	C₂	C₂×#16863742	1	1	yes	characteristic
233	C₂	C₂²×#14523984	1	1	yes	characteristic
234	C₂	C₂²×#14523987	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting, prime radical	C₂×#493×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×#493	D₆₇
hypocenter	C₆₇	C₂×#2240
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#16863801	C₂	semidirect
2. #22562432	C₂	direct (N × H)
3. C₂×#16863197	C₂	semidirect
4. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
9. #16863801	C₂²	semidirect
12. #16863197	C₂²	semidirect
13. C₄×C₆₇⋊C₁₆	C₂²	semidirect
114. C₂×C₆₇	#2240	semidirect
123. C₆₇	C₂×#2240	semidirect
231. C₂	#22562432	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
#22562432	2
C₂×#16863197	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂×#493×C₆₇	2
C₂×#2240	67

Subgroups — C2×#22562432 #31011114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562432 #31011114