Subgroups — C₂×#22562246 #31010072

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523987	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562246	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863197	C₂	1	2	yes	maximal, complemented
6	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₄	—	2	1	no	—
9	C₂×#14523987	—	2	8	no	—
10	C₂×#14523987	C₂²	1	8	yes	complemented
11	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄×Dic₂₆₈	—	2	2	no	—
13	C₄×Dic₂₆₈	C₂²	1	4	yes	complemented
14	C₂×#14523977	C₂²	1	1	yes	characteristic
15	C₂×#14523968	—	2	1	no	—
16	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	complemented
17	#16863197	C₂²	1	8	yes	complemented
18	C₂²×D₈×C₆₇	—	2	1	no	—
19	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
20	C₂×#127×C₆₇	—	2	1	no	—
21	#493×C₆₇	—	2	2	no	—
22	#493×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₃₄	—	2	1	no	—
25	C₂×Dic₂₆₈	—	2	1	no	—
26	C₂×Dic₂₆₈	—	2	2	no	—
27	C₂×Dic₂₆₈	D₄	1	4	yes	complemented
28	C₂×Dic₂₆₈	C₂³	1	2	yes	—
29	C₄×Dic₁₃₄	—	2	4	no	—
30	C₄×Dic₁₃₄	—	4	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×Dic₆₇⋊C₄	—	4	1	no	—
33	C₂×C₄×Dic₆₇	—	4	1	no	—
34	#14523987	—	2	16	no	—
35	C₂×C₆₇⋊C₁₆	—	2	4	no	—
36	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
37	#14523977	—	2	1	no	—
38	#14523977	C₂³	1	2	yes	—
39	#14523968	—	2	4	no	—
40	C₂²×D₄×C₆₇	—	2	2	no	—
41	C₂²×D₄×C₆₇	—	4	1	no	—
42	C₂²×D₄×C₆₇	—	4	1	no	—
43	C₂×D₈×C₆₇	—	2	4	no	—
44	C₂×D₈×C₆₇	—	2	2	no	—
45	C₂×D₈×C₆₇	—	2	4	no	—
46	C₂×D₈×C₆₇	—	2	4	no	—
47	C₂×D₈×C₆₇	—	4	4	no	—
48	C₂×D₈×C₆₇	C₂³	1	4	yes	—
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	#127×C₆₇	—	2	4	no	—
52	#127×C₆₇	—	4	2	no	—
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂×Dic₁₃₄	—	2	4	no	—
57	C₂×Dic₁₃₄	—	2	2	no	—
58	C₂×Dic₁₃₄	—	4	1	no	—
59	C₂×Dic₁₃₄	—	4	2	no	—
60	C₂²×Dic₆₇	—	4	1	no	—
61	C₂²×Dic₆₇	—	4	1	no	—
62	Dic₂₆₈	—	2	4	no	—
63	Dic₂₆₈	—	2	2	no	—
64	Dic₂₆₈	C₂×D₄	1	4	yes	complemented
65	C₄⋊Dic₆₇	—	2	2	no	—
66	C₄⋊Dic₆₇	—	4	1	no	—
67	Dic₆₇⋊C₄	—	4	4	no	—
68	C₄×Dic₆₇	—	4	4	no	—
69	C₆₇⋊C₁₆	—	2	4	no	—
70	C₂⁴×C₆₇	—	4	2	no	—
71	C₂×D₄×C₆₇	—	2	4	no	—
72	C₂×D₄×C₆₇	—	4	4	no	—
73	C₂×D₄×C₆₇	—	4	4	no	—
74	C₂×D₄×C₆₇	—	4	4	no	—
75	C₂×D₄×C₆₇	—	4	2	no	—
76	C₂×D₄×C₆₇	—	4	4	no	—
77	C₂×D₄×C₆₇	—	8	4	no	—
78	C₂×D₄×C₆₇	—	8	4	no	—
79	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	D₈×C₆₇	—	2	4	no	—
83	D₈×C₆₇	—	2	4	no	—
84	D₈×C₆₇	—	4	8	no	—
85	D₈×C₆₇	—	4	4	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	#41	1	2	yes	—
91	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
92	C₄²×C₆₇	D₈	1	2	yes	—
93	C₄²×C₆₇	C₂×D₄	1	2	yes	—
94	C₂³×C₆₇	—	4	4	no	—
95	C₂³×C₆₇	—	8	4	no	—
96	C₂³×C₆₇	—	8	4	no	—
97	C₂³×C₆₇	—	8	4	no	—
98	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
99	D₄×C₆₇	—	4	4	no	—
100	D₄×C₆₇	—	4	4	no	—
101	D₄×C₆₇	—	8	8	no	—
102	D₄×C₆₇	—	8	4	no	—
103	D₄×C₆₇	—	8	8	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	SD₃₂	1	4	yes	complemented
106	C₂×C₄×C₆₇	#121	1	2	yes	—
107	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₆₇	—	4	2	no	—
