Subgroups — C₄×D₈×D₁₉₇ #38751459

Order 25216.

616 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₁₉₇	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄×D₁₉₇	C₂	1	2	yes	maximal, complemented
3	C₄×D₁₅₇₆	C₂	1	1	yes	maximal, characteristic, complemented
4	D₈×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#27161161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#15691343	C₂	1	2	yes	maximal, complemented
7	#27161159	C₂	1	2	yes	maximal, complemented
8	#107×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#102×D₁₉₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₉₇	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₉₇	—	2	2	no	—
14	C₂×D₁₅₇₆	C₂²	1	1	yes	characteristic
15	C₂×#15691343	C₂²	1	2	yes	—
16	D₈×D₁₉₇	C₄	1	8	yes	direct product factor
17	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic
19	C₄×D₇₈₈	C₂²	1	2	yes	complemented
20	D₄×Dic₁₉₇	C₂²	1	2	yes	complemented
21	#20263288	C₂²	1	2	yes	complemented
22	C₄×C₂²⋊D₁₉₇	—	2	2	no	—
23	#20263286	—	2	2	no	—
24	C₄⋊C₄×D₁₉₇	C₂²	1	2	yes	—
25	C₂²⋊C₄×D₁₉₇	—	2	2	no	—
26	C₄²×D₁₉₇	C₂²	1	1	yes	characteristic
27	#20263260	C₂²	1	1	yes	characteristic, complemented
28	#20263259	C₂²	1	2	yes	complemented
29	#20263248	C₂²	1	2	yes	complemented
30	#20263247	C₂²	1	2	yes	complemented
31	#20263246	C₂²	1	2	yes	complemented
32	C₄×#13096802	C₂²	1	1	yes	characteristic, complemented
33	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic, complemented
34	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₈×C₁₉₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₉₇	C₂²	1	2	yes	complemented
37	#107×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
38	#102×C₁₉₇	C₂²	1	2	yes	complemented
39	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₁₉₇	—	2	2	no	—
41	C₂×D₇₈₈	C₂³	1	2	yes	—
42	C₂×C₂²⋊D₁₉₇	—	2	2	no	—
43	D₄×D₁₉₇	—	2	4	no	—
44	D₄×D₁₉₇	C₂×C₄	1	8	yes	complemented
45	C₂²×Dic₁₉₇	—	2	2	no	—
46	C₂×C₄×D₁₉₇	—	2	2	no	—
47	C₂×C₄×D₁₉₇	—	2	2	no	—
48	C₂×C₄×D₁₉₇	—	2	2	no	—
49	C₂×C₄×D₁₉₇	—	4	8	no	—
50	C₂×C₄×D₁₉₇	D₄	1	1	yes	characteristic
51	C₂×C₄×D₁₉₇	D₄	1	1	yes	characteristic
52	C₂×C₄×D₁₉₇	C₂³	1	1	yes	characteristic
53	D₁₅₇₆	C₂×C₄	1	4	yes	complemented
54	#15691343	C₂×C₄	1	8	yes	complemented
55	C₂×#13096802	C₂³	1	1	yes	characteristic
56	C₂×#13096802	C₂³	1	1	yes	characteristic
57	C₈×D₁₉₇	—	2	2	no	—
58	C₈×D₁₉₇	C₂×C₄	1	4	yes	complemented
59	C₄⋊Dic₁₉₇	C₂³	1	2	yes	—
60	#15691337	—	2	2	no	—
61	C₂²⋊Dic₁₉₇	—	2	2	no	—
62	#15691335	—	2	2	no	—
63	#15691335	—	2	2	no	—
64	C₄×Dic₁₉₇	D₄	1	1	yes	characteristic
65	C₄×Dic₁₉₇	D₄	1	1	yes	characteristic
