Subgroups — C₂×#20261537 #27156486

Order 12416.

1512 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690856	C₂	1	2	yes	maximal, complemented
2	#20261537	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15690837	C₂	1	4	yes	maximal, complemented
4	C₂×#15690835	C₂	1	2	yes	maximal, complemented
5	C₂×#15690833	C₂	1	2	yes	maximal, complemented
6	C₂²×D₇₇₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₈×D₉₇	C₂	1	2	yes	maximal, complemented
8	C₂×#15690816	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#15690856	—	2	8	no	—
11	#15690856	C₂²	1	16	yes	complemented
12	C₂×#13095512	C₂²	1	4	yes	complemented
13	C₂×#13095511	—	2	2	no	—
14	C₂×#13095511	C₂²	1	1	yes	characteristic
15	C₂²×D₃₈₈	—	2	2	no	—
16	C₂²×D₃₈₈	C₂²	1	2	yes	complemented
17	C₂×D₄×D₉₇	—	2	4	no	—
18	C₂×D₄×D₉₇	C₂²	1	4	yes	complemented
19	#15690837	C₂²	1	32	yes	complemented
20	#15690835	C₂²	1	16	yes	complemented
21	#15690833	C₂²	1	16	yes	complemented
22	C₂×#13095498	C₂²	1	4	yes	complemented
23	C₂×D₇₇₆	C₂²	1	2	yes	complemented
24	C₂×D₇₇₆	C₂²	1	2	yes	complemented
25	C₂×D₇₇₆	C₂²	1	8	yes	complemented
26	C₂×#13095496	C₂²	1	4	yes	complemented
27	D₈×D₉₇	C₂²	1	16	yes	complemented
28	#15690816	C₂²	1	8	yes	complemented
29	C₂×#13095495	C₂²	1	1	yes	characteristic, complemented
30	C₂×#13095494	C₂²	1	2	yes	complemented
31	C₂×C₈×D₉₇	C₂²	1	2	yes	complemented
32	C₂×#41×C₉₇	C₂²	1	2	yes	complemented
33	#135×C₉₇	C₂²	1	8	yes	complemented
34	C₂×Q₁₆×C₉₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	complemented
36	C₂×D₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
38	C₂³×D₉₇	—	2	2	no	—
39	C₂³×D₉₇	—	2	4	no	—
40	#13095512	—	2	8	no	—
41	#13095512	C₂³	1	16	yes	complemented
42	#13095511	—	2	16	no	—
43	#13095511	D₄	1	4	yes	—
44	#13095511	C₂³	1	4	yes	—
45	C₂×Dic₁₉₄	D₄	1	1	yes	characteristic
46	C₂×D₃₈₈	—	2	4	no	—
47	C₂×D₃₈₈	—	2	2	no	—
48	C₂×D₃₈₈	—	2	4	no	—
49	C₂×D₃₈₈	—	2	16	no	—
50	C₂×D₃₈₈	D₄	1	1	yes	characteristic
51	C₂×D₃₈₈	C₂³	1	4	yes	—
52	C₂×D₃₈₈	C₂³	1	8	yes	complemented
53	C₂×C₂²⋊D₉₇	—	2	4	no	—
54	C₂×C₂²⋊D₉₇	D₄	1	2	yes	—
55	D₄×D₉₇	—	2	32	no	—
56	D₄×D₉₇	—	2	8	no	—
57	D₄×D₉₇	C₂³	1	16	yes	complemented
58	C₂×C₄×D₉₇	—	2	4	no	—
59	C₂×C₄×D₉₇	C₂³	1	2	yes	—
60	#13095498	C₂³	1	16	yes	complemented
61	D₇₇₆	C₂³	1	8	yes	complemented
62	D₇₇₆	C₂³	1	8	yes	complemented
63	#13095496	C₂³	1	16	yes	complemented
64	#13095495	C₂³	1	4	yes	complemented
65	#13095494	C₂³	1	8	yes	complemented
66	C₂×#11715895	C₂³	1	2	yes	—
67	C₈×D₉₇	C₂³	1	8	yes	complemented
68	#41×C₉₇	—	2	4	no	—
69	#41×C₉₇	C₂³	1	8	yes	complemented
70	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
71	C₂×D₄×C₉₇	—	2	2	no	—
