Subgroups — C₂×#22562383 #31010565

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863796	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863513	C₂	1	1	yes	maximal, characteristic
3	#22562383	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863219	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863218	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16863206	C₂	1	1	yes	maximal, characteristic
8	C₂×#360×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14524050	—	2	1	no	—
10	C₂×#14524034	—	2	1	no	—
11	C₂×#14523983	C₂²	1	1	yes	characteristic
12	#16863796	—	2	2	no	—
13	#16863796	C₂²	1	4	yes	complemented
14	#16863513	—	2	2	no	—
15	#16863513	C₂²	1	4	yes	—
16	C₂×#14523979	C₂²	1	1	yes	characteristic
17	C₂×#14523976	—	2	1	no	—
18	C₂×#14523975	C₂²	1	1	yes	characteristic
19	C₂×#14523970	—	2	1	no	—
20	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
21	#16863219	C₂²	1	4	yes	complemented
22	#16863218	C₂²	1	4	yes	complemented
23	#16863216	C₂²	1	4	yes	complemented
24	#16863206	C₂²	1	4	yes	—
25	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
28	#360×C₆₇	C₂²	1	4	yes	—
29	#14524050	—	2	4	no	—
30	#14524050	—	4	2	no	—
31	C₂×#11761612	—	2	1	no	—
32	C₂×#11761612	—	4	1	no	—
33	C₂²×C₆₇⋊C₈	—	4	1	no	—
34	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
35	#14524034	—	2	4	no	—
36	#14524034	—	4	2	no	—
37	C₂×C₄⋊Dic₆₇	—	2	1	no	—
38	C₂×C₄⋊Dic₆₇	—	4	1	no	—
39	C₂×Dic₆₇⋊C₄	—	4	1	no	—
40	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
41	#14523983	—	2	1	no	—
42	#14523983	C₂³	1	2	yes	—
43	C₂×C₆₇⋊C₁₆	—	2	1	no	—
44	#14523979	—	2	1	no	—
45	#14523979	C₂³	1	2	yes	—
46	#14523976	—	2	4	no	—
47	#14523975	—	2	1	no	—
48	#14523975	C₂³	1	2	yes	—
49	#14523970	—	2	4	no	—
50	C₈×Dic₆₇	D₄	1	2	yes	—
51	C₈×Dic₆₇	C₂³	1	2	yes	—
52	C₂×#41×C₆₇	—	2	1	no	—
53	#135×C₆₇	—	2	1	no	—
54	#135×C₆₇	—	2	1	no	—
55	#135×C₆₇	Q₈	1	2	yes	—
56	#135×C₆₇	C₂³	1	2	yes	—
57	C₂×Q₁₆×C₆₇	Q₈	1	1	yes	characteristic
58	C₂×SD₁₆×C₆₇	—	2	1	no	—
59	C₂×D₈×C₆₇	Q₈	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	M₃₂×C₆₇	—	2	1	no	—
63	M₃₂×C₆₇	C₂³	1	2	yes	—
64	C₂×C₁₆×C₆₇	—	2	1	no	—
65	#106×C₆₇	—	2	1	no	—
66	#106×C₆₇	C₂³	1	2	yes	—
67	C₂²×Dic₆₇	—	2	1	no	—
68	C₂²×Dic₆₇	—	4	1	no	—
69	#11761612	—	2	2	no	—
70	#11761612	—	4	1	no	—
71	#11761612	—	4	4	no	—
72	C₂×C₆₇⋊C₈	—	4	1	no	—
73	C₂×C₆₇⋊C₈	—	4	4	no	—
74	C₂×C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
75	C₂×C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
76	C₂×C₆₇⋊C₈	SD₁₆	1	2	yes	—
77	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
78	C₄⋊Dic₆₇	—	2	2	no	—
79	C₄⋊Dic₆₇	—	4	4	no	—
80	C₄⋊Dic₆₇	—	4	1	no	—
81	Dic₆₇⋊C₄	—	4	4	no	—
82	C₄×Dic₆₇	—	2	1	no	—
83	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
84	C₆₇⋊C₁₆	—	2	2	no	—
85	#41×C₆₇	—	2	2	no	—
86	#41×C₆₇	—	2	2	no	—
87	#41×C₆₇	—	4	1	no	—
88	#41×C₆₇	—	4	1	no	—
89	C₂×Q₈×C₆₇	—	2	1	no	—
90	C₂×D₄×C₆₇	—	2	1	no	—
91	C₂×D₄×C₆₇	—	4	1	no	—
92	C₂²×C₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	—	4	1	no	—
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	Q₁₆×C₆₇	—	2	1	no	—
96	Q₁₆×C₆₇	C₂×Q₈	1	2	yes	—
97	SD₁₆×C₆₇	—	2	2	no	—
98	SD₁₆×C₆₇	—	2	2	no	—
99	D₈×C₆₇	—	2	1	no	—
100	D₈×C₆₇	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
105	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	4	1	no	—
