Subgroups — #38745066

Order 25216.

696 conjugacy classes of subgroups.

The subgroup lattice has 290 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	C₂×#20263283	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×D₁₉₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₄×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#15691337	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#27160930	C₂	1	2	yes	maximal, complemented
9	C₄×#15691335	C₂	1	2	yes	maximal
10	#27160926	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₉₇	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
14	#20263284	C₄	1	8	yes	complemented
15	#20263283	C₄	1	8	yes	complemented
16	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
18	C₂×#15691337	C₂²	1	2	yes	—
19	C₂×#15691337	C₂²	1	1	yes	characteristic
20	C₂×#15691337	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×D₁₉₇	C₄	1	8	yes	complemented
22	C₂×#15691335	C₂²	1	1	yes	characteristic
23	C₂×#15691335	C₂²	1	2	yes	—
24	C₂×#15691335	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
30	#20263233	C₂²	1	2	yes	—
31	#20263233	C₂²	1	2	yes	—
32	#20263233	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₉₇	C₂²	1	2	yes	—
36	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
37	#95×C₁₉₇	C₂²	1	2	yes	—
38	C₂³×D₁₉₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
42	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
43	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₁₉₇	—	4	2	no	—
45	C₂×C₄×D₁₉₇	—	4	2	no	—
46	C₂×C₄×D₁₉₇	C₂×C₄	1	8	yes	—
47	C₂×C₄×D₁₉₇	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₁₉₇	C₂×C₄	1	4	yes	—
49	C₂×C₄×D₁₉₇	C₂×C₄	1	4	yes	—
50	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
51	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	complemented
52	#15691337	C₂×C₄	1	8	yes	—
53	#15691337	C₂×C₄	1	4	yes	complemented
54	#15691337	C₂×C₄	1	4	yes	complemented
55	#15691335	C₂×C₄	1	4	yes	—
56	#15691335	C₂×C₄	1	4	yes	—
57	#15691335	C₂×C₄	1	8	yes	—
58	C₄×Dic₁₉₇	—	4	2	no	—
59	C₄×Dic₁₉₇	—	4	2	no	—
60	C₄×Dic₁₉₇	—	4	2	no	—
61	C₄×Dic₁₉₇	C₂×C₄	1	4	yes	complemented
62	C₄×Dic₁₉₇	C₂×C₄	1	8	yes	complemented
63	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	—
69	C₄⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	complemented
70	C₄²×C₁₉₇	—	4	2	no	—
71	C₄²×C₁₉₇	C₂×C₄	1	4	yes	complemented
72	C₂²×D₁₉₇	—	2	1	no	—
73	C₂²×D₁₉₇	—	2	1	no	—
74	C₂²×D₁₉₇	—	2	1	no	—
75	C₂²×D₁₉₇	—	2	1	no	—
76	C₂²×D₁₉₇	C₂²×C₄	1	2	yes	—
77	C₂²×D₁₉₇	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₉₇	C₂²×C₄	1	2	yes	—
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	2	no	—
84	C₂×Dic₁₉₇	—	2	1	no	—
85	C₂×Dic₁₉₇	—	2	1	no	—
86	C₂×Dic₁₉₇	—	2	2	no	—
87	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
90	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
91	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
92	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
93	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
94	C₄×D₁₉₇	—	4	8	no	—
95	C₄×D₁₉₇	C₄²	1	16	yes	complemented
96	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
97	C₂×C₄×C₁₉₇	—	2	1	no	—
98	C₂×C₄×C₁₉₇	—	2	2	no	—
99	C₂×C₄×C₁₉₇	—	2	2	no	—
100	C₂×C₄×C₁₉₇	—	2	1	no	—
