Subgroups — C₂×#22562317 #31010283

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 240 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, complemented
2	C₂×#16863637	C₂	1	2	yes	maximal
3	#22562317	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863224	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863201	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
8	C₂×#14524022	—	2	2	no	—
9	C₂×Dic₅₃₆	—	2	4	no	—
10	C₂×Dic₅₃₆	C₂²	1	4	yes	complemented
11	#16863637	—	2	4	no	—
12	#16863637	C₂²	1	8	yes	—
13	C₂×#14523977	C₂²	1	2	yes	—
14	C₂×#14523968	—	2	2	no	—
15	#16863224	C₂²	1	4	yes	complemented
16	#16863201	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₆₇	—	2	4	no	—
19	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
20	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₃₄	—	2	2	no	—
23	C₂×Dic₂₆₈	—	2	4	no	—
24	C₂×Dic₂₆₈	—	2	2	no	—
25	C₂×Dic₂₆₈	—	2	4	no	—
26	C₂×Dic₂₆₈	C₂³	1	4	yes	—
27	#14524022	—	2	8	no	—
28	#14524022	—	4	4	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×Dic₆₇⋊C₄	—	4	2	no	—
31	C₂×C₂²⋊Dic₆₇	—	4	2	no	—
32	Dic₅₃₆	—	2	8	no	—
33	#14523977	—	2	2	no	—
34	#14523977	C₂³	1	4	yes	—
35	#14523968	—	2	8	no	—
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	M₃₂×C₆₇	—	2	8	no	—
44	C₂×C₁₆×C₆₇	—	2	2	no	—
45	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
46	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
47	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
48	C₂×Dic₁₃₄	—	2	4	no	—
49	C₂×Dic₁₃₄	—	4	4	no	—
50	C₂×Dic₁₃₄	—	4	4	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂²×Dic₆₇	—	4	2	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	Dic₂₆₈	—	2	4	no	—
55	Dic₂₆₈	—	2	4	no	—
56	Dic₂₆₈	—	4	4	no	—
57	C₄⋊Dic₆₇	—	2	4	no	—
58	C₄⋊Dic₆₇	—	4	2	no	—
59	Dic₆₇⋊C₄	—	4	8	no	—
60	C₂²⋊Dic₆₇	—	4	8	no	—
61	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	D₈	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	#41	1	2	yes	—
76	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₁₆×C₆₇	—	2	2	no	—
78	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
79	C₂³×C₆₇	—	2	2	no	—
80	C₂³×C₆₇	—	2	4	no	—
81	C₂³×C₆₇	C₂×D₈	1	2	yes	—
82	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	#121	1	2	yes	—
88	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#135	1	2	yes	—
91	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₆₇	—	4	4	no	—
93	C₂×Dic₆₇	—	4	4	no	—
94	C₂×Dic₆₇	—	8	4	no	—
95	C₂×Dic₆₇	—	8	4	no	—
96	Dic₁₃₄	—	4	4	no	—
97	Dic₁₃₄	—	4	4	no	—
98	Dic₁₃₄	—	8	4	no	—
99	C₈×C₆₇	—	2	2	no	—
100	C₈×C₆₇	#121	1	2	yes	—
101	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
103	C₂²×C₆₇	—	2	4	no	—
104	C₂²×C₆₇	—	2	2	no	—
105	C₂²×C₆₇	—	4	4	no	—
106	C₂²×C₆₇	#466	1	2	yes	—
107	C₂²×C₆₇	#510	1	4	yes	—
108	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	#466	1	2	yes	—
111	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
112	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
113	Dic₆₇	—	8	4	no	—
114	Dic₆₇	—	8	4	no	—
115	C₂×#2219	—	67	1	no	maximal, Sylow
116	C₂×C₆₇	—	4	2	no	—
117	C₂×C₆₇	#2219	1	4	yes	complemented
118	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
119	C₂×C₆₇	C₂×#510	1	2	yes	—
120	C₂²×Q₃₂	—	67	1	no	—
121	C₂²×M₃₂	D₆₇	1	1	yes	characteristic
122	C₂×#467	—	67	2	no	—
123	#2219	—	67	8	no	—
124	C₂×#367	—	67	1	no	—
125	C₂×#358	—	67	2	no	—
126	C₆₇	C₂×#2219	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₁₆	—	67	2	no	—
128	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
129	C₂×C₂²⋊Q₈	—	134	2	no	—
130	C₂×Q₃₂	—	67	4	no	—
131	C₂×Q₃₂	—	134	4	no	—
132	C₂×M₃₂	—	2	4	no	—
133	C₂×M₃₂	C₂×D₆₇	1	4	yes	—
134	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