109	C₁₆×C₆₇	—	2	2	no	—
110	C₂³×C₆₇	—	4	1	no	—
111	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
112	Q₈×C₆₇	—	2	2	no	—
113	Q₈×C₆₇	—	4	1	no	—
114	D₄×C₆₇	—	2	2	no	—
115	D₄×C₆₇	—	4	2	no	—
116	D₄×C₆₇	—	4	2	no	—
117	D₄×C₆₇	—	4	1	no	—
118	C₂×C₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	8	2	no	—
123	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₆₇	—	2	2	no	—
130	C₂×Dic₆₇	—	4	2	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	C₂×Dic₆₇	—	8	2	no	—
133	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
134	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
135	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
136	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₆₇⋊C₈	—	4	2	no	—
138	C₆₇⋊C₈	C₂×SD₁₆	1	2	yes	—
139	C₆₇⋊C₈	C₂×SD₁₆	1	2	yes	—
140	C₂²×C₆₇	—	2	2	no	—
141	C₂²×C₆₇	—	4	2	no	—
142	C₂²×C₆₇	—	8	2	no	—
143	C₂²×C₆₇	#476	1	1	yes	characteristic
144	C₂²×C₆₇	#476	1	1	yes	characteristic
145	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₄×C₆₇	—	4	2	no	—
147	C₄×C₆₇	—	8	2	no	—
148	C₄×C₆₇	#476	1	1	yes	characteristic
149	C₄×C₆₇	#476	1	1	yes	characteristic
150	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
152	Dic₆₇	—	4	2	no	—
153	Dic₆₇	—	8	2	no	—
154	C₂×#2231	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	—	8	2	no	—
158	C₂×C₆₇	#2231	1	2	yes	complemented
159	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
160	C₂×#499	—	67	1	no	—
161	C₂×#443	—	67	1	no	—
162	#2231	—	67	8	no	—
163	C₂×#365	—	67	1	no	—
164	C₂×#365	—	67	1	no	—
165	C₂×#364	—	67	1	no	—
166	C₂×#360	—	67	1	no	—
167	C₂×#360	D₆₇	1	1	yes	characteristic
168	C₆₇	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
169	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
170	C₂×#131	—	134	1	no	—
171	C₂×#125	—	134	1	no	—
172	C₂×M₃₂	—	67	1	no	—
173	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
174	#499	—	67	4	no	—
175	#499	—	134	2	no	—
176	#443	—	67	4	no	—
177	#443	—	134	2	no	—
178	C₂×#108	—	67	1	no	—
179	C₂×#107	—	134	1	no	—
180	C₂×#106	—	67	1	no	—
181	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
182	C₂×#104	—	134	1	no	—
183	C₂×C₄×C₈	—	67	1	no	—
184	#365	—	67	4	no	—
185	#365	—	67	4	no	—
186	#364	—	67	4	no	—
187	#360	—	67	4	no	—
188	#360	C₂×D₆₇	1	4	yes	—
189	C₂×#41	—	2	1	no	—
190	#135	—	2	1	no	—
191	#135	—	2	1	no	—
192	#135	Dic₁₃₄	1	2	yes	—
193	#135	C₂²×D₆₇	1	2	yes	—
194	C₂×Q₁₆	Dic₁₃₄	1	1	yes	characteristic
195	C₂×SD₁₆	—	2	1	no	—
196	C₂×D₈	Dic₁₃₄	1	1	yes	characteristic
197	#131	—	268	2	no	—
198	#131	—	134	4	no	—
199	C₂×M₁₆	—	268	1	no	—
200	C₂×M₁₆	—	134	1	no	—
201	C₂²×C₈	—	268	1	no	—
202	C₂²×C₈	—	67	1	no	—
203	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
204	#125	—	268	2	no	—
205	#125	—	134	4	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	268	1	no	—
208	C₂×C₄⋊C₄	—	268	1	no	—
209	C₂×C₄⋊C₄	—	134	1	no	—
210	C₂×C₄²	—	67	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	67	2	no	—
213	M₃₂	—	134	1	no	—
214	M₃₂	C₂²×D₆₇	1	2	yes	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	134	1	no	—
217	#108	—	67	2	no	—
218	#108	—	134	1	no	—
219	#107	—	134	4	no	—
220	#106	—	2	1	no	—
221	#106	—	67	2	no	—
222	#106	—	134	1	no	—
223	#106	C₂²×D₆₇	1	2	yes	—
224	#104	—	134	4	no	—
225	C₄×C₈	—	67	2	no	—
226	C₄×C₈	—	67	2	no	—