72	C₂×D₄×C₉₇	C₂³	1	2	yes	—
73	C₂²×C₄×C₉₇	—	2	2	no	—
74	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
75	Q₁₆×C₉₇	C₂³	1	4	yes	complemented
76	SD₁₆×C₉₇	C₂³	1	8	yes	complemented
77	D₈×C₉₇	C₂³	1	4	yes	complemented
78	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₉₇	C₂³	1	4	yes	complemented
81	C₂³×C₉₇	—	2	2	no	—
82	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×D₉₇	—	2	4	no	—
84	C₂²×D₉₇	—	2	8	no	—
85	C₂²×D₉₇	—	2	16	no	—
86	C₂²×D₉₇	—	8	8	no	—
87	C₂²×D₉₇	—	8	4	no	—
88	C₂²×D₉₇	C₂×D₄	1	2	yes	—
89	Q₈×C₉₇	—	2	2	no	—
90	Q₈×C₉₇	C₂⁴	1	4	yes	—
91	D₄×C₉₇	—	2	8	no	—
92	D₄×C₉₇	—	2	2	no	—
93	D₄×C₉₇	C₂⁴	1	4	yes	—
94	C₂×C₄×C₉₇	—	2	2	no	—
95	C₂×C₄×C₉₇	—	2	8	no	—
96	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
100	C₂²⋊D₉₇	—	2	16	no	—
101	C₂²⋊D₉₇	C₂×D₄	1	8	yes	—
102	C₂×Dic₉₇	C₂×D₄	1	2	yes	—
103	D₃₈₈	—	2	16	no	—
104	D₃₈₈	—	2	4	no	—
105	D₃₈₈	—	2	8	no	—
106	D₃₈₈	C₂×D₄	1	4	yes	—
107	D₃₈₈	C₂⁴	1	8	yes	—
108	C₄×D₉₇	—	2	16	no	—
109	C₄×D₉₇	C₂×D₄	1	4	yes	—
110	C₄×D₉₇	C₂⁴	1	4	yes	—
111	Dic₁₉₄	C₂×D₄	1	4	yes	—
112	C₈×C₉₇	C₂⁴	1	2	yes	—
113	C₈×C₉₇	C₂⁴	1	2	yes	—
114	#11715895	C₂⁴	1	4	yes	—
115	C₂²×C₉₇	—	2	2	no	—
116	C₂²×C₉₇	—	2	4	no	—
117	C₂²×C₉₇	—	4	4	no	—
118	C₂²×C₉₇	C₂²×D₄	1	2	yes	—
119	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₉₇	—	2	4	no	—
121	C₂×D₉₇	—	2	8	no	—
122	C₂×D₉₇	—	4	8	no	—
123	C₂×D₉₇	—	8	16	no	—
124	C₂×D₉₇	—	8	8	no	—
125	C₂×D₉₇	C₂²×D₄	1	4	yes	—
126	C₄×C₉₇	—	2	4	no	—
127	C₄×C₉₇	C₂²×D₄	1	2	yes	—
128	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₉₇	C₂⁵	1	1	yes	characteristic
130	Dic₉₇	C₂²×D₄	1	4	yes	—
131	C₂×C₉₇	—	2	2	no	—
132	C₂×C₉₇	—	4	4	no	—
133	C₂×C₉₇	#576	1	2	yes	complemented
134	C₂×C₉₇	C₂³×D₄	1	1	yes	characteristic
135	D₉₇	—	2	4	no	—
136	D₉₇	—	4	8	no	—
137	C₂×#576	—	97	1	no	maximal, Sylow
138	C₉₇	C₂×#576	1	1	yes	characteristic, complemented, Sylow
139	C₂×H(5,𝔽₂)	—	97	2	no	—
140	#576	—	97	16	no	—
141	C₂×AGL(1,ℤ/8)	—	97	2	no	—
142	C₂×AGL(1,ℤ/8)	—	97	4	no	—
143	C₂×#135	—	97	2	no	—
144	C₂×#135	D₉₇	1	1	yes	characteristic, complemented
145	C₂²×D₈	—	97	2	no	—
146	C₂²×D₈	—	97	1	no	—
147	C₂×#131	—	97	1	no	—
148	H(5,𝔽₂)	—	97	16	no	—
149	H(5,𝔽₂)	—	194	8	no	—
150	C₂×#41	—	97	1	no	—
151	C₂×#41	—	97	4	no	—
152	C₂×#41	—	194	2	no	—
153	C₂×#41	C₂×D₉₇	1	2	yes	complemented
154	C₂²×D₄	—	97	4	no	—
155	C₂²×D₄	—	97	2	no	—
156	C₂²×D₄	—	194	2	no	—
157	C₂²×D₄	—	194	4	no	—
158	AGL(1,ℤ/8)	—	97	16	no	—
159	AGL(1,ℤ/8)	—	97	32	no	—
160	#135	—	97	16	no	—
161	#135	C₂×D₉₇	1	8	yes	complemented
162	C₂×Q₁₆	C₂×D₉₇	1	1	yes	characteristic, complemented
163	C₂×SD₁₆	—	97	4	no	—
164	C₂×SD₁₆	C₂×D₉₇	1	2	yes	complemented
165	C₂×D₈	—	97	4	no	—
