Subgroups — C₂×#22562380 #31010561

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863794	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562380	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863220	C₂	1	2	yes	maximal, complemented
4	C₂×#16863217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863206	C₂	1	2	yes	maximal
6	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524043	—	2	1	no	—
8	C₂×#14523983	C₂²	1	2	yes	—
9	#16863794	—	2	2	no	—
10	#16863794	C₂²	1	4	yes	complemented
11	C₂×#14523975	—	2	1	no	—
12	C₂×#14523975	C₂²	1	2	yes	—
13	#16863220	C₂²	1	8	yes	complemented
14	#16863217	C₂²	1	4	yes	complemented
15	#16863206	C₂²	1	8	yes	—
16	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#131×C₆₇	—	2	1	no	—
18	#443×C₆₇	—	2	2	no	—
19	#443×C₆₇	C₂²	1	4	yes	—
20	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#104×C₆₇	—	2	1	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₃₄	—	4	1	no	—
24	#14524043	—	2	4	no	—
25	#14524043	—	4	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	4	1	no	—
28	#14523983	—	2	2	no	—
29	#14523983	C₂³	1	4	yes	—
30	C₂×C₆₇⋊C₁₆	—	2	2	no	—
31	#14523975	—	2	2	no	—
32	#14523975	—	2	4	no	—
33	#14523975	C₂³	1	4	yes	—
34	C₂×#41×C₆₇	—	2	1	no	—
35	#135×C₆₇	—	2	1	no	—
36	#135×C₆₇	—	2	1	no	—
37	#135×C₆₇	Q₈	1	2	yes	complemented
38	#135×C₆₇	C₂³	1	2	yes	—
39	C₂×Q₁₆×C₆₇	Q₈	1	1	yes	characteristic, complemented
40	C₂×SD₁₆×C₆₇	—	2	1	no	—
41	C₂×D₈×C₆₇	Q₈	1	1	yes	characteristic, complemented
42	#131×C₆₇	—	2	4	no	—
43	#131×C₆₇	—	4	2	no	—
44	C₂×M₁₆×C₆₇	—	2	1	no	—
45	C₂×M₁₆×C₆₇	—	4	1	no	—
46	C₂²×C₈×C₆₇	—	4	1	no	—
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	#108×C₆₇	—	2	1	no	—
51	#108×C₆₇	C₂³	1	2	yes	—
52	#104×C₆₇	—	2	4	no	—
53	C₄×C₈×C₆₇	D₄	1	2	yes	—
54	C₄×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂×Dic₁₃₄	—	4	4	no	—
56	C₂×Dic₁₃₄	—	8	4	no	—
57	C₂²×Dic₆₇	—	4	2	no	—
58	C₄⋊Dic₆₇	—	2	4	no	—
59	C₄⋊Dic₆₇	—	4	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₆₇⋊C₁₆	—	2	4	no	—
62	#41×C₆₇	—	2	2	no	—
63	#41×C₆₇	—	2	2	no	—
64	#41×C₆₇	—	4	1	no	—
65	#41×C₆₇	—	4	1	no	—
66	C₂×Q₈×C₆₇	—	2	1	no	—
67	C₂×D₄×C₆₇	—	2	1	no	—
68	C₂×D₄×C₆₇	—	4	1	no	—
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	4	1	no	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₆₇	—	2	1	no	—
73	Q₁₆×C₆₇	C₂×Q₈	1	2	yes	complemented
74	SD₁₆×C₆₇	—	2	2	no	—
75	SD₁₆×C₆₇	—	2	2	no	—
76	D₈×C₆₇	—	2	1	no	—
77	D₈×C₆₇	C₂×Q₈	1	2	yes	complemented
78	M₁₆×C₆₇	—	2	2	no	—
79	M₁₆×C₆₇	—	4	1	no	—
80	M₁₆×C₆₇	—	4	4	no	—
81	C₂×C₈×C₆₇	—	4	1	no	—
82	C₂×C₈×C₆₇	—	4	4	no	—
83	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
84	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
85	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
86	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
91	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	C₂×D₄	1	2	yes	—
95	C₂³×C₆₇	—	4	1	no	—
96	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
97	Q₈×C₆₇	—	2	2	no	—
98	Q₈×C₆₇	—	4	1	no	—
99	D₄×C₆₇	—	2	2	no	—
100	D₄×C₆₇	—	4	2	no	—
101	D₄×C₆₇	—	4	2	no	—