227	#41	—	2	2	no	—
228	#41	—	2	2	no	—
229	#41	—	4	1	no	—
230	#41	—	4	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	268	1	no	—
237	C₂²×C₄	—	134	1	no	—
238	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
239	Q₁₆	—	2	1	no	—
240	Q₁₆	C₂×Dic₁₃₄	1	2	yes	—
241	SD₁₆	—	2	2	no	—
242	SD₁₆	—	2	2	no	—
243	D₈	—	2	1	no	—
244	D₈	C₂×Dic₁₃₄	1	2	yes	—
245	M₁₆	—	268	1	no	—
246	M₁₆	—	268	4	no	—
247	M₁₆	—	134	2	no	—
248	C₂×C₈	—	268	1	no	—
249	C₂×C₈	—	268	4	no	—
250	C₂×C₈	—	67	1	no	—
251	C₂×C₈	—	67	1	no	—
252	C₂×C₈	—	67	2	no	—
253	C₂×C₈	—	67	2	no	—
254	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
255	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
256	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
257	C₂×C₈	#11761627	1	1	yes	characteristic
258	C₂×C₈	#11761627	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	268	4	no	—
263	C₄⋊C₄	—	268	4	no	—
264	C₄⋊C₄	—	268	1	no	—
265	C₄⋊C₄	—	134	2	no	—
266	C₄²	—	67	2	no	—
267	C₄²	—	134	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	134	2	no	—
270	C₂³	—	4	1	no	—
271	C₂³	#14524067	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	4	1	no	—
274	D₄	—	2	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	268	2	no	—
284	C₂×C₄	—	268	2	no	—
285	C₂×C₄	—	536	2	no	—
286	C₂×C₄	—	134	2	no	—
287	C₂×C₄	#14524020	1	1	yes	characteristic
288	C₂×C₄	#14524020	1	1	yes	characteristic
289	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
290	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
291	C₂×C₄	#14524067	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
293	C₈	—	268	2	no	—
294	C₈	—	67	2	no	—
295	C₈	—	67	2	no	—
296	C₈	#14524020	1	1	yes	characteristic
297	C₈	#14524020	1	1	yes	characteristic
298	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
299	C₈	C₂×#11761627	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	#16863684	1	1	yes	characteristic
304	C₂²	#16863684	1	1	yes	characteristic
305	C₂²	C₂×#14524067	1	1	yes	characteristic
306	C₄	—	4	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	268	2	no	—
309	C₄	—	536	2	no	—
310	C₄	#16863684	1	1	yes	characteristic
311	C₄	#16863684	1	1	yes	characteristic
312	C₄	C₂×#14524020	1	1	yes	characteristic
313	C₄	C₂×SD₁₆×D₆₇	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	8	2	no	—
317	C₂	#22562383	1	2	yes	direct product factor
318	C₂	C₂×#16863684	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863684
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#360×C₆₇	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂×#135×C₆₇	C₂²
hypercenter	C₂×#360	D₆₇
hypocenter	C₆₇	C₂×#2231
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
1. C₂×#16863796	C₂	semidirect
3. #22562383	C₂	direct (N × H)
4. C₂×#16863219	C₂	semidirect
5. C₂×#16863218	C₂	semidirect
6. C₂×#16863216	C₂	semidirect
13. #16863796	C₂²	semidirect
21. #16863219	C₂²	semidirect
22. #16863218	C₂²	semidirect
23. #16863216	C₂²	semidirect
158. C₂×C₆₇	#2231	semidirect
168. C₆₇	C₂×#2231	semidirect
317. C₂	#22562383	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×#16863796	2
C₂×#16863513	2
#22562383	2
C₂×#16863219	2
C₂×#16863218	2
C₂×#16863216	2
C₂×#16863206	2
C₂×#360×C₆₇	2
C₂×#2231	67

Subgroups — C2×#22562383 #31010565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562383 #31010565