66	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₉₇	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₉₇	—	2	2	no	—
69	D₈×C₁₉₇	C₂×C₄	1	4	yes	complemented
70	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉₇	C₂³	1	2	yes	—
73	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
74	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
75	C₂²×D₁₉₇	—	2	2	no	—
76	C₂²×D₁₉₇	—	2	8	no	—
77	C₂²×D₁₉₇	—	4	4	no	—
78	C₂²×D₁₉₇	—	4	4	no	—
79	C₂²×D₁₉₇	C₂×D₄	1	1	yes	characteristic
80	D₇₈₈	—	2	2	no	—
81	D₇₈₈	C₂²×C₄	1	4	yes	complemented
82	C₂²⋊D₁₉₇	—	2	8	no	—
83	C₂×Dic₁₉₇	—	2	2	no	—
84	C₂×Dic₁₉₇	—	4	4	no	—
85	C₂×Dic₁₉₇	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₁₉₇	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₉₇	C₂×D₄	1	1	yes	characteristic
88	C₄×D₁₉₇	—	2	2	no	—
89	C₄×D₁₉₇	—	4	4	no	—
90	C₄×D₁₉₇	—	4	4	no	—
91	C₄×D₁₉₇	—	4	4	no	—
92	C₄×D₁₉₇	D₈	1	4	yes	direct product factor
93	C₄×D₁₉₇	C₂²×C₄	1	2	yes	—
94	C₄×D₁₉₇	#41	1	2	yes	—
95	#13096802	—	2	1	no	—
96	#13096802	C₂²×C₄	1	2	yes	complemented
97	C₂³×C₁₉₇	—	2	2	no	—
98	D₄×C₁₉₇	—	2	2	no	—
99	D₄×C₁₉₇	C₂²×C₄	1	4	yes	complemented
100	C₂×C₄×C₁₉₇	—	2	2	no	—
101	C₂×C₄×C₁₉₇	—	4	4	no	—
102	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
105	C₈×C₁₉₇	—	2	1	no	—
106	C₈×C₁₉₇	C₂²×C₄	1	2	yes	complemented
107	C₂²×C₁₉₇	—	2	4	no	—
108	C₂²×C₁₉₇	—	4	2	no	—
109	C₂²×C₁₉₇	—	4	2	no	—
110	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×D₁₉₇	—	2	4	no	—
112	C₂×D₁₉₇	—	4	2	no	—
113	C₂×D₁₉₇	—	4	2	no	—
114	C₂×D₁₉₇	—	4	16	no	—
115	C₂×D₁₉₇	C₄×D₄	1	2	yes	—
116	C₂×D₁₉₇	C₂×D₈	1	2	yes	—
117	C₂×D₁₉₇	#135	1	2	yes	—
118	C₄×C₁₉₇	—	2	1	no	—
119	C₄×C₁₉₇	—	4	2	no	—
120	C₄×C₁₉₇	C₂×D₈	1	2	yes	complemented
121	C₄×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
123	Dic₁₉₇	—	2	1	no	—
124	Dic₁₉₇	—	4	2	no	—
125	Dic₁₉₇	C₄×D₄	1	2	yes	—
126	Dic₁₉₇	C₂×D₈	1	2	yes	complemented
127	C₂×C₁₉₇	—	4	4	no	—
128	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂×C₁₉₇	C₂²×D₈	1	1	yes	characteristic
130	C₂×C₁₉₇	C₂×#135	1	1	yes	characteristic
131	D₁₉₇	—	4	4	no	—
132	D₁₉₇	C₄×D₈	1	4	yes	direct product factor
133	C₁₉₇	C₂×C₄×D₈	1	1	yes	characteristic, complemented, Sylow
134	C₂×C₄×D₈	—	197	1	no	maximal, Sylow
135	C₂²×D₈	—	197	1	no	—
136	C₂×C₄×D₄	—	197	2	no	—
137	C₄×D₈	—	197	1	no	—
138	C₄×D₈	—	197	2	no	—
139	C₄×D₈	—	197	1	no	—
140	C₄×D₈	—	197	2	no	—
141	C₄×D₈	—	197	1	no	—
142	C₄×D₈	D₁₉₇	1	1	yes	characteristic, direct product factor
143	C₂×#107	—	197	1	no	—
144	C₂×#102	—	197	2	no	—
145	C₂×C₄×C₈	—	197	1	no	—
146	C₂²×D₄	—	197	2	no	—
147	C₂³×C₄	—	394	2	no	—
148	C₂×D₈	—	197	2	no	—
149	C₂×D₈	—	197	8	no	—
150	C₂×D₈	—	197	1	no	—
