Subgroups — C₂×#22562470 #31011230

Order 17152.

854 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	C₂²×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14523988	C₂	1	2	yes	maximal, complemented
3	C₂²×D₁₀₇₂	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863780	C₂	1	2	yes	maximal, complemented
5	#22562470	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
8	C₂²×#11761617	—	2	2	no	—
9	C₂²×D₅₃₆	C₂²	1	2	yes	—
10	C₂×#14524031	—	2	2	no	—
11	C₂×Dic₅₃₆	—	2	4	no	—
12	C₂×Dic₅₃₆	C₂²	1	4	yes	complemented
13	C₂×#14523988	—	2	8	no	—
14	C₂×#14523988	C₂²	1	8	yes	complemented
15	C₂×D₁₀₇₂	—	2	4	no	—
16	C₂×D₁₀₇₂	C₂²	1	4	yes	—
17	#16863780	—	2	4	no	—
18	#16863780	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₆₇	C₂²	1	4	yes	complemented
23	C₂²×Dic₁₃₄	—	2	2	no	—
24	C₂²×D₂₆₈	—	2	2	no	—
25	C₂×Dic₂₆₈	—	2	4	no	—
26	C₂×Dic₂₆₈	—	2	4	no	—
27	C₂×Dic₂₆₈	—	2	2	no	—
28	C₂×Dic₂₆₈	C₂³	1	4	yes	—
29	C₂×#11761617	—	2	8	no	—
30	C₂×#11761617	—	4	8	no	—
31	C₂×D₅₃₆	—	2	4	no	—
32	C₂×D₅₃₆	—	2	4	no	—
33	C₂×D₅₃₆	—	2	2	no	—
34	C₂×D₅₃₆	C₂³	1	4	yes	—
35	#14524031	—	2	8	no	—
36	#14524031	—	4	4	no	—
37	C₂×(C₂×D₆₇)⋊C₄	—	4	4	no	—
38	Dic₅₃₆	—	2	8	no	—
39	#14523988	—	2	16	no	—
40	D₁₀₇₂	—	2	8	no	—
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
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	C₄×C₈×C₆₇	D₄	1	2	yes	—
49	C₄×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂³×D₆₇	—	4	2	no	—
51	C₂×Dic₁₃₄	—	2	4	no	—
52	C₂×Dic₁₃₄	—	4	4	no	—
53	C₂×Dic₁₃₄	—	4	4	no	—
54	C₂×Dic₁₃₄	—	4	2	no	—
55	C₂×D₂₆₈	—	2	4	no	—
56	C₂×D₂₆₈	—	4	4	no	—
57	C₂×D₂₆₈	—	4	4	no	—
58	C₂×D₂₆₈	—	4	2	no	—
59	C₂²×Dic₆₇	—	4	2	no	—
60	Dic₂₆₈	—	2	4	no	—
61	Dic₂₆₈	—	2	4	no	—
62	Dic₂₆₈	—	4	4	no	—
63	#11761617	—	4	16	no	—
64	D₅₃₆	—	2	4	no	—
65	D₅₃₆	—	2	4	no	—
66	D₅₃₆	—	4	4	no	—
67	(C₂×D₆₇)⋊C₄	—	4	16	no	—
68	C₂²×C₄×C₆₇	D₈	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	D₈	1	4	yes	—
71	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
75	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
76	C₄²×C₆₇	D₈	1	2	yes	—
77	C₄²×C₆₇	C₂×D₄	1	2	yes	—
78	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
79	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
80	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₆₇	—	4	4	no	—
82	C₂²×D₆₇	—	8	4	no	—
83	C₂²×D₆₇	—	8	8	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	#127	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
87	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×Dic₆₇	—	4	4	no	—
91	C₂×Dic₆₇	—	8	4	no	—
92	D₂₆₈	—	4	4	no	—
93	D₂₆₈	—	4	4	no	—
94	D₂₆₈	—	8	4	no	—
95	Dic₁₃₄	—	4	4	no	—
96	Dic₁₃₄	—	4	4	no	—
97	Dic₁₃₄	—	8	4	no	—
98	C₈×C₆₇	#127	1	2	yes	—
99	C₈×C₆₇	C₂×D₈	1	4	yes	—
100	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	C₂²×C₆₇	#493	1	2	yes	—
103	C₂²×C₆₇	#508	1	4	yes	—
104	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
105	C₂×D₆₇	—	8	4	no	—
106	C₂×D₆₇	—	8	8	no	—
107	C₂×D₆₇	—	16	8	no	—
108	C₄×C₆₇	—	2	4	no	—
109	C₄×C₆₇	#493	1	2	yes	—
110	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
111	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
112	Dic₆₇	—	8	4	no	—
113	C₂×#2249	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	#2249	1	4	yes	complemented
115	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
116	C₂×C₆₇	C₂×#508	1	2	yes	—
117	D₆₇	—	16	4	no	—
118	C₂²×Q₃₂	—	67	1	no	—
119	C₂²×SD₃₂	—	67	2	no	—
120	C₂²×D₁₆	—	67	1	no	—
121	C₂×#495	—	67	2	no	—
122	#2249	—	67	8	no	—
123	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic, complemented
124	C₆₇	C₂×#2249	1	1	yes	characteristic, complemented, Sylow
125	C₂²×Q₁₆	—	67	2	no	—
126	C₂²×SD₁₆	—	134	2	no	—
127	C₂²×D₈	—	67	2	no	—
128	C₂×#124	—	134	2	no	—
129	C₂×Q₃₂	—	67	4	no	—
130	C₂×Q₃₂	—	134	4	no	—
131	C₂×SD₃₂	—	67	8	no	—
132	C₂×SD₃₂	—	134	8	no	—
133	C₂×D₁₆	—	67	4	no	—
134	C₂×D₁₆	—	134	4	no	—
135	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
