Subgroups — #38751530

Order 25216.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263359	C₂	1	1	yes	maximal, characteristic
2	#27161547	C₂	1	1	yes	maximal, characteristic
3	C₂×#20263291	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161185	C₂	1	2	yes	maximal, complemented
5	#27161178	C₂	1	2	yes	maximal, complemented
6	#27161173	C₂	1	2	yes	maximal, complemented
7	#27161166	C₂	1	2	yes	maximal, complemented
8	#27161087	C₂	1	1	yes	maximal, characteristic, complemented
9	#27161078	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#428×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691358	C₂²	1	1	yes	characteristic
13	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
14	#20263359	C₄	1	8	yes	complemented
15	C₂×Dic₇₈₈	C₂²	1	2	yes	—
16	C₂×#15691347	C₂²	1	2	yes	—
17	C₂×#15691342	C₂²	1	1	yes	characteristic
18	#20263291	—	2	1	no	—
19	#20263291	—	2	1	no	—
20	#20263291	C₂²	1	2	yes	—
21	#20263291	C₂²	1	1	yes	characteristic
22	#20263291	C₂²	1	2	yes	—
23	#20263291	C₂²	1	1	yes	characteristic
24	#20263288	C₂²	1	1	yes	characteristic
25	#20263288	C₂²	1	1	yes	characteristic
26	#20263286	—	2	1	no	—
27	#20263286	—	2	1	no	—
28	#20263283	—	2	1	no	—
29	C₂×#15691337	—	2	1	no	—
30	C₂×C₄×Dic₁₉₇	—	2	1	no	—
31	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
32	C₄²×D₁₉₇	—	2	1	no	—
33	#20263249	C₂²	1	2	yes	—
34	#20263249	C₂²	1	2	yes	—
35	#20263249	C₂²	1	2	yes	—
36	#20263246	C₂²	1	2	yes	—
37	#20263236	C₂²	1	2	yes	—
38	C₈×Dic₁₉₇	C₂²	1	2	yes	—
39	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
40	#117×C₁₉₇	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
42	#107×C₁₉₇	C₂²	1	2	yes	—
43	#106×C₁₉₇	C₂²	1	2	yes	—
44	#15691358	—	2	2	no	—
45	#15691358	C₂×C₄	1	4	yes	—
46	C₂×Dic₃₉₄	—	2	1	no	—
47	C₂×Dic₃₉₄	—	2	2	no	—
48	C₂×Dic₃₉₄	C₂×C₄	1	4	yes	complemented
49	C₂×Dic₃₉₄	C₂³	1	2	yes	—
50	C₂×Dic₃₉₄	C₂³	1	1	yes	characteristic
51	C₂×D₇₈₈	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
53	C₂²×Dic₁₉₇	—	2	1	no	—
54	C₂²×Dic₁₉₇	D₄	1	1	yes	characteristic
55	C₂²×Dic₁₉₇	D₄	1	1	yes	characteristic
56	C₂×C₄×D₁₉₇	—	2	1	no	—
57	C₂×C₄×D₁₉₇	—	2	1	no	—
58	C₂×C₄×D₁₉₇	—	2	1	no	—
59	Dic₇₈₈	C₂×C₄	1	8	yes	complemented
60	#15691347	C₂×C₄	1	8	yes	complemented
61	#15691342	—	2	2	no	—
62	C₂×#13096802	C₂³	1	2	yes	—
63	#15691337	—	2	2	no	—
64	#15691337	—	2	2	no	—
65	#15691337	—	2	1	no	—
66	#15691337	—	2	1	no	—
67	#15691335	—	2	1	no	—
68	#15691335	—	2	1	no	—
69	C₄×Dic₁₉₇	—	2	1	no	—
70	C₄×Dic₁₉₇	—	2	1	no	—
71	C₄×Dic₁₉₇	—	2	2	no	—
72	C₄×Dic₁₉₇	—	2	2	no	—
73	C₄×Dic₁₉₇	D₄	1	1	yes	characteristic
74	C₄×Dic₁₉₇	D₄	1	1	yes	characteristic
75	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
76	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₉₇	—	2	1	no	—
78	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
79	M₁₆×C₁₉₇	C₂×C₄	1	4	yes	complemented
80	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
81	C₄⋊C₄×C₁₉₇	—	2	1	no	—
82	C₄⋊C₄×C₁₉₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₉₇	C₂³	1	2	yes	—
84	C₄⋊C₄×C₁₉₇	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₉₇	—	2	1	no	—
86	C₄²×C₁₉₇	—	2	1	no	—
87	C₂²×D₁₉₇	—	2	1	no	—
88	Dic₃₉₄	—	2	4	no	—
89	Dic₃₉₄	—	2	2	no	—
90	Dic₃₉₄	—	2	1	no	—
91	Dic₃₉₄	C₂²×C₄	1	4	yes	—
92	Dic₃₉₄	C₂²×C₄	1	2	yes	—
93	D₇₈₈	—	2	1	no	—
94	D₇₈₈	C₂²×C₄	1	2	yes	—
95	C₂²⋊D₁₉₇	—	2	4	no	—
96	C₂×Dic₁₉₇	—	2	4	no	—
97	C₂×Dic₁₉₇	—	2	1	no	—
98	C₂×Dic₁₉₇	—	2	2	no	—
99	C₂×Dic₁₉₇	—	2	2	no	—
100	C₂×Dic₁₉₇	—	4	1	no	—
101	C₂×Dic₁₉₇	C₂×D₄	1	1	yes	characteristic
102	C₂×Dic₁₉₇	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₁₉₇	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₁₉₇	C₂×D₄	1	1	yes	characteristic