102	D₄×C₆₇	—	4	1	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	4	1	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₆₇	—	4	4	no	—
115	C₂×Dic₆₇	—	8	4	no	—
116	Dic₁₃₄	—	8	8	no	—
117	C₈×C₆₇	—	4	2	no	—
118	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
121	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
122	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	—	4	2	no	—
126	C₂²×C₆₇	—	8	2	no	—
127	C₂²×C₆₇	#476	1	1	yes	characteristic
128	C₂²×C₆₇	#476	1	1	yes	characteristic
129	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₄×C₆₇	—	2	4	no	—
131	C₄×C₆₇	—	4	2	no	—
132	C₄×C₆₇	#476	1	1	yes	characteristic
133	C₄×C₆₇	#476	1	1	yes	characteristic
134	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
135	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
136	Dic₆₇	—	8	4	no	—
137	C₂×#2230	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	—	2	1	no	—
139	C₂×C₆₇	—	2	1	no	—
140	C₂×C₆₇	—	8	2	no	—
141	C₂×C₆₇	#2230	1	2	yes	complemented
142	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
143	C₂×#498	—	67	1	no	—
144	C₂×#443	D₆₇	1	1	yes	characteristic
145	#2230	—	67	8	no	—
146	C₂×#365	—	67	2	no	—
147	C₂×#364	—	67	1	no	—
148	C₂×#360	—	67	2	no	—
149	C₆₇	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
150	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
151	C₂×#131	—	2	1	no	—
152	C₂×C₄⋊Q₈	—	134	1	no	—
153	C₂×M₃₂	—	67	2	no	—
154	#498	—	67	4	no	—
155	#498	—	134	2	no	—
156	#443	—	2	2	no	—
157	#443	C₂×D₆₇	1	4	yes	—
158	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
159	C₂×#106	—	67	2	no	—
160	C₂×#106	—	134	1	no	—
161	C₂×#104	—	2	1	no	—
162	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
163	#365	—	67	8	no	—
164	#364	—	67	4	no	—
165	#360	—	67	8	no	—
166	C₂×#41	—	2	1	no	—
167	C₂²×Q₈	—	268	1	no	—
168	#135	—	2	1	no	—
169	#135	—	2	1	no	—
170	#135	Dic₁₃₄	1	2	yes	complemented
171	#135	C₂²×D₆₇	1	2	yes	—
172	C₂×Q₁₆	Dic₁₃₄	1	1	yes	characteristic, complemented
173	C₂×SD₁₆	—	2	1	no	—
174	C₂×D₈	Dic₁₃₄	1	1	yes	characteristic, complemented
175	#131	—	2	4	no	—
176	#131	—	4	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	4	1	no	—
179	C₂²×C₈	—	4	1	no	—
180	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
182	C₄⋊Q₈	—	268	2	no	—
183	C₄⋊Q₈	—	134	4	no	—
184	C₂×C₄⋊C₄	—	268	1	no	—
185	C₂×C₄⋊C₄	—	134	2	no	—
186	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
187	M₃₂	—	67	4	no	—
188	M₃₂	—	134	2	no	—
189	C₂×C₁₆	—	134	2	no	—
190	#108	—	2	1	no	—
191	#108	C₂²×D₆₇	1	2	yes	—
192	#106	—	67	4	no	—
193	#106	—	134	2	no	—
194	#106	—	134	4	no	—
195	#104	—	2	4	no	—
196	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
197	C₄×C₈	C₂²×D₆₇	1	2	yes	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	—	4	1	no	—
201	#41	—	4	1	no	—
202	C₂×Q₈	—	2	1	no	—
203	C₂×Q₈	—	268	4	no	—
204	C₂×Q₈	—	536	4	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	268	2	no	—
210	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
211	Q₁₆	—	2	1	no	—
212	Q₁₆	C₂×Dic₁₃₄	1	2	yes	complemented
213	SD₁₆	—	2	2	no	—
214	SD₁₆	—	2	2	no	—
215	D₈	—	2	1	no	—
216	D₈	C₂×Dic₁₃₄	1	2	yes	complemented