101	C₂×C₄×C₁₉₇	—	2	2	no	—
102	C₂×C₄×C₁₉₇	—	2	1	no	—
103	C₂×C₄×C₁₉₇	—	2	2	no	—
104	C₂×C₄×C₁₉₇	—	2	1	no	—
105	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₉₇	C₂²×C₄	1	4	yes	—
108	C₂×C₄×C₁₉₇	C₂²×C₄	1	4	yes	—
109	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
111	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
113	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
114	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
115	C₂²×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×C₁₉₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂²×C₁₉₇	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×D₁₉₇	—	2	2	no	—
120	C₂×D₁₉₇	—	2	2	no	—
121	C₂×D₁₉₇	—	2	2	no	—
122	C₂×D₁₉₇	—	2	2	no	—
123	C₂×D₁₉₇	—	2	2	no	—
124	C₂×D₁₉₇	—	2	2	no	—
125	C₂×D₁₉₇	C₂×C₄²	1	4	yes	—
126	C₄×C₁₉₇	—	2	4	no	—
127	C₄×C₁₉₇	—	2	4	no	—
128	C₄×C₁₉₇	—	2	2	no	—
129	C₄×C₁₉₇	—	2	2	no	—
130	C₄×C₁₉₇	C₂×C₄²	1	4	yes	—
131	Dic₁₉₇	—	2	4	no	—
132	Dic₁₉₇	—	2	2	no	—
133	Dic₁₉₇	—	2	2	no	—
134	Dic₁₉₇	—	2	4	no	—
135	Dic₁₉₇	C₂×C₄²	1	4	yes	—
136	C₂×C₁₉₇	C₂²×C₄²	1	1	yes	characteristic
137	C₂×C₁₉₇	#518	1	1	yes	characteristic
138	C₂×C₁₉₇	#519	1	1	yes	characteristic
139	C₂×C₁₉₇	#520	1	1	yes	characteristic
140	C₂×C₁₉₇	#520	1	1	yes	characteristic
141	C₂×C₁₉₇	#520	1	1	yes	characteristic
142	C₂×C₁₉₇	#520	1	1	yes	characteristic
143	D₁₉₇	—	2	4	no	—
144	C₁₉₇	#2274	1	1	yes	characteristic, complemented, Sylow
145	#2274	—	197	1	no	maximal, Sylow
146	C₂×#117	—	197	1	no	—
147	C₂×#117	—	197	1	no	—
148	C₂²×C₄⋊C₄	—	197	1	no	—
149	C₄×C₄⋊C₄	—	197	1	no	—
150	C₄×C₄⋊C₄	—	197	1	no	—
151	C₄×C₄⋊C₄	—	197	1	no	—
152	C₄×C₄⋊C₄	—	197	2	no	—
153	C₄×C₄⋊C₄	—	197	2	no	—
154	C₄×C₄⋊C₄	D₁₉₇	1	1	yes	characteristic, complemented
155	C₄×C₂²⋊C₄	—	197	2	no	—
156	C₄×C₂²⋊C₄	—	197	2	no	—
157	C₂³×C₄	—	197	2	no	—
158	C₂³×C₄	—	197	1	no	—
159	#117	—	197	8	no	—
160	#117	—	197	8	no	—
161	C₂×C₄⋊C₄	—	197	2	no	—
162	C₂×C₄⋊C₄	—	197	1	no	—
163	C₂×C₄⋊C₄	—	197	1	no	—
164	C₂×C₄⋊C₄	—	197	1	no	—
165	C₂×C₄⋊C₄	—	197	1	no	—
166	C₂×C₄⋊C₄	—	197	8	no	—
167	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	197	1	no	—
170	C₂×C₂²⋊C₄	—	197	2	no	—
171	C₂×C₂²⋊C₄	—	197	1	no	—
172	C₂×C₄²	—	197	2	no	—
173	C₂×C₄²	—	197	2	no	—
174	C₂×C₄²	—	197	2	no	—
175	C₂×C₄²	—	197	1	no	—
176	C₂×C₄²	—	197	2	no	—
177	C₂×C₄²	C₂×D₁₉₇	1	2	yes	—
178	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
179	#95	—	197	2	no	—
180	#95	—	197	2	no	—
181	#95	—	197	2	no	—
182	#95	C₂×D₁₉₇	1	2	yes	—
183	C₂⁴	—	197	1	no	—
184	C₂²×C₄	—	788	2	no	—
185	C₂²×C₄	—	788	2	no	—
186	C₂²×C₄	—	197	8	no	—
187	C₂²×C₄	—	197	1	no	—
188	C₂²×C₄	—	197	1	no	—
189	C₂²×C₄	—	197	2	no	—
190	C₂²×C₄	—	197	2	no	—
191	C₂²×C₄	—	197	1	no	—
192	C₂²×C₄	—	197	4	no	—
193	C₂²×C₄	—	197	4	no	—
194	C₂²×C₄	—	197	4	no	—
195	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
197	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
198	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
200	C₄⋊C₄	—	197	8	no	—
201	C₄⋊C₄	—	197	4	no	—
202	C₄⋊C₄	—	197	4	no	—
203	C₄⋊C₄	—	197	4	no	—
204	C₄⋊C₄	—	197	4	no	—
205	C₄⋊C₄	C₄×D₁₉₇	1	4	yes	—
206	C₄⋊C₄	C₄×D₁₉₇	1	4	yes	complemented
207	C₂²⋊C₄	—	197	8	no	—
208	C₂²⋊C₄	—	197	4	no	—
209	C₂²⋊C₄	—	197	4	no	—