166	C₂×D₈	—	97	16	no	—
167	C₂×D₈	—	97	2	no	—
168	C₂×D₈	—	97	2	no	—
169	C₂×D₈	—	97	8	no	—
170	C₂×D₈	C₂×D₉₇	1	1	yes	characteristic, complemented
171	#131	—	97	8	no	—
172	C₂×M₁₆	—	97	1	no	—
173	C₂×M₁₆	—	97	2	no	—
174	C₂²×C₈	—	97	2	no	—
175	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic, complemented
176	C₂⁴	—	194	4	no	—
177	C₂⁴	—	194	2	no	—
178	#41	—	2	4	no	—
179	#41	—	97	4	no	—
180	#41	—	97	4	no	—
181	#41	—	97	16	no	—
182	#41	—	194	16	no	—
183	#41	—	194	8	no	—
184	#41	C₂²×D₉₇	1	8	yes	complemented
185	C₂×Q₈	—	97	1	no	—
186	C₂×Q₈	C₂²×D₉₇	1	2	yes	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	97	2	no	—
189	C₂×D₄	—	97	16	no	—
190	C₂×D₄	—	97	1	no	—
191	C₂×D₄	—	97	4	no	—
192	C₂×D₄	—	97	8	no	—
193	C₂×D₄	—	194	16	no	—
194	C₂×D₄	—	194	4	no	—
195	C₂×D₄	—	194	8	no	—
196	C₂×D₄	—	194	4	no	—
197	C₂×D₄	—	194	32	no	—
198	C₂×D₄	—	194	2	no	—
199	C₂×D₄	—	194	4	no	—
200	C₂×D₄	C₂²×D₉₇	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	97	2	no	—
203	C₂²×C₄	—	194	4	no	—
204	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
205	Q₁₆	C₂²×D₉₇	1	4	yes	complemented
206	SD₁₆	—	97	16	no	—
207	SD₁₆	C₂²×D₉₇	1	8	yes	complemented
208	D₈	—	97	16	no	—
209	D₈	—	97	8	no	—
210	D₈	—	97	8	no	—
211	D₈	C₂²×D₉₇	1	4	yes	complemented
212	M₁₆	—	97	4	no	—
213	M₁₆	—	97	8	no	—
214	C₂×C₈	—	97	2	no	—
215	C₂×C₈	—	97	8	no	—
216	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
217	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
218	C₂×C₈	C₂²×D₉₇	1	4	yes	complemented
219	C₂³	—	2	2	no	—
220	C₂³	—	776	4	no	—
221	C₂³	—	776	8	no	—
222	C₂³	—	97	2	no	—
223	C₂³	—	194	16	no	—
224	C₂³	—	194	8	no	—
225	C₂³	—	194	4	no	—
226	C₂³	D₄×D₉₇	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	97	4	no	—
229	Q₈	C₂³×D₉₇	1	4	yes	—
230	D₄	—	2	8	no	—
231	D₄	—	2	2	no	—
232	D₄	—	97	8	no	—
233	D₄	—	97	4	no	—
234	D₄	—	97	8	no	—
235	D₄	—	194	16	no	—
236	D₄	—	194	16	no	—
237	D₄	—	194	8	no	—
238	D₄	—	194	4	no	—
239	D₄	C₂³×D₉₇	1	4	yes	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	97	2	no	—
243	C₂×C₄	—	97	4	no	—
244	C₂×C₄	—	97	4	no	—
245	C₂×C₄	—	194	16	no	—
246	C₂×C₄	D₄×D₉₇	1	2	yes	—
247	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
248	C₂×C₄	C₂³×D₉₇	1	2	yes	—
249	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
250	C₈	—	97	4	no	—
251	C₈	C₂³×D₉₇	1	2	yes	—
252	C₈	C₂³×D₉₇	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	776	8	no	—
257	C₂²	—	776	16	no	—
258	C₂²	—	97	4	no	—
259	C₂²	—	388	8	no	—
260	C₂²	—	194	8	no	—
261	C₂²	—	194	4	no	—
262	C₂²	C₂×D₄×D₉₇	1	2	yes	—
263	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	97	4	no	—
266	C₄	C₂×D₄×D₉₇	1	2	yes	—
