Subgroups — C₂×#22561997 #31008865

Order 17152.

718 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523977	C₂	1	1	yes	maximal, characteristic
3	#22561997	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863224	C₂	1	2	yes	maximal, complemented
5	C₂×#16863222	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524002	—	2	1	no	—
8	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
9	#16863434	—	2	2	no	—
10	#16863434	C₂²	1	4	yes	—
11	C₂×#14523977	—	2	1	no	—
12	C₂×#14523977	—	2	2	no	—
13	C₂×#14523977	C₂²	1	2	yes	—
14	C₂×#14523977	C₂²	1	4	yes	—
15	C₂×#14523977	C₂²	1	1	yes	characteristic
16	C₂×#14523977	C₂²	1	1	yes	characteristic
17	C₂×#14523975	—	2	1	no	—
18	#16863224	C₂²	1	8	yes	complemented
19	#16863222	C₂²	1	8	yes	complemented
20	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₆₇	C₄	1	8	yes	complemented
22	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
23	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
24	C₂³×Dic₆₇	—	4	1	no	—
25	#14524002	—	2	4	no	—
26	#14524002	—	4	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	4	no	—
29	C₂×C₄⋊Dic₆₇	—	2	1	no	—
30	C₂×C₄⋊Dic₆₇	—	2	1	no	—
31	C₂×C₄⋊Dic₆₇	—	4	1	no	—
32	C₂×C₄⋊Dic₆₇	—	4	2	no	—
33	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
34	C₂×C₄×Dic₆₇	—	4	1	no	—
35	#14523977	—	2	4	no	—
36	#14523977	—	2	2	no	—
37	#14523977	—	2	1	no	—
38	#14523977	—	2	1	no	—
39	#14523977	C₂³	1	4	yes	—
40	#14523977	C₂³	1	2	yes	—
41	#14523977	C₂³	1	2	yes	—
42	#14523975	—	2	4	no	—
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	D₄	1	2	yes	—
47	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	M₃₂×C₆₇	C₂×C₄	1	16	yes	complemented
51	C₂×C₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
53	C₂²×Dic₆₇	—	4	1	no	—
54	C₂²×Dic₆₇	—	4	1	no	—
55	C₂²×Dic₆₇	—	4	4	no	—
56	C₂²×Dic₆₇	—	4	2	no	—
57	C₂²×Dic₆₇	—	8	4	no	—
58	C₄⋊Dic₆₇	—	2	2	no	—
59	C₄⋊Dic₆₇	—	2	2	no	—
60	C₄⋊Dic₆₇	—	2	4	no	—
61	C₄⋊Dic₆₇	—	4	4	no	—
62	C₄⋊Dic₆₇	—	4	2	no	—
63	C₄⋊Dic₆₇	—	4	1	no	—
64	C₄⋊Dic₆₇	—	4	1	no	—
65	C₂²⋊Dic₆₇	—	4	4	no	—
66	C₄×Dic₆₇	—	4	4	no	—
67	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	D₈	1	2	yes	—
71	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
75	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₁₆×C₆₇	C₂²×C₄	1	8	yes	complemented
84	C₂³×C₆₇	—	2	2	no	—
85	C₂³×C₆₇	—	2	2	no	—
86	C₂³×C₆₇	#107	1	4	yes	—
87	C₂³×C₆₇	C₂×D₈	1	2	yes	—
88	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#107	1	8	yes	—
90	C₂×C₄×C₆₇	#107	1	4	yes	—
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₆₇	—	4	2	no	—
99	C₂×Dic₆₇	—	4	2	no	—
100	C₂×Dic₆₇	—	4	4	no	—
101	C₂×Dic₆₇	—	8	8	no	—
102	C₂×Dic₆₇	—	8	2	no	—
103	C₂×Dic₆₇	—	8	4	no	—
104	C₂×Dic₆₇	—	8	2	no	—
105	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
107	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	2	8	no	—
112	C₂²×C₆₇	C₂×#107	1	4	yes	—
113	C₂²×C₆₇	C₂×#107	1	2	yes	—
114	C₂²×C₆₇	#509	1	2	yes	—
115	C₂²×C₆₇	#510	1	2	yes	—
116	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
117	C₄×C₆₇	C₂×#107	1	4	yes	—
118	C₄×C₆₇	C₂×#107	1	2	yes	—
119	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
121	Dic₆₇	—	8	2	no	—
122	Dic₆₇	—	8	4	no	—
123	Dic₆₇	—	8	2	no	—
124	C₂×#2159	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	—	2	4	no	—
126	C₂×C₆₇	#2159	1	4	yes	complemented
127	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
128	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
129	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
130	C₂²×M₃₂	D₆₇	1	1	yes	characteristic
131	C₂×#427	—	67	1	no	—
132	C₂²×#107	—	67	1	no	—
133	#2159	—	67	8	no	—
134	C₂×#367	—	67	2	no	—
135	C₂×#366	—	67	2	no	—
136	C₆₇	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
138	C₂×#117	—	134	1	no	—
139	C₂²×C₄⋊C₄	—	134	1	no	—
140	C₂×M₃₂	Dic₆₇	1	8	yes	complemented
141	C₂×M₃₂	C₂×D₆₇	1	4	yes	—
142	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
143	#427	—	67	4	no	—
144	#427	—	134	2	no	—
145	C₂×#107	—	67	2	no	—
146	C₂×#107	—	67	4	no	—
147	C₂×#107	—	67	1	no	—
148	C₂×#107	—	67	1	no	—
149	C₂×#107	—	134	1	no	—
150	C₂×#107	—	134	2	no	—
151	C₂×#106	—	134	1	no	—
152	#367	—	67	8	no	—
153	#366	—	67	8	no	—
154	C₂³×C₄	—	268	1	no	—
155	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
156	C₂²×C₈	Dic₁₃₄	1	4	yes	—
157	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
158	C₂²×C₈	D₂₆₈	1	2	yes	—
