Subgroups — #444×D₁₉₇ #38751604

Order 25216.

464 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₁₉₇	C₂	1	1	yes	maximal, characteristic
2	#131×D₁₉₇	C₂	1	2	yes	maximal, complemented
3	#27161202	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161201	C₂	1	1	yes	maximal, characteristic, complemented
5	#27161200	C₂	1	1	yes	maximal, characteristic, complemented
6	#27161199	C₂	1	2	yes	maximal, complemented
7	#27161198	C₂	1	2	yes	maximal, complemented
8	#108×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₁₉₇	C₂	1	2	yes	maximal, complemented
10	#97×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#444×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₉₇	C₂²	1	2	yes	—
13	#20263349	C₄	1	1	yes	characteristic, complemented
14	#20263348	C₂²	1	1	yes	characteristic
15	#20263347	C₄	1	2	yes	complemented
16	#20263346	C₂²	1	2	yes	—
17	#20263345	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₁₉₇	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₁₉₇	C₄	1	2	yes	complemented
20	D₈×D₁₉₇	C₄	1	1	yes	characteristic, complemented
21	#20263334	—	2	2	no	—
22	#20263334	C₂²	1	2	yes	complemented
23	#20263333	C₂²	1	2	yes	complemented
24	#20263332	—	2	2	no	—
25	#20263332	C₂²	1	2	yes	complemented
26	M₁₆×D₁₉₇	—	2	2	no	—
27	M₁₆×D₁₉₇	C₂²	1	2	yes	—
28	C₂×C₈×D₁₉₇	—	2	2	no	—
29	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic
31	C₄²×D₁₉₇	C₂²	1	1	yes	characteristic
32	#20263262	C₂²	1	1	yes	characteristic, complemented
33	#20263261	C₂²	1	2	yes	complemented
34	#20263254	C₂²	1	2	yes	complemented
35	#20263253	C₂²	1	2	yes	complemented
36	#20263252	C₂²	1	2	yes	complemented
37	#20263237	C₂²	1	1	yes	characteristic, complemented
38	#20263236	C₂²	1	1	yes	characteristic, complemented
39	#20263236	C₂²	1	1	yes	characteristic, complemented
40	#135×C₁₉₇	C₂²	1	1	yes	characteristic
41	#131×C₁₉₇	C₂²	1	2	yes	complemented
42	#108×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
43	#104×C₁₉₇	C₂²	1	2	yes	complemented
44	#97×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
45	#15691359	C₂×C₄	1	2	yes	—
46	#15691358	—	2	2	no	—
47	#15691358	C₂³	1	2	yes	—
48	#15691357	—	2	2	no	—
49	#15691357	C₂×C₄	1	2	yes	—
50	Q₈×D₁₉₇	C₂×C₄	1	2	yes	—
51	D₄×D₁₉₇	—	2	2	no	—
52	D₄×D₁₉₇	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₉₇	—	2	2	no	—
54	C₂×C₄×D₁₉₇	—	2	1	no	—
55	C₂×C₄×D₁₉₇	C₂³	1	1	yes	characteristic
56	Dic₇₈₈	C₂×C₄	1	1	yes	characteristic, complemented
57	#15691348	C₂×C₄	1	2	yes	complemented
58	#15691347	C₂×C₄	1	2	yes	complemented
59	#15691346	C₂×C₄	1	2	yes	complemented
60	#15691345	C₂×C₄	1	2	yes	complemented
61	D₁₅₇₆	C₂×C₄	1	1	yes	characteristic, complemented
62	#15691343	C₂×C₄	1	2	yes	complemented
63	#15691342	—	2	2	no	—
64	#15691342	C₂³	1	2	yes	—
65	#15691341	—	2	2	no	—
66	#15691341	—	2	2	no	—
67	#15691341	—	2	2	no	—
68	C₂×#13096802	—	2	2	no	—
69	C₂×#13096802	C₂³	1	1	yes	characteristic
