Subgroups — #38745593

Order 25216.

504 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263284	C₂	1	1	yes	maximal, characteristic
2	#27160977	C₂	1	1	yes	maximal, characteristic, complemented
3	#27160975	C₂	1	1	yes	maximal, characteristic, complemented
4	#27160970	C₂	1	2	yes	maximal
5	#27160968	C₂	1	2	yes	maximal
6	#27160943	C₂	1	2	yes	maximal, complemented
7	#27160936	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#15691335	C₂	1	1	yes	maximal, characteristic
10	#27160925	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
13	#20263284	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
16	C₂×#15691337	C₂²	1	2	yes	—
17	C₂×#15691337	C₂²	1	2	yes	—
18	C₂×#15691335	C₂²	1	1	yes	characteristic
19	C₂×#15691335	C₂²	1	1	yes	characteristic
20	C₂×#15691335	C₂²	1	2	yes	—
21	C₂×#15691335	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
25	#20263233	C₂²	1	1	yes	characteristic
26	#20263233	C₂²	1	2	yes	—
27	#20263233	C₂²	1	2	yes	—
28	#20263233	C₂²	1	2	yes	—
29	#20263233	C₂²	1	1	yes	characteristic
30	#20263233	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₉₇	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
35	#95×C₁₉₇	C₂²	1	2	yes	—
36	C₂³×D₁₉₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
38	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
39	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
41	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₁₉₇	—	2	2	no	—
43	C₂×C₄×D₁₉₇	—	2	2	no	—
44	C₂×C₄×D₁₉₇	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₁₉₇	—	2	2	no	—
46	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
47	#15691337	—	4	2	no	—
48	#15691337	—	4	2	no	—
49	#15691335	—	2	2	no	—
50	#15691335	—	2	2	no	—
51	#15691335	—	4	2	no	—
52	#15691335	C₂×C₄	1	8	yes	—
53	C₄×Dic₁₉₇	—	2	4	no	—
54	C₄×Dic₁₉₇	—	2	2	no	—
55	C₄×Dic₁₉₇	C₂×C₄	1	8	yes	—
56	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₁₉₇	—	2	2	no	—
62	C₄⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	—
63	C₄²×C₁₉₇	—	2	4	no	—
64	C₄²×C₁₉₇	—	2	2	no	—
65	C₂²×D₁₉₇	—	2	1	no	—
66	C₂²×D₁₉₇	—	2	1	no	—
67	C₂²×D₁₉₇	—	2	1	no	—
68	C₂²×D₁₉₇	—	2	1	no	—
69	C₂²×D₁₉₇	—	2	1	no	—
70	C₂²×D₁₉₇	—	2	1	no	—
71	C₂²×D₁₉₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₉₇	—	2	1	no	—
73	C₂×Dic₁₉₇	—	2	1	no	—
74	C₂×Dic₁₉₇	—	2	1	no	—
75	C₂×Dic₁₉₇	—	2	2	no	—
76	C₂×Dic₁₉₇	—	2	2	no	—
77	C₂×Dic₁₉₇	—	2	1	no	—
78	C₂×Dic₁₉₇	—	2	2	no	—
79	C₂×Dic₁₉₇	—	2	1	no	—
80	C₂×Dic₁₉₇	—	2	1	no	—
81	C₂×Dic₁₉₇	—	2	2	no	—
82	C₂×Dic₁₉₇	—	2	2	no	—
83	C₂×Dic₁₉₇	—	2	1	no	—
84	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
86	C₂×Dic₁₉₇	#41	1	2	yes	—
87	C₄×D₁₉₇	—	2	8	no	—
88	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₄×C₁₉₇	—	2	2	no	—
90	C₂×C₄×C₁₉₇	—	2	1	no	—
91	C₂×C₄×C₁₉₇	—	2	2	no	—
92	C₂×C₄×C₁₉₇	—	2	1	no	—