267	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
268	C₄	C₂⁴×D₉₇	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	4	4	no	—
271	C₂	—	388	8	no	—
272	C₂	—	194	4	no	—
273	C₂	#20261537	1	2	yes	direct product factor
274	C₂	C₂²×D₄×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₉₇
commutator, cosocle	C₄×C₉₇	C₂⁵
Frattini	C₄	C₂⁴×D₉₇
Fitting	C₂×#135×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
hypercenter	C₂×#135	D₉₇
hypocenter	C₉₇	C₂×#576
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#15690856	C₂	semidirect
2. #20261537	C₂	direct (N × H)
3. C₂×#15690837	C₂	semidirect
4. C₂×#15690835	C₂	semidirect
5. C₂×#15690833	C₂	semidirect
6. C₂²×D₇₇₆	C₂	semidirect
7. C₂×D₈×D₉₇	C₂	semidirect
8. C₂×#15690816	C₂	semidirect
9. C₂×#135×C₉₇	C₂	semidirect
11. #15690856	C₂²	semidirect
12. C₂×#13095512	C₂²	semidirect
16. C₂²×D₃₈₈	C₂²	semidirect
18. C₂×D₄×D₉₇	C₂²	semidirect
19. #15690837	C₂²	semidirect
20. #15690835	C₂²	semidirect
21. #15690833	C₂²	semidirect
22. C₂×#13095498	C₂²	semidirect
23. C₂×D₇₇₆	C₂²	semidirect
24. C₂×D₇₇₆	C₂²	semidirect
25. C₂×D₇₇₆	C₂²	semidirect
26. C₂×#13095496	C₂²	semidirect
27. D₈×D₉₇	C₂²	semidirect
28. #15690816	C₂²	semidirect
29. C₂×#13095495	C₂²	semidirect
30. C₂×#13095494	C₂²	semidirect
31. C₂×C₈×D₉₇	C₂²	semidirect
32. C₂×#41×C₉₇	C₂²	semidirect
33. #135×C₉₇	C₂²	semidirect
34. C₂×Q₁₆×C₉₇	C₂²	semidirect
35. C₂×SD₁₆×C₉₇	C₂²	semidirect
36. C₂×D₈×C₉₇	C₂²	semidirect
37. C₂²×C₈×C₉₇	C₂²	semidirect
41. #13095512	C₂³	semidirect
52. C₂×D₃₈₈	C₂³	semidirect
57. D₄×D₉₇	C₂³	semidirect
60. #13095498	C₂³	semidirect
61. D₇₇₆	C₂³	semidirect
62. D₇₇₆	C₂³	semidirect
63. #13095496	C₂³	semidirect
64. #13095495	C₂³	semidirect
65. #13095494	C₂³	semidirect
67. C₈×D₉₇	C₂³	semidirect
69. #41×C₉₇	C₂³	semidirect
75. Q₁₆×C₉₇	C₂³	semidirect
76. SD₁₆×C₉₇	C₂³	semidirect
77. D₈×C₉₇	C₂³	semidirect
80. C₂×C₈×C₉₇	C₂³	semidirect
133. C₂×C₉₇	#576	semidirect
138. C₉₇	C₂×#576	semidirect
144. C₂×#135	D₉₇	semidirect
153. C₂×#41	C₂×D₉₇	semidirect
161. #135	C₂×D₉₇	semidirect
162. C₂×Q₁₆	C₂×D₉₇	semidirect
164. C₂×SD₁₆	C₂×D₉₇	semidirect
170. C₂×D₈	C₂×D₉₇	semidirect
175. C₂²×C₈	C₂×D₉₇	semidirect
184. #41	C₂²×D₉₇	semidirect
205. Q₁₆	C₂²×D₉₇	semidirect
207. SD₁₆	C₂²×D₉₇	semidirect
211. D₈	C₂²×D₉₇	semidirect
218. C₂×C₈	C₂²×D₉₇	semidirect
273. C₂	#20261537	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×#15690856	2
#20261537	2
C₂×#15690837	2
C₂×#15690835	2
C₂×#15690833	2
C₂²×D₇₇₆	2
C₂×D₈×D₉₇	2
C₂×#15690816	2
C₂×#135×C₉₇	2
C₂×#576	97

Subgroups — C2×#20261537 #27156486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261537 #27156486