70	C₂×#13096802	C₂³	1	1	yes	characteristic
71	C₈×D₁₉₇	—	2	2	no	—
72	C₈×D₁₉₇	—	2	2	no	—
73	C₈×D₁₉₇	—	2	2	no	—
74	C₈×D₁₉₇	C₂×C₄	1	1	yes	characteristic
75	C₈×D₁₉₇	C₂×C₄	1	1	yes	characteristic
76	C₈×D₁₉₇	C₂×C₄	1	1	yes	characteristic
77	C₈×D₁₉₇	C₂×C₄	1	1	yes	characteristic
78	C₈×D₁₉₇	D₄	1	2	yes	—
79	C₈×D₁₉₇	D₄	1	2	yes	—
80	C₄×Dic₁₉₇	—	2	1	no	—
81	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
82	#41×C₁₉₇	C₂³	1	2	yes	—
83	Q₁₆×C₁₉₇	C₂×C₄	1	1	yes	characteristic, complemented
84	SD₁₆×C₁₉₇	C₂×C₄	1	2	yes	complemented
85	D₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic, complemented
86	M₁₆×C₁₉₇	—	2	2	no	—
87	M₁₆×C₁₉₇	C₂³	1	2	yes	—
88	C₂×C₈×C₁₉₇	—	2	2	no	—
89	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
90	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
91	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
92	C₂²×D₁₉₇	—	2	2	no	—
93	C₂²×D₁₉₇	#41	1	1	yes	characteristic
94	Dic₃₉₄	—	2	2	no	—
95	Dic₃₉₄	C₂²×C₄	1	2	yes	—
96	D₇₈₈	—	2	2	no	—
97	D₇₈₈	C₂²×C₄	1	2	yes	—
98	C₂²⋊D₁₉₇	—	2	2	no	—
99	C₂²⋊D₁₉₇	—	2	2	no	—
100	C₂²⋊D₁₉₇	—	2	2	no	—
101	C₂×Dic₁₉₇	—	2	2	no	—
102	C₂×Dic₁₉₇	—	2	1	no	—
103	C₂×Dic₁₉₇	#41	1	1	yes	characteristic
104	C₄×D₁₉₇	—	2	2	no	—
105	C₄×D₁₉₇	—	2	2	no	—
106	C₄×D₁₉₇	—	2	2	no	—
107	C₄×D₁₉₇	—	4	2	no	—
108	C₄×D₁₉₇	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₁₉₇	C₂²×C₄	1	1	yes	characteristic
110	C₄×D₁₉₇	C₂×D₄	1	1	yes	characteristic
111	C₄×D₁₉₇	C₂×D₄	1	1	yes	characteristic
112	#13096802	—	2	2	no	—
113	#13096802	C₂²×C₄	1	1	yes	characteristic
114	#13096802	C₂²×C₄	1	1	yes	characteristic
115	#13096802	C₂×D₄	1	2	yes	—
116	Q₈×C₁₉₇	C₂²×C₄	1	2	yes	—
117	D₄×C₁₉₇	—	2	2	no	—
118	D₄×C₁₉₇	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₁₉₇	—	2	2	no	—
120	C₂×C₄×C₁₉₇	—	2	1	no	—
121	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
122	C₈×C₁₉₇	—	2	2	no	—
123	C₈×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
124	C₈×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
125	C₈×C₁₉₇	C₂×D₄	1	2	yes	—
126	C₂²×C₁₉₇	—	2	2	no	—
127	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
128	C₂×D₁₉₇	—	2	2	no	—
129	C₂×D₁₉₇	—	2	2	no	—
130	C₂×D₁₉₇	—	4	4	no	—
131	C₂×D₁₉₇	C₄×D₄	1	1	yes	characteristic
132	C₂×D₁₉₇	C₄×D₄	1	1	yes	characteristic
133	C₄×C₁₉₇	—	2	2	no	—
134	C₄×C₁₉₇	—	4	1	no	—
135	C₄×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
136	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
137	Dic₁₉₇	—	2	2	no	—
138	Dic₁₉₇	—	4	1	no	—
139	Dic₁₉₇	C₄×D₄	1	1	yes	characteristic
140	Dic₁₉₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₁₉₇	—	2	1	no	—
142	C₂×C₁₉₇	—	4	2	no	—
143	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
144	D₁₉₇	—	2	1	no	—
145	D₁₉₇	—	4	2	no	—
146	D₁₉₇	#444	1	2	yes	direct product factor
147	C₁₉₇	C₂×#444	1	1	yes	characteristic, complemented, Sylow
148	C₂×#444	—	197	1	no	maximal, Sylow
149	C₂×#135	—	197	1	no	—
150	C₂×#131	—	197	2	no	—