151	C₂×D₈	C₂×D₁₉₇	1	1	yes	characteristic
152	C₂²×C₈	—	197	1	no	—
153	C₂²×C₈	—	197	1	no	—
154	C₄×D₄	—	394	2	no	—
155	C₄×D₄	—	394	2	no	—
156	C₄×D₄	—	197	2	no	—
157	C₄×D₄	—	197	2	no	—
158	C₄×D₄	—	197	2	no	—
159	C₄×D₄	C₂×D₁₉₇	1	2	yes	complemented
160	C₂×C₄⋊C₄	—	197	2	no	—
161	C₂×C₂²⋊C₄	—	394	2	no	—
162	C₂×C₄²	—	197	1	no	—
163	#107	—	197	1	no	—
164	#107	—	197	2	no	—
165	#107	C₂×D₁₉₇	1	1	yes	characteristic, complemented
166	#102	—	197	2	no	—
167	#102	—	197	2	no	—
168	#102	—	197	2	no	—
169	#102	C₂×D₁₉₇	1	2	yes	complemented
170	C₄×C₈	—	197	1	no	—
171	C₄×C₈	—	197	1	no	—
172	C₄×C₈	—	197	1	no	—
173	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
174	C₂⁴	—	394	2	no	—
175	C₂×D₄	—	394	4	no	—
176	C₂×D₄	—	394	2	no	—
177	C₂×D₄	—	197	8	no	—
178	C₂×D₄	—	197	2	no	—
179	C₂×D₄	C₂²×D₁₉₇	1	2	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	788	8	no	—
182	C₂²×C₄	—	394	2	no	—
183	C₂²×C₄	—	394	2	no	—
184	C₂²×C₄	—	394	2	no	—
185	C₂²×C₄	—	394	2	no	—
186	C₂²×C₄	—	197	1	no	—
187	C₂²×C₄	—	197	1	no	—
188	C₂²×C₄	—	197	1	no	—
189	D₈	—	197	8	no	—
190	D₈	—	197	4	no	—
191	D₈	C₄×D₁₉₇	1	4	yes	direct product factor
192	C₂×C₈	—	394	2	no	—
193	C₂×C₈	—	197	1	no	—
194	C₂×C₈	—	197	1	no	—
195	C₂×C₈	—	197	4	no	—
196	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
198	C₄⋊C₄	—	394	2	no	—
199	C₄⋊C₄	—	197	2	no	—
200	C₄⋊C₄	C₂²×D₁₉₇	1	2	yes	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	394	2	no	—
203	C₂²⋊C₄	—	394	2	no	—
204	C₂²⋊C₄	—	394	2	no	—
205	C₄²	—	197	1	no	—
206	C₄²	—	197	1	no	—
207	C₄²	—	197	1	no	—
208	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
209	C₂³	—	2	2	no	—
210	C₂³	—	788	4	no	—
211	C₂³	—	788	4	no	—
212	C₂³	—	394	8	no	—
213	C₂³	—	394	2	no	—
214	C₂³	—	197	1	no	—
215	D₄	—	2	2	no	—
216	D₄	—	394	8	no	—
217	D₄	—	394	2	no	—
218	D₄	—	197	4	no	—
219	D₄	C₂×C₄×D₁₉₇	1	4	yes	complemented
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	788	4	no	—
223	C₂×C₄	—	788	4	no	—
224	C₂×C₄	—	788	4	no	—
225	C₂×C₄	—	788	4	no	—
226	C₂×C₄	—	394	2	no	—
227	C₂×C₄	—	394	2	no	—
228	C₂×C₄	—	197	1	no	—
229	C₂×C₄	—	197	1	no	—
230	C₂×C₄	—	197	1	no	—
231	C₂×C₄	—	197	4	no	—
232	C₂×C₄	—	197	2	no	—
233	C₂×C₄	—	197	2	no	—
234	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
235	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
236	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	394	1	no	—
239	C₈	—	197	2	no	—
240	C₈	C₂×C₄×D₁₉₇	1	2	yes	complemented
241	C₂²	—	2	4	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	788	16	no	—
245	C₂²	—	788	2	no	—
246	C₂²	—	788	2	no	—
247	C₂²	—	394	4	no	—
248	C₂²	—	197	2	no	—
249	C₂²	—	197	2	no	—
