Subgroups — C₂×#22561812 #31008487

Order 17152.

738 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₆₇	C₂	1	2	yes	maximal
2	C₂×#16863353	C₂	1	1	yes	maximal, characteristic
3	#22561812	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863174	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863173	C₂	1	2	yes	maximal, complemented
6	C₂×#347×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11761611	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	C₂×#14524000	C₄	1	1	yes	characteristic
10	C₂×#14523983	C₂²	1	2	yes	—
11	C₂×#14523982	C₄	1	4	yes	complemented
12	M₃₂×D₆₇	C₄	1	8	yes	complemented
13	M₃₂×D₆₇	C₂²	1	8	yes	—
14	C₂×C₁₆×D₆₇	C₄	1	4	yes	complemented
15	#16863353	C₄	1	4	yes	—
16	#16863353	C₂²	1	4	yes	—
17	C₂×#14523953	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
20	#16863174	C₂²	1	4	yes	complemented
21	#16863173	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
23	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
24	#347×C₆₇	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×#11761611	—	2	2	no	—
27	C₂×#11761611	—	2	2	no	—
28	C₂×#11761611	C₂×C₄	1	4	yes	—
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₆₇	C₂³	1	2	yes	—
36	#14524000	—	2	2	no	—
37	#14524000	C₂×C₄	1	4	yes	—
38	C₂×Dic₆₇⋊C₄	—	2	1	no	—
39	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
40	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
41	#14523983	C₂×C₄	1	4	yes	—
42	#14523983	C₂³	1	4	yes	—
43	#14523982	C₂×C₄	1	16	yes	complemented
44	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
45	C₁₆×D₆₇	C₂×C₄	1	16	yes	complemented
46	#14523953	C₂×C₄	1	2	yes	—
47	#14523953	C₂³	1	2	yes	—
48	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
49	C₈×Dic₆₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₆₇	C₂×C₄	1	4	yes	—
54	M₃₂×C₆₇	C₂³	1	4	yes	—
55	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
56	#97×C₆₇	C₂×C₄	1	2	yes	—
57	#97×C₆₇	C₂³	1	2	yes	—
58	C₂³×D₆₇	M₁₆	1	1	yes	characteristic
59	C₂²×Dic₆₇	—	2	1	no	—
60	C₂²×Dic₆₇	M₁₆	1	1	yes	characteristic
61	C₂×C₄×D₆₇	C₄²	1	2	yes	—
62	C₂×C₄×D₆₇	C₄²	1	2	yes	—
63	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
64	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
65	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
66	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
67	#11761611	—	2	8	no	—
68	C₂×C₆₇⋊C₈	—	2	1	no	—
69	C₂×C₆₇⋊C₈	—	2	1	no	—
70	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
71	C₂×C₆₇⋊C₈	C₄²	1	1	yes	characteristic
72	C₂×C₆₇⋊C₈	C₄²	1	1	yes	characteristic
73	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
75	C₈×D₆₇	C₄²	1	4	yes	—
76	C₈×D₆₇	C₄²	1	4	yes	—
77	C₈×D₆₇	C₂²×C₄	1	4	yes	—
78	C₈×D₆₇	C₂²×C₄	1	4	yes	—
79	Dic₆₇⋊C₄	—	2	4	no	—
80	(C₂×D₆₇)⋊C₄	—	2	4	no	—
81	C₄×Dic₆₇	—	2	1	no	—
82	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
83	C₆₇⋊C₁₆	C₂²×C₄	1	8	yes	—
84	C₂²×C₄×C₆₇	—	2	1	no	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
95	C₁₆×C₆₇	C₂²×C₄	1	8	yes	—
96	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
97	C₂²×D₆₇	—	4	2	no	—
98	C₂²×D₆₇	#97	1	2	yes	—
99	C₂²×D₆₇	#97	1	2	yes	—