159	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
160	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
162	#117	—	268	2	no	—
163	#117	—	134	4	no	—
164	C₂×C₄⋊C₄	—	268	2	no	—
165	C₂×C₄⋊C₄	—	268	1	no	—
166	C₂×C₄⋊C₄	—	134	4	no	—
167	C₂×C₄⋊C₄	—	134	2	no	—
168	C₂×C₄⋊C₄	—	134	1	no	—
169	C₂×C₄⋊C₄	—	134	1	no	—
170	C₂×C₂²⋊C₄	—	268	1	no	—
171	C₂×C₄²	—	268	1	no	—
172	M₃₂	C₂×Dic₆₇	1	16	yes	complemented
173	C₂×C₁₆	C₂×Dic₆₇	1	8	yes	complemented
174	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
175	#107	—	67	2	no	—
176	#107	—	67	4	no	—
177	#107	—	67	2	no	—
178	#107	—	134	4	no	—
179	#107	—	134	2	no	—
180	#107	—	134	1	no	—
181	#107	—	134	1	no	—
182	#106	—	134	4	no	—
183	C₂⁴	D₅₃₆	1	1	yes	characteristic
184	C₂²×C₄	—	268	4	no	—
185	C₂²×C₄	—	268	1	no	—
186	C₂²×C₄	—	268	1	no	—
187	C₂²×C₄	—	268	2	no	—
188	C₂²×C₄	—	536	4	no	—
189	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
190	C₂²×C₄	D₅₃₆	1	2	yes	—
191	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
192	C₂²×C₄	Dic₂₆₈	1	4	yes	—
193	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
194	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
195	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
196	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	—
197	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
198	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
199	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
200	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
201	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
202	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
203	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
204	C₄⋊C₄	—	268	4	no	—
205	C₄⋊C₄	—	268	2	no	—
206	C₄⋊C₄	—	268	1	no	—
207	C₄⋊C₄	—	268	1	no	—
208	C₄⋊C₄	—	134	2	no	—
209	C₄⋊C₄	—	134	2	no	—
210	C₄⋊C₄	—	134	4	no	—
211	C₂²⋊C₄	—	268	4	no	—
212	C₄²	—	268	4	no	—
213	C₁₆	C₂²×Dic₆₇	1	8	yes	complemented
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	#14523977	1	4	yes	—
217	C₂³	C₂×D₅₃₆	1	2	yes	—
218	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
219	C₂×C₄	—	268	2	no	—
220	C₂×C₄	—	268	2	no	—
221	C₂×C₄	—	268	4	no	—
222	C₂×C₄	—	536	8	no	—
223	C₂×C₄	—	536	4	no	—
224	C₂×C₄	—	536	2	no	—
225	C₂×C₄	—	536	2	no	—
226	C₂×C₄	#14523977	1	8	yes	—
227	C₂×C₄	#14523977	1	4	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
230	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
231	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
232	C₂×C₄	C₂×Dic₂₆₈	1	4	yes	—
233	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
234	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
235	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
236	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
237	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
238	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	C₂×#14523977	1	4	yes	—
243	C₂²	C₂×#14523977	1	2	yes	—
244	C₂²	#16863834	1	2	yes	—
245	C₂²	#16863839	1	2	yes	—
246	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
247	C₄	—	536	4	no	—
248	C₄	—	536	2	no	—
249	C₄	—	536	2	no	—
250	C₄	C₂×#14523977	1	4	yes	—
251	C₄	C₂×#14523977	1	2	yes	—
252	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
253	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#22561997	1	4	yes	direct product factor
256	C₂	C₂²×#14523977	1	1	yes	characteristic
257	C₂	C₂×#16863834	1	1	yes	characteristic
258	C₂	C₂×#16863839	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×M₃₂×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×M₃₂	D₆₇
hypocenter	C₆₇	C₂×#2159
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22561997	C₂	direct (N × H)
4. C₂×#16863224	C₂	semidirect
5. C₂×#16863222	C₂	semidirect
18. #16863224	C₂²	semidirect
19. #16863222	C₂²	semidirect
21. C₂×M₃₂×C₆₇	C₄	semidirect
50. M₃₂×C₆₇	C₂×C₄	semidirect
51. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
83. C₁₆×C₆₇	C₂²×C₄	semidirect
126. C₂×C₆₇	#2159	semidirect
136. C₆₇	C₂×#2159	semidirect
140. C₂×M₃₂	Dic₆₇	semidirect
172. M₃₂	C₂×Dic₆₇	semidirect
173. C₂×C₁₆	C₂×Dic₆₇	semidirect
213. C₁₆	C₂²×Dic₆₇	semidirect
255. C₂	#22561997	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×#16863434	2
C₂²×#14523977	2
#22561997	2
C₂×#16863224	2
C₂×#16863222	2
C₂²×M₃₂×C₆₇	2
C₂×#2159	67

Subgroups — C2×#22561997 #31008865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22561997 #31008865