136	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
137	#495	—	67	8	no	—
138	#495	—	134	4	no	—
139	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
140	C₄×C₁₆	C₂×D₆₇	1	4	yes	complemented
141	C₂²×Q₈	—	134	2	no	—
142	C₂²×D₄	—	134	2	no	—
143	C₂×Q₁₆	—	67	4	no	—
144	C₂×Q₁₆	—	134	2	no	—
145	C₂×Q₁₆	—	134	4	no	—
146	C₂×Q₁₆	—	134	4	no	—
147	C₂×SD₁₆	—	268	8	no	—
148	C₂×SD₁₆	—	134	8	no	—
149	C₂×D₈	—	67	4	no	—
150	C₂×D₈	—	134	2	no	—
151	C₂×D₈	—	134	4	no	—
152	C₂×D₈	—	134	4	no	—
153	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
155	#124	—	268	4	no	—
156	#124	—	134	8	no	—
157	C₂×C₂²⋊C₄	—	268	4	no	—
158	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
159	Q₃₂	—	134	8	no	—
160	SD₃₂	—	134	16	no	—
161	D₁₆	—	134	8	no	—
162	C₂×C₁₆	D₂₆₈	1	4	yes	—
163	C₂×C₁₆	D₂₆₈	1	4	yes	—
164	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
165	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
166	C₄×C₈	D₂₆₈	1	2	yes	—
167	C₄×C₈	C₂²×D₆₇	1	2	yes	—
168	C₂⁴	—	268	2	no	—
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	C₂×D₄	—	268	4	no	—
174	C₂×D₄	—	268	4	no	—
175	C₂×D₄	—	268	2	no	—
176	C₂×D₄	—	134	4	no	—
177	C₂²×C₄	—	268	2	no	—
178	C₂²×C₄	D₅₃₆	1	2	yes	—
179	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
180	Q₁₆	—	268	4	no	—
181	Q₁₆	—	134	4	no	—
182	Q₁₆	—	134	4	no	—
183	SD₁₆	—	268	16	no	—
184	D₈	—	268	4	no	—
185	D₈	—	134	4	no	—
186	D₈	—	134	4	no	—
187	C₂×C₈	D₅₃₆	1	4	yes	—
188	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
189	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
190	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
191	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
192	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
193	C₂²⋊C₄	—	268	16	no	—
194	C₄²	D₅₃₆	1	2	yes	—
195	C₄²	C₂×D₂₆₈	1	2	yes	—
196	C₁₆	C₂×D₂₆₈	1	4	yes	—
197	C₁₆	C₂×D₂₆₈	1	4	yes	—
198	C₂³	—	268	4	no	—
199	C₂³	—	536	4	no	—
200	C₂³	—	536	8	no	—
201	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
202	Q₈	—	268	4	no	—
203	Q₈	—	268	4	no	—
204	Q₈	—	536	4	no	—
205	D₄	—	268	4	no	—
206	D₄	—	268	4	no	—
207	D₄	—	536	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	268	4	no	—
210	C₂×C₄	—	536	4	no	—
211	C₂×C₄	#14524040	1	2	yes	—
212	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
213	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
214	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
215	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
216	C₈	#14524040	1	2	yes	—
217	C₈	C₂×D₅₃₆	1	4	yes	—
218	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
219	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
220	C₂²	—	536	8	no	—
221	C₂²	—	536	4	no	—
222	C₂²	—	1072	8	no	—
223	C₂²	#16863773	1	2	yes	—
224	C₂²	#16863828	1	4	yes	—
225	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	536	4	no	—
228	C₄	#16863773	1	2	yes	—
229	C₄	C₂×#14524040	1	1	yes	characteristic
230	C₄	C₂²×D₅₃₆	1	1	yes	characteristic
231	C₂	—	1072	4	no	—
232	C₂	#22562470	1	4	yes	direct product factor
233	C₂	C₂×#16863773	1	1	yes	characteristic
234	C₂	C₂×#16863828	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂²×D₁₀₇₂	C₂
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	C₂×#2249
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₅₃₆	C₂	semidirect
2. C₂²×#14523988	C₂	semidirect
4. C₂×#16863780	C₂	semidirect
5. #22562470	C₂	direct (N × H)
6. C₂×C₄×C₁₆×C₆₇	C₂	semidirect
12. C₂×Dic₅₃₆	C₂²	semidirect
14. C₂×#14523988	C₂²	semidirect
18. #16863780	C₂²	semidirect
22. C₄×C₁₆×C₆₇	C₂²	semidirect
114. C₂×C₆₇	#2249	semidirect
123. C₂×C₄×C₁₆	D₆₇	semidirect
124. C₆₇	C₂×#2249	semidirect
140. C₄×C₁₆	C₂×D₆₇	semidirect
232. C₂	#22562470	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
C₂²×#14523988	2
C₂²×D₁₀₇₂	2
C₂×#16863780	2
#22562470	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#2249	67

Subgroups — C2×#22562470 #31011230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562470 #31011230