Subgroups — C₂×#22570723 #31725436

Order 17152.

1942 conjugacy classes of subgroups.

The subgroup lattice has 248 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	#22570723	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16863801	C₂	1	2	yes	maximal, complemented
4	C₂×#16863798	C₂	1	4	yes	maximal, complemented
5	C₂×#16863794	C₂	1	2	yes	maximal, complemented
6	C₂×#16863434	C₂	1	4	yes	maximal
7	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863953	—	2	8	no	—
9	#16863953	C₂²	1	16	yes	—
10	C₂×#14524043	C₂²	1	4	yes	—
11	C₂×#14524035	C₂²	1	4	yes	—
12	C₂×#14524022	—	2	4	no	—
13	C₂×C₄×Dic₁₃₄	—	2	4	no	—
14	C₂×#14524002	C₂²	1	4	yes	—
15	#16863801	C₂²	1	16	yes	complemented
16	#16863798	C₂²	1	32	yes	complemented
17	#16863794	C₂²	1	16	yes	complemented
18	#16863434	C₂²	1	32	yes	—
19	C₂×#14523977	C₂²	1	8	yes	—
20	C₂×#14523975	C₂²	1	8	yes	—
21	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	8	yes	—
27	C₂²×Dic₁₃₄	—	2	4	no	—
28	#14524043	—	2	8	no	—
29	#14524043	C₂³	1	16	yes	—
30	#14524035	—	2	8	no	—
31	#14524035	C₂³	1	16	yes	—
32	#14524022	—	2	32	no	—
33	C₄×Dic₁₃₄	—	2	32	no	—
34	#14524002	—	2	8	no	—
35	#14524002	C₂³	1	16	yes	—
36	C₂×C₄⋊Dic₆₇	—	2	4	no	—
37	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
38	C₂×Dic₆₇⋊C₄	—	2	8	no	—
39	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
40	C₂×C₄×Dic₆₇	—	2	4	no	—
41	#14523977	C₂³	1	32	yes	—
42	#14523975	C₂³	1	32	yes	—
43	C₂³×C₄×C₆₇	D₄	1	2	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	Q₈	1	16	yes	complemented
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
49	C₂×C₄²×C₆₇	D₄	1	2	yes	—
50	C₂×C₄²×C₆₇	D₄	1	4	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
54	C₄×C₈×C₆₇	C₂³	1	4	yes	—
55	C₄×C₈×C₆₇	C₂³	1	8	yes	—
56	C₄×C₈×C₆₇	C₂³	1	4	yes	—
57	C₂×Dic₁₃₄	—	2	16	no	—
58	C₂×Dic₁₃₄	—	4	16	no	—
59	C₂²×Dic₆₇	—	2	8	no	—
60	C₄⋊Dic₆₇	—	2	16	no	—
61	C₄⋊Dic₆₇	—	2	8	no	—
62	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
63	Dic₆₇⋊C₄	—	2	32	no	—
64	C₂²⋊Dic₆₇	—	2	16	no	—
65	C₄×Dic₆₇	—	2	16	no	—
66	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	8	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
74	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
76	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
77	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
78	C₂×C₈×C₆₇	C₂×Q₈	1	32	yes	complemented
79	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
80	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
81	C₄²×C₆₇	C₂×D₄	1	8	yes	—
82	C₄²×C₆₇	C₂×D₄	1	2	yes	—
83	C₄²×C₆₇	C₂×D₄	1	2	yes	—
84	C₄²×C₆₇	C₂⁴	1	2	yes	—
85	C₄²×C₆₇	C₂⁴	1	2	yes	—
86	C₂³×C₆₇	—	2	4	no	—
87	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
88	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
89	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	16	no	—
92	C₂×C₄×C₆₇	C₄⋊Q₈	1	8	yes	—
93	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