210	C₄²	—	4	2	no	—
211	C₄²	—	788	2	no	—
212	C₄²	—	788	2	no	—
213	C₄²	—	788	2	no	—
214	C₄²	—	197	4	no	—
215	C₄²	—	197	8	no	—
216	C₄²	C₄×D₁₉₇	1	4	yes	complemented
217	C₂³	—	394	1	no	—
218	C₂³	—	394	1	no	—
219	C₂³	—	394	1	no	—
220	C₂³	—	394	1	no	—
221	C₂³	—	197	2	no	—
222	C₂³	—	197	2	no	—
223	C₂³	—	197	2	no	—
224	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	788	8	no	—
234	C₂×C₄	—	394	1	no	—
235	C₂×C₄	—	394	2	no	—
236	C₂×C₄	—	394	1	no	—
237	C₂×C₄	—	394	2	no	—
238	C₂×C₄	—	394	2	no	—
239	C₂×C₄	—	394	1	no	—
240	C₂×C₄	—	394	1	no	—
241	C₂×C₄	—	394	2	no	—
242	C₂×C₄	—	197	2	no	—
243	C₂×C₄	—	197	2	no	—
244	C₂×C₄	—	197	4	no	—
245	C₂×C₄	—	197	4	no	—
246	C₂×C₄	—	197	2	no	—
247	C₂×C₄	—	197	2	no	—
248	C₂×C₄	—	197	2	no	—
249	C₂×C₄	—	197	16	no	—
250	C₂×C₄	C₂×C₄×D₁₉₇	1	4	yes	—
251	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
252	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
253	C₂×C₄	C₂×C₄×D₁₉₇	1	4	yes	—
254	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
255	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
256	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
257	C₂²	—	394	2	no	—
258	C₂²	—	394	2	no	—
259	C₂²	—	394	2	no	—
260	C₂²	—	394	2	no	—
261	C₂²	—	394	2	no	—
262	C₂²	—	394	2	no	—
263	C₂²	—	197	4	no	—
264	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
265	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
266	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
267	C₂²	#20263371	1	1	yes	characteristic
268	C₂²	#20263372	1	1	yes	characteristic
269	C₂²	#20263373	1	1	yes	characteristic
270	C₂²	#20263374	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	394	4	no	—
276	C₄	—	394	2	no	—
277	C₄	—	394	2	no	—
278	C₄	—	394	4	no	—
279	C₄	—	197	4	no	—
280	C₄	C₄²×D₁₉₇	1	4	yes	—
281	C₂	—	394	4	no	—
282	C₂	C₂×C₄²×D₁₉₇	1	1	yes	characteristic
283	C₂	#27161551	1	1	yes	characteristic
284	C₂	#27161555	1	1	yes	characteristic
285	C₂	#27161558	1	1	yes	characteristic
286	C₂	#27161559	1	1	yes	characteristic
287	C₂	#27161560	1	1	yes	characteristic
288	C₂	#27161561	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₇
commutator	C₂×C₁₉₇	C₂²×C₄²
Fitting	C₄×C₄⋊C₄×C₁₉₇	C₂
socle, cosocle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂³×D₁₉₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₉₇
hypocenter	C₁₉₇	#2274
LCS	G ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×Dic₁₉₇	C₂	semidirect
5. C₄⋊C₄×Dic₁₉₇	C₂	semidirect
6. C₄×#15691337	C₂	semidirect
7. C₄×C₄⋊Dic₁₉₇	C₂	semidirect
8. #27160930	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₉₇	C₂	semidirect
14. #20263284	C₄	semidirect
15. #20263283	C₄	semidirect
21. C₄⋊C₄×D₁₉₇	C₄	semidirect
51. C₄⋊Dic₁₉₇	C₂×C₄	semidirect
53. #15691337	C₂×C₄	semidirect
54. #15691337	C₂×C₄	semidirect
61. C₄×Dic₁₉₇	C₂×C₄	semidirect
62. C₄×Dic₁₉₇	C₂×C₄	semidirect
69. C₄⋊C₄×C₁₉₇	C₂×C₄	semidirect
71. C₄²×C₁₉₇	C₂×C₄	semidirect
95. C₄×D₁₉₇	C₄²	semidirect
144. C₁₉₇	#2274	semidirect
154. C₄×C₄⋊C₄	D₁₉₇	semidirect
206. C₄⋊C₄	C₄×D₁₉₇	semidirect
216. C₄²	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
C₂×#20263283	2
C₂×C₄⋊C₄×D₁₉₇	2
C₄⋊C₄×Dic₁₉₇	2
C₄×#15691337	2
C₄×C₄⋊Dic₁₉₇	2
#27160930	2
C₄×#15691335	2
#27160926	2
C₄×C₄⋊C₄×C₁₉₇	2
#2274	197