151	#444	—	197	1	no	—
152	#444	—	197	2	no	—
153	#444	—	197	1	no	—
154	#444	—	197	2	no	—
155	#444	—	197	1	no	—
156	#444	D₁₉₇	1	1	yes	characteristic, direct product factor
157	C₂×#108	—	197	1	no	—
158	C₂×#104	—	197	2	no	—
159	C₂×#97	—	197	1	no	—
160	C₂×#41	—	197	2	no	—
161	#135	—	197	2	no	—
162	#135	—	197	1	no	—
163	#135	—	197	2	no	—
164	#135	—	197	1	no	—
165	#135	—	197	1	no	—
166	#135	C₂×D₁₉₇	1	1	yes	characteristic
167	C₂×Q₁₆	—	197	1	no	—
168	C₂×SD₁₆	—	197	2	no	—
169	C₂×D₈	—	197	1	no	—
170	#131	—	394	2	no	—
171	#131	—	394	2	no	—
172	#131	—	197	2	no	—
173	#131	—	197	2	no	—
174	#131	—	197	2	no	—
175	#131	C₂×D₁₉₇	1	2	yes	complemented
176	C₂×M₁₆	—	394	2	no	—
177	C₂×M₁₆	—	197	2	no	—
178	C₂²×C₈	—	394	2	no	—
179	C₂²×C₈	—	197	1	no	—
180	C₂²×C₈	—	197	1	no	—
181	C₂×C₄²	—	197	1	no	—
182	#108	—	197	2	no	—
183	#108	—	197	1	no	—
184	#108	C₂×D₁₉₇	1	1	yes	characteristic, complemented
185	#104	—	197	2	no	—
186	#104	—	197	2	no	—
187	#104	—	197	2	no	—
188	#104	C₂×D₁₉₇	1	2	yes	complemented
189	#97	—	197	1	no	—
190	#97	—	197	1	no	—
191	#97	—	197	1	no	—
192	#97	C₂×D₁₉₇	1	1	yes	characteristic, complemented
193	#41	—	394	2	no	—
194	#41	—	394	2	no	—
195	#41	—	197	2	no	—
196	#41	—	197	2	no	—
197	#41	—	197	2	no	—
198	#41	C₂²×D₁₉₇	1	2	yes	—
199	C₂×Q₈	—	197	2	no	—
200	C₂×D₄	—	394	2	no	—
201	C₂×D₄	—	197	2	no	—
202	C₂²×C₄	—	394	1	no	—
203	C₂²×C₄	—	394	2	no	—
204	C₂²×C₄	—	197	1	no	—
205	Q₁₆	—	197	2	no	—
206	Q₁₆	—	197	1	no	—
207	Q₁₆	C₄×D₁₉₇	1	1	yes	characteristic, complemented
208	SD₁₆	—	197	2	no	—
209	SD₁₆	—	197	2	no	—
210	SD₁₆	—	197	2	no	—
211	SD₁₆	C₄×D₁₉₇	1	2	yes	complemented
212	D₈	—	197	2	no	—
213	D₈	—	197	1	no	—
214	D₈	C₄×D₁₉₇	1	1	yes	characteristic, complemented
215	M₁₆	—	2	2	no	—
216	M₁₆	—	394	2	no	—
217	M₁₆	—	394	2	no	—
218	M₁₆	—	394	2	no	—
219	M₁₆	—	394	2	no	—
220	M₁₆	—	197	2	no	—
221	M₁₆	C₂²×D₁₉₇	1	2	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	394	2	no	—
224	C₂×C₈	—	394	2	no	—
225	C₂×C₈	—	394	2	no	—
226	C₂×C₈	—	394	2	no	—
227	C₂×C₈	—	197	1	no	—
228	C₂×C₈	—	197	1	no	—
229	C₂×C₈	—	197	1	no	—
230	C₂×C₈	—	197	1	no	—
231	C₂×C₈	—	197	1	no	—
232	C₂×C₈	—	197	1	no	—
233	C₂×C₈	—	197	2	no	—
234	C₂×C₈	—	197	2	no	—
235	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
237	C₄²	—	394	1	no	—
238	C₄²	—	197	1	no	—
239	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
240	C₂³	—	394	2	no	—
241	C₂³	—	197	1	no	—
242	Q₈	—	394	2	no	—
243	Q₈	—	197	2	no	—
244	Q₈	C₂×C₄×D₁₉₇	1	2	yes	—
245	D₄	—	2	2	no	—
246	D₄	—	394	2	no	—
247	D₄	—	394	2	no	—
248	D₄	—	394	2	no	—
249	D₄	—	394	2	no	—
250	D₄	—	197	2	no	—
251	D₄	C₂×C₄×D₁₉₇	1	2	yes	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	788	2	no	—
255	C₂×C₄	—	394	2	no	—
256	C₂×C₄	—	394	2	no	—
257	C₂×C₄	—	394	1	no	—
258	C₂×C₄	—	394	2	no	—
259	C₂×C₄	—	394	2	no	—
260	C₂×C₄	—	197	1	no	—
261	C₂×C₄	—	197	1	no	—
262	C₂×C₄	—	197	1	no	—
263	C₂×C₄	—	197	1	no	—