94	C₂×C₄×C₆₇	#135	1	8	yes	—
95	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
96	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
97	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
99	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
100	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
101	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
102	C₂×Dic₆₇	—	2	16	no	—
103	C₂×Dic₆₇	—	4	16	no	—
104	Dic₁₃₄	—	4	32	no	—
105	C₈×C₆₇	C₂²×Q₈	1	8	yes	—
106	C₈×C₆₇	C₂²×Q₈	1	8	yes	—
107	C₂²×C₆₇	—	2	8	no	—
108	C₂²×C₆₇	—	2	4	no	—
109	C₂²×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
110	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
111	C₂²×C₆₇	C₂×#135	1	4	yes	—
112	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
113	C₄×C₆₇	—	2	4	no	—
114	C₄×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
115	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
116	C₄×C₆₇	C₂×#135	1	8	yes	—
117	C₄×C₆₇	C₂³×Q₈	1	2	yes	—
118	Dic₆₇	—	4	16	no	—
119	C₂×#3160	—	67	1	no	maximal, Sylow
120	C₂×C₆₇	—	2	4	no	—
121	C₂×C₆₇	#3160	1	4	yes	complemented
122	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₆₇	C₂²×#135	1	2	yes	—
124	C₂×#533	—	67	2	no	—
125	#3160	—	67	16	no	—
126	C₂×#500	—	67	2	no	—
127	C₂×#499	—	67	4	no	—
128	C₂×#498	—	67	2	no	—
129	C₂×#427	—	67	4	no	—
130	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
131	C₆₇	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₈	C₂×D₆₇	1	2	yes	—
133	#533	—	67	16	no	—
134	#533	—	134	8	no	—
135	C₂×C₄⋊Q₈	—	67	4	no	—
136	C₂×#125	—	67	4	no	—
137	C₂×C₂²⋊Q₈	—	134	4	no	—
138	C₂×C₄×Q₈	—	134	4	no	—
139	C₂×#117	—	67	4	no	—
140	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
141	#500	—	67	16	no	—
142	#499	—	67	32	no	—
143	#498	—	67	16	no	—
144	#427	—	67	32	no	—
145	C₂×#107	—	67	8	no	—
146	C₂×#106	—	67	8	no	—
147	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
149	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂×D₆₇	1	8	yes	—
151	C₂²×Q₈	—	134	4	no	—
152	C₂³×C₄	D₂₆₈	1	2	yes	—
153	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
154	C₂²×C₈	Dic₁₃₄	1	16	yes	complemented
155	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
156	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
157	C₂²×C₈	C₂²×D₆₇	1	8	yes	—
158	C₄⋊Q₈	—	67	16	no	—
159	C₄⋊Q₈	—	134	8	no	—
160	#125	—	67	16	no	—
161	#125	—	134	8	no	—
162	C₂²⋊Q₈	—	134	32	no	—
163	C₄×Q₈	—	134	32	no	—
164	#117	—	67	16	no	—
165	#117	—	134	8	no	—
166	C₂×C₄⋊C₄	—	67	8	no	—
167	C₂×C₄⋊C₄	—	134	8	no	—
168	C₂×C₄⋊C₄	—	134	4	no	—
169	C₂×C₂²⋊C₄	—	134	4	no	—
170	C₂×C₄²	—	134	4	no	—
171	C₂×C₄²	D₂₆₈	1	4	yes	—
172	C₂×C₄²	D₂₆₈	1	2	yes	—
173	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
176	#107	—	67	32	no	—
177	#106	—	67	32	no	—
178	C₄×C₈	C₂²×D₆₇	1	4	yes	—
179	C₄×C₈	C₂²×D₆₇	1	8	yes	—
180	C₄×C₈	C₂²×D₆₇	1	4	yes	—
181	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