113	C₂×Dic₆₇	—	4	4	no	—
114	C₂×Dic₆₇	—	4	2	no	—
115	C₂×Dic₆₇	—	8	2	no	—
116	Dic₁₃₄	—	2	4	no	—
117	Dic₁₃₄	—	4	2	no	—
118	Dic₁₃₄	—	4	4	no	—
119	C₈×C₆₇	—	2	2	no	—
120	C₈×C₆₇	#121	1	2	yes	—
121	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
123	C₂²×C₆₇	—	8	4	no	—
124	C₂²×C₆₇	—	8	4	no	—
125	C₂²×C₆₇	—	8	4	no	—
126	C₂²×C₆₇	—	16	8	no	—
127	C₂²×C₆₇	#459	1	2	yes	—
128	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
129	C₂²×C₆₇	#509	1	2	yes	—
130	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
131	C₄×C₆₇	—	2	2	no	—
132	C₄×C₆₇	#459	1	2	yes	—
133	C₄×C₆₇	C₂×SD₃₂	1	4	yes	complemented
134	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
135	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
136	Dic₆₇	—	4	4	no	—
137	Dic₆₇	—	8	2	no	—
138	C₂×#2207	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	—	16	4	no	—
140	C₂×C₆₇	#2207	1	4	yes	complemented
141	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
142	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
143	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
144	C₂²×SD₃₂	—	67	2	no	—
145	C₂×#493	D₆₇	1	1	yes	characteristic
146	C₂×C₄×Q₁₆	—	67	1	no	—
147	#2207	—	67	8	no	—
148	C₂×C₄⋊C₁₆	—	67	1	no	—
149	C₂×#357	—	67	2	no	—
150	C₆₇	C₂×#2207	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₁₆	—	67	1	no	—
152	C₂²×D₈	—	2	1	no	—
153	C₂²×D₈	C₂×D₆₇	1	2	yes	—
154	C₂×#127	—	2	1	no	—
155	C₂×C₄×Q₈	—	134	1	no	—
156	C₂×SD₃₂	—	67	8	no	—
157	C₂×SD₃₂	—	134	8	no	—
158	C₂²×C₁₆	—	67	2	no	—
159	#493	—	2	2	no	—
160	#493	C₂×D₆₇	1	4	yes	—
161	C₄×Q₁₆	—	67	4	no	—
162	C₄×Q₁₆	—	134	2	no	—
163	C₂×#107	—	67	1	no	—
164	C₂×#103	—	134	1	no	—
165	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
166	C₄⋊C₁₆	—	67	4	no	—
167	#357	—	67	8	no	—
168	C₂²×Q₈	—	134	1	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	—	4	1	no	—
171	C₂²×D₄	—	4	1	no	—
172	C₂×Q₁₆	—	67	4	no	—
173	C₂×Q₁₆	—	67	2	no	—
174	C₂×Q₁₆	—	134	1	no	—
175	C₂×Q₁₆	—	134	2	no	—
176	C₂×D₈	—	2	4	no	—
177	C₂×D₈	—	2	4	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	—	2	4	no	—
180	C₂×D₈	—	4	4	no	—
181	C₂×D₈	C₂²×D₆₇	1	4	yes	—
182	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
184	#127	—	2	4	no	—
185	#127	—	4	2	no	—
186	C₄×Q₈	—	268	2	no	—
187	C₄×Q₈	—	134	4	no	—
188	C₂×C₄⋊C₄	—	268	1	no	—
189	C₂×C₄⋊C₄	—	134	1	no	—
190	C₂×C₄²	—	268	1	no	—
191	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
192	SD₃₂	—	134	16	no	—
193	C₂×C₁₆	—	67	4	no	—
194	C₂×C₁₆	—	134	4	no	—
195	#107	—	67	2	no	—
196	#107	—	134	1	no	—
197	#103	—	134	4	no	—
198	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
199	C₄×C₈	C₂²×D₆₇	1	2	yes	—
200	C₂⁴	—	4	2	no	—
201	C₂×Q₈	—	268	1	no	—
202	C₂×Q₈	—	268	2	no	—
203	C₂×Q₈	—	134	4	no	—
204	C₂×Q₈	—	134	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	4	no	—
211	C₂×D₄	—	8	4	no	—
212	C₂×D₄	—	8	4	no	—
213	C₂²×C₄	—	268	1	no	—
214	C₂²×C₄	—	268	1	no	—
215	C₂²×C₄	#11761613	1	1	yes	characteristic
216	C₂²×C₄	#11761613	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
218	Q₁₆	—	67	4	no	—
219	Q₁₆	—	134	4	no	—