217	M₁₆	—	2	2	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	4	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	#11761616	1	1	yes	characteristic
223	C₂×C₈	#11761616	1	1	yes	characteristic
224	C₂×C₈	#11761616	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
227	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
228	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
229	C₂×C₈	#11761628	1	1	yes	characteristic
230	C₂×C₈	#11761628	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
232	C₄⋊C₄	—	268	4	no	—
233	C₄⋊C₄	—	268	2	no	—
234	C₄⋊C₄	—	134	4	no	—
235	C₄²	—	2	1	no	—
236	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
237	C₁₆	—	134	4	no	—
238	C₂³	—	4	1	no	—
239	C₂³	#14524068	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	4	1	no	—
242	Q₈	—	536	8	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	4	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	268	4	no	—
253	C₂×C₄	—	536	4	no	—
254	C₂×C₄	#14524022	1	1	yes	characteristic
255	C₂×C₄	#14524022	1	1	yes	characteristic
256	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
257	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
258	C₂×C₄	#14524068	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
260	C₈	—	4	2	no	—
261	C₈	#14524022	1	1	yes	characteristic
262	C₈	#14524022	1	1	yes	characteristic
263	C₈	C₂×#11761616	1	2	yes	—
264	C₈	C₂×#11761616	1	2	yes	—
265	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
266	C₈	C₂×#11761628	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	#16863686	1	1	yes	characteristic
271	C₂²	#16863686	1	1	yes	characteristic
272	C₂²	C₂×#14524068	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	536	4	no	—
276	C₄	#16863686	1	1	yes	characteristic
277	C₄	#16863686	1	1	yes	characteristic
278	C₄	C₂×#14524022	1	1	yes	characteristic
279	C₄	C₂²×#11761616	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	8	2	no	—
283	C₂	#22562380	1	2	yes	direct product factor
284	C₂	C₂×#16863686	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863686
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#443×C₆₇	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂×#135×C₆₇	C₂²
hypercenter	C₂×#443	D₆₇
hypocenter	C₆₇	C₂×#2230
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#16863794	C₂	semidirect
2. #22562380	C₂	direct (N × H)
3. C₂×#16863220	C₂	semidirect
4. C₂×#16863217	C₂	semidirect
10. #16863794	C₂²	semidirect
13. #16863220	C₂²	semidirect
14. #16863217	C₂²	semidirect
37. #135×C₆₇	Q₈	semidirect
39. C₂×Q₁₆×C₆₇	Q₈	semidirect
41. C₂×D₈×C₆₇	Q₈	semidirect
73. Q₁₆×C₆₇	C₂×Q₈	semidirect
77. D₈×C₆₇	C₂×Q₈	semidirect
141. C₂×C₆₇	#2230	semidirect
149. C₆₇	C₂×#2230	semidirect
170. #135	Dic₁₃₄	semidirect
172. C₂×Q₁₆	Dic₁₃₄	semidirect
174. C₂×D₈	Dic₁₃₄	semidirect
212. Q₁₆	C₂×Dic₁₃₄	semidirect
216. D₈	C₂×Dic₁₃₄	semidirect
283. C₂	#22562380	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×#16863794	2
#22562380	2
C₂×#16863220	2
C₂×#16863217	2
C₂×#16863206	2
C₂×#443×C₆₇	2
C₂×#2230	67

Subgroups — C2×#22562380 #31010561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562380 #31010561