250	C₂²	—	197	2	no	—
251	C₂²	C₂×D₄×D₁₉₇	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	788	2	no	—
255	C₄	—	394	1	no	—
256	C₄	—	197	2	no	—
257	C₄	—	197	2	no	—
258	C₄	D₈×D₁₉₇	1	2	yes	direct product factor
259	C₄	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
260	C₄	#41×D₁₉₇	1	1	yes	characteristic
261	C₂	—	4	4	no	—
262	C₂	—	788	4	no	—
263	C₂	—	197	4	no	—
264	C₂	C₄×D₄×D₁₉₇	1	1	yes	characteristic
265	C₂	C₂×D₈×D₁₉₇	1	1	yes	characteristic
266	C₂	#135×D₁₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉₇
commutator	C₄×C₁₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₄×D₈×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
prime radical	C₂×D₈×D₁₉₇	C₂
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	C₄×D₈	D₁₉₇
hypocenter	C₁₉₇	C₂×C₄×D₈
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×D₁₉₇	C₂	semidirect
3. C₄×D₁₅₇₆	C₂	semidirect
4. D₈×Dic₁₉₇	C₂	semidirect
5. #27161161	C₂	semidirect
6. C₄×#15691343	C₂	semidirect
7. #27161159	C₂	semidirect
8. #107×D₁₉₇	C₂	semidirect
9. #102×D₁₉₇	C₂	semidirect
10. C₄×C₈×D₁₉₇	C₂	semidirect
11. C₄×D₈×C₁₉₇	C₂	semidirect
16. D₈×D₁₉₇	C₄	direct (N × H)
19. C₄×D₇₈₈	C₂²	semidirect
20. D₄×Dic₁₉₇	C₂²	semidirect
21. #20263288	C₂²	semidirect
27. #20263260	C₂²	semidirect
28. #20263259	C₂²	semidirect
29. #20263248	C₂²	semidirect
30. #20263247	C₂²	semidirect
31. #20263246	C₂²	semidirect
32. C₄×#13096802	C₂²	semidirect
33. C₈×Dic₁₉₇	C₂²	semidirect
34. C₈×Dic₁₉₇	C₂²	semidirect
36. C₄×D₄×C₁₉₇	C₂²	semidirect
37. #107×C₁₉₇	C₂²	semidirect
38. #102×C₁₉₇	C₂²	semidirect
39. C₄×C₈×C₁₉₇	C₂²	semidirect
44. D₄×D₁₉₇	C₂×C₄	semidirect
53. D₁₅₇₆	C₂×C₄	semidirect
54. #15691343	C₂×C₄	semidirect
58. C₈×D₁₉₇	C₂×C₄	semidirect
69. D₈×C₁₉₇	C₂×C₄	semidirect
81. D₇₈₈	C₂²×C₄	semidirect
92. C₄×D₁₉₇	D₈	direct (N × H)
96. #13096802	C₂²×C₄	semidirect
99. D₄×C₁₉₇	C₂²×C₄	semidirect
106. C₈×C₁₉₇	C₂²×C₄	semidirect
120. C₄×C₁₉₇	C₂×D₈	semidirect
126. Dic₁₉₇	C₂×D₈	semidirect
132. D₁₉₇	C₄×D₈	direct (N × H)
133. C₁₉₇	C₂×C₄×D₈	semidirect
142. C₄×D₈	D₁₉₇	direct (N × H)
159. C₄×D₄	C₂×D₁₉₇	semidirect
165. #107	C₂×D₁₉₇	semidirect
169. #102	C₂×D₁₉₇	semidirect
173. C₄×C₈	C₂×D₁₉₇	semidirect
191. D₈	C₄×D₁₉₇	direct (N × H)
219. D₄	C₂×C₄×D₁₉₇	semidirect
240. C₈	C₂×C₄×D₁₉₇	semidirect
258. C₄	D₈×D₁₉₇	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×D₈×D₁₉₇	2
C₄×D₄×D₁₉₇	2
C₄×D₁₅₇₆	2
D₈×Dic₁₉₇	2
#27161161	2
C₄×#15691343	2
#27161159	2
#107×D₁₉₇	2
#102×D₁₉₇	2
C₄×C₈×D₁₉₇	2
C₄×D₈×C₁₉₇	2
C₂×C₄×D₈	197

Subgroups — C4×D8×D197 #38751459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₈×D₁₉₇ #38751459