93	C₂×C₄×C₁₉₇	—	2	2	no	—
94	C₂×C₄×C₁₉₇	—	2	1	no	—
95	C₂×C₄×C₁₉₇	—	2	2	no	—
96	C₂×C₄×C₁₉₇	—	2	1	no	—
97	C₂×C₄×C₁₉₇	—	2	2	no	—
98	C₂×C₄×C₁₉₇	—	2	1	no	—
99	C₂×C₄×C₁₉₇	C₂²×C₄	1	4	yes	—
100	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₉₇	#41	1	2	yes	—
102	C₂×C₄×C₁₉₇	#41	1	2	yes	—
103	C₂×C₄×C₁₉₇	#41	1	2	yes	—
104	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
106	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
107	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
108	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
109	C₂²×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₁₉₇	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×D₁₉₇	—	2	2	no	—
112	C₂×D₁₉₇	—	2	2	no	—
113	C₂×D₁₉₇	—	2	2	no	—
114	C₂×D₁₉₇	—	4	1	no	—
115	C₂×D₁₉₇	—	4	1	no	—
116	C₂×D₁₉₇	—	4	1	no	—
117	C₂×D₁₉₇	—	4	1	no	—
118	C₄×C₁₉₇	—	2	4	no	—
119	C₄×C₁₉₇	—	2	2	no	—
120	C₄×C₁₉₇	—	4	1	no	—
121	C₄×C₁₉₇	—	4	2	no	—
122	C₄×C₁₉₇	#117	1	4	yes	—
123	Dic₁₉₇	—	2	2	no	—
124	Dic₁₉₇	—	2	4	no	—
125	Dic₁₉₇	—	2	2	no	—
126	Dic₁₉₇	—	4	2	no	—
127	Dic₁₉₇	—	4	1	no	—
128	C₂×C₁₉₇	C₂×#117	1	1	yes	characteristic
129	C₂×C₁₉₇	C₄×#41	1	1	yes	characteristic
130	C₂×C₁₉₇	#520	1	1	yes	characteristic
131	C₂×C₁₉₇	#552	1	1	yes	characteristic
132	C₂×C₁₉₇	#552	1	1	yes	characteristic
133	C₂×C₁₉₇	#555	1	1	yes	characteristic
134	C₂×C₁₉₇	#555	1	1	yes	characteristic
135	D₁₉₇	—	4	2	no	—
136	C₁₉₇	#2346	1	1	yes	characteristic, complemented, Sylow
137	#2346	—	197	1	no	maximal, Sylow
138	C₂×#117	—	197	1	no	—
139	#389	—	197	1	no	—
140	#389	—	197	1	no	—
141	#388	—	197	2	no	—
142	#388	—	197	2	no	—
143	#382	—	197	2	no	—
144	#380	—	197	1	no	—
145	C₄×C₄⋊C₄	—	197	1	no	—
146	C₄×C₄⋊C₄	D₁₉₇	1	1	yes	characteristic, complemented
147	C₄×C₂²⋊C₄	—	197	1	no	—
148	#376	—	197	2	no	—
149	C₂³×C₄	—	197	1	no	—
150	#117	—	197	8	no	—
151	C₂×C₄⋊C₄	—	197	2	no	—
152	C₂×C₄⋊C₄	—	197	2	no	—
153	C₂×C₄⋊C₄	—	197	1	no	—
154	C₂×C₄⋊C₄	—	197	1	no	—
155	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	—	197	2	no	—
158	C₂×C₂²⋊C₄	—	197	1	no	—
159	C₂×C₂²⋊C₄	—	197	1	no	—
160	C₂×C₂²⋊C₄	—	197	2	no	—
161	C₂×C₄²	—	197	2	no	—
162	C₂×C₄²	—	197	1	no	—
163	C₂×C₄²	—	197	2	no	—
164	C₂×C₄²	C₂×D₁₉₇	1	2	yes	—
165	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
166	#95	—	197	1	no	—
167	#95	—	197	2	no	—
168	#95	—	197	2	no	—
169	#95	—	197	2	no	—
170	#95	—	197	1	no	—
171	#95	—	197	2	no	—
172	#95	C₂×D₁₉₇	1	2	yes	—
173	C₂⁴	—	197	1	no	—
174	C₂²×C₄	—	394	2	no	—
175	C₂²×C₄	—	394	2	no	—
176	C₂²×C₄	—	197	2	no	—
177	C₂²×C₄	—	197	1	no	—
178	C₂²×C₄	—	197	1	no	—
179	C₂²×C₄	—	197	1	no	—
180	C₂²×C₄	—	197	2	no	—
181	C₂²×C₄	—	197	4	no	—
182	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
183	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
185	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	788	2	no	—