220	Q₁₆	—	134	2	no	—
221	D₈	—	2	4	no	—
222	D₈	—	2	4	no	—
223	D₈	—	4	8	no	—
224	D₈	—	4	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	D₄×D₆₇	1	2	yes	—
227	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
228	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
229	C₂×C₈	#11761627	1	2	yes	—
230	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
231	C₄⋊C₄	—	268	4	no	—
232	C₄⋊C₄	—	268	1	no	—
233	C₄⋊C₄	—	134	2	no	—
234	C₄²	—	268	4	no	—
235	C₄²	#11761613	1	2	yes	—
236	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
237	C₁₆	—	134	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	8	4	no	—
240	C₂³	—	8	4	no	—
241	C₂³	—	8	4	no	—
242	C₂³	C₂×#11761613	1	1	yes	characteristic
243	Q₈	—	268	4	no	—
244	Q₈	—	268	2	no	—
245	Q₈	—	134	4	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	4	no	—
248	D₄	—	8	8	no	—
249	D₄	—	8	8	no	—
250	D₄	—	8	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	268	4	no	—
253	C₂×C₄	—	268	2	no	—
254	C₂×C₄	—	268	2	no	—
255	C₂×C₄	—	536	2	no	—
256	C₂×C₄	#14523987	1	4	yes	complemented
257	C₂×C₄	#14524014	1	2	yes	—
258	C₂×C₄	C₂×#11761613	1	2	yes	—
259	C₂×C₄	C₂×#11761613	1	2	yes	—
260	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
261	C₂×C₄	#14524069	1	2	yes	—
262	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	#14524014	1	2	yes	—
265	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
266	C₈	C₂×#11761627	1	1	yes	characteristic
267	C₂²	—	8	4	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	16	8	no	—
271	C₂²	#16863588	1	2	yes	—
272	C₂²	C₂×#14523987	1	2	yes	—
273	C₂²	#16863832	1	2	yes	—
274	C₂²	C₂²×#11761613	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	268	4	no	—
277	C₄	—	536	2	no	—
278	C₄	#16863588	1	2	yes	—
279	C₄	C₂×#14523987	1	4	yes	complemented
280	C₄	C₂×#14524014	1	1	yes	characteristic
281	C₄	C₂×#14524069	1	1	yes	characteristic
282	C₂	—	16	4	no	—
283	C₂	#22562246	1	4	yes	direct product factor
284	C₂	C₂×#16863588	1	1	yes	characteristic
285	C₂	C₂²×#14523987	1	1	yes	characteristic
286	C₂	C₂×#16863832	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×#2207
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. C₂²×#14523987	C₂	semidirect
2. C₂×C₄×Dic₂₆₈	C₂	semidirect
3. #22562246	C₂	direct (N × H)
4. C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
5. C₂×#16863197	C₂	semidirect
10. C₂×#14523987	C₂²	semidirect
13. C₄×Dic₂₆₈	C₂²	semidirect
16. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
17. #16863197	C₂²	semidirect
27. C₂×Dic₂₆₈	D₄	semidirect
64. Dic₂₆₈	C₂×D₄	semidirect
105. C₂×C₄×C₆₇	SD₃₂	semidirect
133. C₄×C₆₇	C₂×SD₃₂	semidirect
140. C₂×C₆₇	#2207	semidirect
150. C₆₇	C₂×#2207	semidirect
256. C₂×C₄	#14523987	semidirect
279. C₄	C₂×#14523987	semidirect
283. C₂	#22562246	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×#14523987	2
C₂×C₄×Dic₂₆₈	2
#22562246	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×#16863197	2
C₂×#493×C₆₇	2
C₂×#2207	67

Subgroups — C2×#22562246 #31010072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562246 #31010072