182	C₂×Q₈	—	268	16	no	—
183	C₂×Q₈	—	134	16	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	134	8	no	—
186	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
187	C₂²×C₄	C₂×D₂₆₈	1	8	yes	—
188	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
189	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
190	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
191	C₂²×C₄	C₂×Dic₁₃₄	1	8	yes	—
192	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
194	C₂×C₈	C₂×Dic₁₃₄	1	8	yes	—
195	C₂×C₈	C₂×Dic₁₃₄	1	8	yes	—
196	C₂×C₈	C₂×Dic₁₃₄	1	32	yes	complemented
197	C₂×C₈	C₂³×D₆₇	1	4	yes	—
198	C₂×C₈	C₂³×D₆₇	1	4	yes	—
199	C₄⋊C₄	—	67	16	no	—
200	C₄⋊C₄	—	134	8	no	—
201	C₄⋊C₄	—	134	32	no	—
202	C₄⋊C₄	—	134	16	no	—
203	C₂²⋊C₄	—	134	16	no	—
204	C₄²	—	134	16	no	—
205	C₄²	C₂×D₂₆₈	1	8	yes	—
206	C₄²	C₂×D₂₆₈	1	2	yes	—
207	C₄²	C₂×D₂₆₈	1	2	yes	—
208	C₄²	C₂³×D₆₇	1	2	yes	—
209	C₄²	C₂³×D₆₇	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	#14524043	1	4	yes	—
212	C₂³	C₂²×D₂₆₈	1	2	yes	—
213	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
214	Q₈	—	268	32	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	16	no	—
217	C₂×C₄	—	268	16	no	—
218	C₂×C₄	—	134	16	no	—
219	C₂×C₄	#14524043	1	8	yes	—
220	C₂×C₄	#14524043	1	4	yes	—
221	C₂×C₄	#14524065	1	8	yes	—
222	C₂×C₄	C₂²×D₂₆₈	1	4	yes	—
223	C₂×C₄	C₂²×D₂₆₈	1	4	yes	—
224	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
225	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
226	C₂×C₄	C₂²×Dic₁₃₄	1	4	yes	—
227	C₂×C₄	C₂²×Dic₁₃₄	1	8	yes	—
228	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
229	C₈	C₂²×Dic₁₃₄	1	8	yes	—
230	C₈	C₂²×Dic₁₃₄	1	8	yes	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	8	no	—
233	C₂²	C₂×#14524043	1	4	yes	—
234	C₂²	C₂×#14524043	1	2	yes	—
235	C₂²	C₂×#14524065	1	4	yes	—
236	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	268	16	no	—
239	C₄	C₂×#14524043	1	2	yes	—
240	C₄	C₂×#14524043	1	4	yes	—
241	C₄	C₂×#14524065	1	8	yes	—
242	C₄	C₂³×Dic₁₃₄	1	2	yes	—
243	C₂	—	2	4	no	—
244	C₂	#22570723	1	4	yes	direct product factor
245	C₂	C₂²×#14524043	1	1	yes	characteristic
246	C₂	C₂²×#14524065	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	C₂×#3160
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22570723	C₂	direct (N × H)
3. C₂×#16863801	C₂	semidirect
4. C₂×#16863798	C₂	semidirect
5. C₂×#16863794	C₂	semidirect
15. #16863801	C₂²	semidirect
16. #16863798	C₂²	semidirect
17. #16863794	C₂²	semidirect
45. C₂²×C₈×C₆₇	Q₈	semidirect
78. C₂×C₈×C₆₇	C₂×Q₈	semidirect
121. C₂×C₆₇	#3160	semidirect
131. C₆₇	C₂×#3160	semidirect
154. C₂²×C₈	Dic₁₃₄	semidirect
196. C₂×C₈	C₂×Dic₁₃₄	semidirect
244. C₂	#22570723	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
#22570723	2
C₂×#16863801	2
C₂×#16863798	2
C₂×#16863794	2
C₂×#16863434	2
C₂²×C₄×C₈×C₆₇	2
C₂×#3160	67

Subgroups — C2×#22570723 #31725436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22570723 #31725436