135	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
136	#467	—	67	8	no	—
137	#467	—	134	4	no	—
138	C₂×#107	—	67	2	no	—
139	C₂×#103	—	134	2	no	—
140	#367	—	67	4	no	—
141	#358	—	67	8	no	—
142	C₂²×Q₈	—	134	2	no	—
143	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
144	C₂×Q₁₆	—	67	4	no	—
145	C₂×Q₁₆	—	134	4	no	—
146	C₂×Q₁₆	—	134	2	no	—
147	C₂×Q₁₆	—	134	4	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
151	C₂²×C₈	D₂₆₈	1	2	yes	—
152	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
153	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
154	C₂²⋊Q₈	—	268	4	no	—
155	C₂²⋊Q₈	—	134	8	no	—
156	C₂×C₄⋊C₄	—	268	2	no	—
157	C₂×C₄⋊C₄	—	134	2	no	—
158	C₂×C₂²⋊C₄	—	268	2	no	—
159	Q₃₂	—	134	8	no	—
160	M₃₂	—	2	8	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	C₂²⋊D₆₇	1	4	yes	—
163	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
164	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
165	#107	—	67	4	no	—
166	#107	—	134	2	no	—
167	#103	—	134	8	no	—
168	C₂⁴	D₅₃₆	1	1	yes	characteristic
169	C₂×Q₈	—	268	2	no	—
170	C₂×Q₈	—	268	4	no	—
171	C₂×Q₈	—	268	4	no	—
172	C₂×Q₈	—	134	4	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	268	2	no	—
176	C₂²×C₄	—	268	2	no	—
177	C₂²×C₄	D₅₃₆	1	2	yes	—
178	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
180	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
181	Q₁₆	—	268	4	no	—
182	Q₁₆	—	134	4	no	—
183	Q₁₆	—	134	4	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
189	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
190	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
191	C₂×C₈	#11761628	1	2	yes	—
192	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
193	C₄⋊C₄	—	268	8	no	—
194	C₄⋊C₄	—	268	2	no	—
195	C₄⋊C₄	—	134	4	no	—
196	C₂²⋊C₄	—	268	8	no	—
197	C₁₆	—	2	2	no	—
198	C₁₆	C₂×C₂²⋊D₆₇	1	4	yes	—
199	C₂³	—	2	4	no	—
200	C₂³	—	2	2	no	—
201	C₂³	C₂×D₅₃₆	1	2	yes	—
202	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
203	Q₈	—	268	4	no	—
204	Q₈	—	268	4	no	—
205	Q₈	—	536	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	268	4	no	—
211	C₂×C₄	—	268	4	no	—
212	C₂×C₄	—	536	4	no	—
213	C₂×C₄	—	536	4	no	—
214	C₂×C₄	#14524017	1	2	yes	—
215	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
216	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
217	C₂×C₄	#14524065	1	2	yes	—
218	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	#14524017	1	2	yes	—
221	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
222	C₈	C₂×#11761628	1	1	yes	characteristic
223	C₂²	—	2	4	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	#16863632	1	2	yes	—
227	C₂²	#16863839	1	4	yes	—
228	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	536	4	no	—
231	C₄	—	536	4	no	—
232	C₄	#16863632	1	2	yes	—
233	C₄	C₂×#14524017	1	1	yes	characteristic
234	C₄	C₂×#14524065	1	1	yes	characteristic
235	C₂	—	4	2	no	—
236	C₂	#22562317	1	4	yes	direct product factor
237	C₂	C₂×#16863632	1	1	yes	characteristic
238	C₂	C₂×#16863839	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×M₃₂×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
hypercenter	C₂²×M₃₂	D₆₇
hypocenter	C₆₇	C₂×#2219
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₅₃₆	C₂	semidirect
3. #22562317	C₂	direct (N × H)
4. C₂×#16863224	C₂	semidirect
5. C₂×#16863201	C₂	semidirect
10. C₂×Dic₅₃₆	C₂²	semidirect
15. #16863224	C₂²	semidirect
16. #16863201	C₂²	semidirect
117. C₂×C₆₇	#2219	semidirect
126. C₆₇	C₂×#2219	semidirect
236. C₂	#22562317	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×#16863637	2
#22562317	2
C₂×#16863224	2
C₂×#16863201	2
C₂²×M₃₂×C₆₇	2
C₂×#2219	67

Subgroups — C2×#22562317 #31010283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562317 #31010283