264	C₂×C₄	—	197	1	no	—
265	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	394	2	no	—
268	C₈	—	197	1	no	—
269	C₈	—	197	1	no	—
270	C₈	—	197	2	no	—
271	C₈	C₂×C₄×D₁₉₇	1	1	yes	characteristic
272	C₈	C₂×C₄×D₁₉₇	1	1	yes	characteristic
273	C₈	D₄×D₁₉₇	1	2	yes	—
274	C₂²	—	2	2	no	—
275	C₂²	—	788	4	no	—
276	C₂²	—	394	2	no	—
277	C₂²	—	394	2	no	—
278	C₂²	—	197	1	no	—
279	C₂²	—	197	1	no	—
280	C₂²	#41×D₁₉₇	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	4	1	no	—
283	C₄	—	788	1	no	—
284	C₄	—	394	2	no	—
285	C₄	—	197	1	no	—
286	C₄	—	197	1	no	—
287	C₄	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
288	C₄	C₂×D₄×D₁₉₇	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	4	2	no	—
291	C₂	—	788	2	no	—
292	C₂	—	394	1	no	—
293	C₂	—	197	2	no	—
294	C₂	C₄×D₄×D₁₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₉₇
commutator	C₄×C₁₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#444×C₁₉₇	C₂
socle	C₂×C₁₉₇	C₂×C₄×D₄
prime radical	#135×D₁₉₇	C₂
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	#444	D₁₉₇
hypocenter	C₁₉₇	C₂×#444
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #131×D₁₉₇	C₂	semidirect
3. #27161202	C₂	semidirect
4. #27161201	C₂	semidirect
5. #27161200	C₂	semidirect
6. #27161199	C₂	semidirect
7. #27161198	C₂	semidirect
8. #108×D₁₉₇	C₂	semidirect
9. #104×D₁₉₇	C₂	semidirect
10. #97×D₁₉₇	C₂	semidirect
11. #444×C₁₉₇	C₂	semidirect
13. #20263349	C₄	semidirect
15. #20263347	C₄	semidirect
17. #20263345	C₄	semidirect
18. Q₁₆×D₁₉₇	C₄	semidirect
19. SD₁₆×D₁₉₇	C₄	semidirect
20. D₈×D₁₉₇	C₄	semidirect
22. #20263334	C₂²	semidirect
23. #20263333	C₂²	semidirect
25. #20263332	C₂²	semidirect
32. #20263262	C₂²	semidirect
33. #20263261	C₂²	semidirect
34. #20263254	C₂²	semidirect
35. #20263253	C₂²	semidirect
36. #20263252	C₂²	semidirect
37. #20263237	C₂²	semidirect
38. #20263236	C₂²	semidirect
39. #20263236	C₂²	semidirect
41. #131×C₁₉₇	C₂²	semidirect
42. #108×C₁₉₇	C₂²	semidirect
43. #104×C₁₉₇	C₂²	semidirect
44. #97×C₁₉₇	C₂²	semidirect
56. Dic₇₈₈	C₂×C₄	semidirect
57. #15691348	C₂×C₄	semidirect
58. #15691347	C₂×C₄	semidirect
59. #15691346	C₂×C₄	semidirect
60. #15691345	C₂×C₄	semidirect
61. D₁₅₇₆	C₂×C₄	semidirect
62. #15691343	C₂×C₄	semidirect
83. Q₁₆×C₁₉₇	C₂×C₄	semidirect
84. SD₁₆×C₁₉₇	C₂×C₄	semidirect
85. D₈×C₁₉₇	C₂×C₄	semidirect
146. D₁₉₇	#444	direct (N × H)
147. C₁₉₇	C₂×#444	semidirect
156. #444	D₁₉₇	direct (N × H)
175. #131	C₂×D₁₉₇	semidirect
184. #108	C₂×D₁₉₇	semidirect
188. #104	C₂×D₁₉₇	semidirect
192. #97	C₂×D₁₉₇	semidirect
207. Q₁₆	C₄×D₁₉₇	semidirect
211. SD₁₆	C₄×D₁₉₇	semidirect
214. D₈	C₄×D₁₉₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#135×D₁₉₇	2
#131×D₁₉₇	2
#27161202	2
#27161201	2
#27161200	2
#27161199	2
#27161198	2
#108×D₁₉₇	2
#104×D₁₉₇	2
#97×D₁₉₇	2
#444×C₁₉₇	2
C₂×#444	197

Subgroups — #444×D197 #38751604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #444×D₁₉₇ #38751604