189	C₄⋊C₄	—	788	2	no	—
190	C₄⋊C₄	—	394	2	no	—
191	C₄⋊C₄	—	197	4	no	—
192	C₄⋊C₄	C₄×D₁₉₇	1	4	yes	—
193	C₂²⋊C₄	—	788	2	no	—
194	C₂²⋊C₄	—	394	2	no	—
195	C₂²⋊C₄	—	394	2	no	—
196	C₂²⋊C₄	—	197	8	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	394	4	no	—
200	C₄²	—	394	2	no	—
201	C₄²	—	197	8	no	—
202	C₂³	—	394	1	no	—
203	C₂³	—	394	1	no	—
204	C₂³	—	394	1	no	—
205	C₂³	—	394	1	no	—
206	C₂³	—	394	1	no	—
207	C₂³	—	394	1	no	—
208	C₂³	—	197	2	no	—
209	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	394	2	no	—
221	C₂×C₄	—	394	1	no	—
222	C₂×C₄	—	394	1	no	—
223	C₂×C₄	—	394	2	no	—
224	C₂×C₄	—	394	2	no	—
225	C₂×C₄	—	394	1	no	—
226	C₂×C₄	—	394	1	no	—
227	C₂×C₄	—	394	1	no	—
228	C₂×C₄	—	394	1	no	—
229	C₂×C₄	—	394	2	no	—
230	C₂×C₄	—	394	1	no	—
231	C₂×C₄	—	394	2	no	—
232	C₂×C₄	—	394	8	no	—
233	C₂×C₄	—	197	2	no	—
234	C₂×C₄	—	197	4	no	—
235	C₂×C₄	—	197	2	no	—
236	C₂×C₄	C₂×C₄×D₁₉₇	1	4	yes	—
237	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
238	C₂×C₄	#15691357	1	2	yes	—
239	C₂×C₄	#15691358	1	2	yes	—
240	C₂×C₄	#15691358	1	2	yes	—
241	C₂²	—	788	1	no	—
242	C₂²	—	788	1	no	—
243	C₂²	—	788	1	no	—
244	C₂²	—	788	1	no	—
245	C₂²	—	394	2	no	—
246	C₂²	—	394	2	no	—
247	C₂²	—	394	2	no	—
248	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
249	C₂²	#41×D₁₉₇	1	1	yes	characteristic
250	C₂²	C₂×#15691357	1	1	yes	characteristic
251	C₂²	C₂×#15691358	1	1	yes	characteristic
252	C₂²	C₂×#15691358	1	1	yes	characteristic
253	C₂²	#20263372	1	1	yes	characteristic
254	C₂²	#20263374	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	—	788	2	no	—
260	C₄	—	788	1	no	—
261	C₄	—	394	2	no	—
262	C₄	—	394	2	no	—
263	C₄	—	394	4	no	—
264	C₄	#20263283	1	4	yes	—
265	C₂	—	788	2	no	—
266	C₂	C₂×#20263283	1	1	yes	characteristic
267	C₂	C₄×#15691357	1	1	yes	characteristic
268	C₂	#27161561	1	1	yes	characteristic
269	C₂	#27161764	1	1	yes	characteristic
270	C₂	#27161770	1	1	yes	characteristic
271	C₂	#27161792	1	1	yes	characteristic
272	C₂	#27161794	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₇
commutator	C₂²×C₁₉₇	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₁₉₇	C₂
socle, cosocle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂³×D₁₉₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₉₇
hypocenter	C₁₉₇	#2346
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #27160977	C₂	semidirect
3. #27160975	C₂	semidirect
6. #27160943	C₂	semidirect
8. C₄⋊C₄×Dic₁₉₇	C₂	semidirect
10. #27160925	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₉₇	C₂	semidirect
13. #20263284	C₄	semidirect
136. C₁₉₇	#2346	semidirect
146. C₄×C₄⋊C₄	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263284	2
#27160977	2
#27160975	2
#27160970	2
#27160968	2
#27160943	2
#27160936	2
C₄⋊C₄×Dic₁₉₇	2
C₄×#15691335	2
#27160925	2
C₄×C₄⋊C₄×C₁₉₇	2
#2346	197