100	C₂²×D₆₇	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×Dic₆₇	—	2	2	no	—
110	C₂×Dic₆₇	—	4	2	no	—
111	C₂×Dic₆₇	#97	1	2	yes	—
112	C₂×Dic₆₇	#97	1	2	yes	—
113	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
114	C₄×D₆₇	#97	1	4	yes	—
115	C₄×D₆₇	#97	1	4	yes	—
116	C₄×D₆₇	C₂×C₄²	1	4	yes	—
117	C₄×D₆₇	C₂×M₁₆	1	4	yes	—
118	C₈×C₆₇	—	2	2	no	—
119	C₈×C₆₇	C₂×C₄²	1	2	yes	—
120	C₈×C₆₇	C₂³×C₄	1	2	yes	—
121	C₆₇⋊C₈	—	2	2	no	—
122	C₆₇⋊C₈	C₂×C₄²	1	2	yes	—
123	C₆₇⋊C₈	C₂×C₄²	1	2	yes	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
126	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
127	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
128	C₂×D₆₇	—	2	4	no	—
129	C₂×D₆₇	—	4	2	no	—
130	C₂×D₆₇	—	4	2	no	—
131	C₂×D₆₇	C₂×#97	1	4	yes	—
132	C₄×C₆₇	—	4	2	no	—
133	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
134	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
135	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
136	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
137	Dic₆₇	—	4	2	no	—
138	Dic₆₇	C₂×#97	1	4	yes	—
139	C₂×#2110	—	67	1	no	maximal, Sylow
140	C₂×C₆₇	—	2	1	no	—
141	C₂×C₆₇	—	2	1	no	—
142	C₂×C₆₇	#2110	1	2	yes	complemented
143	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
144	D₆₇	—	2	4	no	—
145	C₂²×M₃₂	—	67	2	no	—
146	C₂×#405	—	67	1	no	—
147	#2110	—	67	8	no	—
148	C₂×#347	—	67	2	no	—
149	C₂×#347	—	67	1	no	—
150	C₂×#347	D₆₇	1	1	yes	characteristic, complemented
151	C₆₇	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
152	C₂²×M₁₆	—	67	1	no	—
153	C₂³×C₈	—	67	1	no	—
154	C₂×#117	—	67	1	no	—
155	C₂×M₃₂	—	67	2	no	—
156	C₂×M₃₂	—	67	8	no	—
157	C₂×M₃₂	—	67	8	no	—
158	C₂×M₃₂	—	67	4	no	—
159	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
160	C₂²×C₁₆	—	67	4	no	—
161	#405	—	67	4	no	—
162	#405	—	67	4	no	—
163	C₂×#97	—	67	1	no	—
164	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
165	C₂×C₄×C₈	—	67	1	no	—
166	C₂×C₄×C₈	—	67	1	no	—
167	#347	—	67	8	no	—
168	#347	—	67	4	no	—
169	#347	C₂×D₆₇	1	4	yes	complemented
170	C₂³×C₄	—	67	1	no	—
171	C₂×M₁₆	—	67	4	no	—
172	C₂×M₁₆	—	134	2	no	—
173	C₂×M₁₆	—	134	2	no	—
174	C₂²×C₈	—	67	1	no	—
175	C₂²×C₈	—	67	2	no	—
176	C₂²×C₈	—	67	2	no	—
177	C₂²×C₈	—	67	4	no	—
178	C₂²×C₈	—	67	2	no	—
179	C₂²×C₈	—	67	2	no	—
180	C₂²×C₈	—	67	1	no	—
181	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
183	#117	—	67	4	no	—
184	#117	—	134	2	no	—
185	C₂×C₄⋊C₄	—	134	1	no	—
186	C₂×C₂²⋊C₄	—	134	1	no	—
187	C₂×C₄²	—	67	1	no	—
188	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
189	M₃₂	—	67	4	no	—
190	M₃₂	—	67	4	no	—
191	M₃₂	—	67	16	no	—
192	M₃₂	C₄×D₆₇	1	4	yes	—
193	M₃₂	C₂²×D₆₇	1	4	yes	—
194	C₂×C₁₆	—	67	4	no	—
195	C₂×C₁₆	—	67	16	no	—
196	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
197	#97	—	67	2	no	—
198	#97	—	67	2	no	—
199	#97	C₄×D₆₇	1	2	yes	—
200	#97	C₂²×D₆₇	1	2	yes	—
201	C₄×C₈	—	67	4	no	—
202	C₄×C₈	—	67	4	no	—
203	C₂⁴	—	67	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	67	1	no	—
206	C₂²×C₄	—	67	2	no	—
207	C₂²×C₄	—	67	2	no	—
208	C₂²×C₄	—	67	2	no	—
209	C₂²×C₄	—	67	2	no	—
210	C₂²×C₄	—	67	2	no	—
211	C₂²×C₄	—	67	2	no	—
212	C₂²×C₄	—	134	1	no	—
213	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
214	M₁₆	—	134	8	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	67	2	no	—
218	C₂×C₈	—	67	1	no	—
219	C₂×C₈	—	67	1	no	—
220	C₂×C₈	—	67	2	no	—
221	C₂×C₈	—	67	4	no	—
222	C₂×C₈	—	67	4	no	—
223	C₂×C₈	—	67	4	no	—
224	C₂×C₈	—	67	4	no	—
225	C₂×C₈	—	67	2	no	—
226	C₂×C₈	—	134	1	no	—
227	C₂×C₈	—	134	1	no	—
228	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
231	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
232	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
233	C₄⋊C₄	—	134	4	no	—
234	C₂²⋊C₄	—	134	4	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	67	2	no	—
237	C₄²	—	134	1	no	—
238	C₄²	C₂×C₄×D₆₇	1	2	yes	—
239	C₁₆	—	67	8	no	—
240	C₁₆	C₂×C₄×D₆₇	1	8	yes	—
241	C₂³	—	268	2	no	—
242	C₂³	—	67	2	no	—
243	C₂³	—	67	2	no	—
244	C₂³	—	67	2	no	—
245	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	268	2	no	—
249	C₂×C₄	—	67	2	no	—
250	C₂×C₄	—	67	2	no	—
251	C₂×C₄	—	67	2	no	—
252	C₂×C₄	—	67	4	no	—
253	C₂×C₄	—	67	4	no	—
254	C₂×C₄	—	67	4	no	—
255	C₂×C₄	—	67	4	no	—
256	C₂×C₄	—	134	2	no	—
257	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
258	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
259	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
260	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
261	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	67	2	no	—
265	C₈	—	67	2	no	—
266	C₈	—	134	2	no	—
267	C₈	C₄²×D₆₇	1	2	yes	—
268	C₈	C₂²×C₄×D₆₇	1	2	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	268	2	no	—
271	C₂²	—	268	2	no	—
272	C₂²	—	67	4	no	—
273	C₂²	—	134	4	no	—
274	C₂²	#97×D₆₇	1	1	yes	characteristic
275	C₂²	#97×D₆₇	1	1	yes	characteristic
276	C₂²	C₂×M₁₆×D₆₇	1	1	yes	characteristic
277	C₄	—	4	2	no	—
278	C₄	—	268	2	no	—
279	C₄	—	67	4	no	—
280	C₄	#97×D₆₇	1	1	yes	characteristic
281	C₄	#97×D₆₇	1	1	yes	characteristic
282	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
283	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	134	4	no	—
287	C₂	#22561812	1	2	yes	direct product factor
288	C₂	C₂×#97×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#347×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂³×D₆₇	M₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#347	D₆₇
hypocenter	C₆₇	C₂×#2110
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. #22561812	C₂	direct (N × H)
4. C₂×#16863174	C₂	semidirect
5. C₂×#16863173	C₂	semidirect
6. C₂×#347×C₆₇	C₂	semidirect
11. C₂×#14523982	C₄	semidirect
12. M₃₂×D₆₇	C₄	semidirect
14. C₂×C₁₆×D₆₇	C₄	semidirect
20. #16863174	C₂²	semidirect
21. #16863173	C₂²	semidirect
24. #347×C₆₇	C₂²	semidirect
43. #14523982	C₂×C₄	semidirect
45. C₁₆×D₆₇	C₂×C₄	semidirect
142. C₂×C₆₇	#2110	semidirect
150. C₂×#347	D₆₇	semidirect
151. C₆₇	C₂×#2110	semidirect
169. #347	C₂×D₆₇	semidirect
287. C₂	#22561812	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×M₃₂×D₆₇	2
C₂×#16863353	2
#22561812	2
C₂×#16863174	2
C₂×#16863173	2
C₂×#347×C₆₇	2
C₂×#2110	67

Subgroups — C2×#22561812 #31008487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22561812 #31008487