105	C₄×D₁₉₇	—	2	4	no	—
106	C₄×D₁₉₇	—	4	2	no	—
107	C₄×D₁₉₇	—	4	2	no	—
108	#13096802	—	2	2	no	—
109	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉₇	—	2	1	no	—
111	C₂×C₄×C₁₉₇	—	2	1	no	—
112	C₂×C₄×C₁₉₇	—	2	2	no	—
113	C₂×C₄×C₁₉₇	—	4	2	no	—
114	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉₇	#41	1	2	yes	—
117	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
118	C₈×C₁₉₇	C₂²×C₄	1	4	yes	—
119	C₂²×C₁₉₇	—	2	1	no	—
120	C₂²×C₁₉₇	—	2	1	no	—
121	C₂²×C₁₉₇	C₄×D₄	1	2	yes	—
122	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₉₇	—	2	2	no	—
124	C₂×D₁₉₇	—	4	1	no	—
125	C₂×D₁₉₇	—	4	1	no	—
126	C₄×C₁₉₇	—	4	1	no	—
127	C₄×C₁₉₇	—	4	2	no	—
128	C₄×C₁₉₇	—	4	1	no	—
129	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
130	C₄×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
132	Dic₁₉₇	—	2	2	no	—
133	Dic₁₉₇	—	2	4	no	—
134	Dic₁₉₇	—	2	1	no	—
135	Dic₁₉₇	—	2	1	no	—
136	Dic₁₉₇	—	2	1	no	—
137	Dic₁₉₇	#137	1	2	yes	complemented
138	C₂×C₁₉₇	—	2	2	no	—
139	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₉₇	C₂×#137	1	1	yes	characteristic
141	C₂×C₁₉₇	#575	1	1	yes	characteristic
142	D₁₉₇	—	4	2	no	—
143	C₁₉₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
144	C₄×#137	—	197	1	no	maximal, Sylow
145	C₂×#137	—	197	1	no	—
146	C₄×#41	—	197	1	no	—
147	C₂×C₄×Q₈	—	197	1	no	—
148	#441	—	197	2	no	—
149	#440	—	197	2	no	—
150	C₄×Q₁₆	—	197	2	no	—
151	C₄×SD₁₆	—	197	2	no	—
152	#428	D₁₉₇	1	1	yes	characteristic, complemented
153	#419	—	197	1	no	—
154	#417	—	197	1	no	—
155	C₄×M₁₆	—	197	1	no	—
156	C₂×#41	—	197	1	no	—
157	C₂²×Q₈	—	197	1	no	—
158	#137	—	197	8	no	—
159	C₂×Q₁₆	—	197	2	no	—
160	C₂×SD₁₆	—	197	2	no	—
161	C₂×M₁₆	—	197	1	no	—
162	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
163	C₄×Q₈	—	394	1	no	—
164	C₄×Q₈	—	394	1	no	—
165	C₄×Q₈	—	197	2	no	—
166	C₄×Q₈	—	197	2	no	—
167	C₄×Q₈	—	197	1	no	—
168	C₄×Q₈	—	197	1	no	—
169	C₄×D₄	—	394	1	no	—
170	C₄×D₄	—	394	1	no	—
171	C₄×D₄	—	197	1	no	—
172	C₄×D₄	—	197	1	no	—
173	#117	—	394	1	no	—
174	#117	C₂×D₁₉₇	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	394	1	no	—
176	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
177	C₂×C₄²	—	394	1	no	—
178	C₂×C₄²	—	394	1	no	—
179	C₂×C₄²	—	197	1	no	—
180	#107	C₂×D₁₉₇	1	2	yes	—
181	#106	C₂×D₁₉₇	1	2	yes	—
182	#103	—	197	2	no	—
183	#103	—	197	2	no	—
184	#103	—	197	2	no	—
185	#102	—	197	2	no	—
186	#97	—	197	2	no	—
187	C₄×C₈	—	197	2	no	—
188	#41	—	394	2	no	—
189	#41	—	197	4	no	—
190	C₂×Q₈	—	394	1	no	—
191	C₂×Q₈	—	394	2	no	—
192	C₂×Q₈	—	197	4	no	—
193	C₂×Q₈	—	197	2	no	—
194	C₂×Q₈	—	197	1	no	—
195	C₂×D₄	—	394	1	no	—
196	C₂×D₄	—	197	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	394	1	no	—
199	C₂²×C₄	—	394	1	no	—
200	C₂²×C₄	—	394	1	no	—
201	C₂²×C₄	—	394	1	no	—
202	C₂²×C₄	—	197	1	no	—
203	C₂²×C₄	—	197	1	no	—
204	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
205	Q₁₆	—	197	8	no	—
206	SD₁₆	—	197	8	no	—
207	M₁₆	—	394	2	no	—
208	M₁₆	C₄×D₁₉₇	1	4	yes	complemented
209	C₂×C₈	—	197	2	no	—
210	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	394	2	no	—
213	C₄⋊C₄	—	394	2	no	—
214	C₄⋊C₄	—	394	1	no	—
215	C₄⋊C₄	—	394	1	no	—
216	C₄⋊C₄	C₂²×D₁₉₇	1	2	yes	—
217	C₄⋊C₄	C₂²×D₁₉₇	1	1	yes	characteristic
218	C₄⋊C₄	C₂²×D₁₉₇	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	394	1	no	—
221	C₂²⋊C₄	—	394	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	394	1	no	—
224	C₄²	—	394	1	no	—
225	C₄²	—	394	2	no	—
226	C₄²	—	394	2	no	—
227	C₄²	—	197	1	no	—
228	C₄²	—	197	1	no	—
229	C₄²	—	197	1	no	—
230	C₄²	—	197	1	no	—
231	C₂³	—	394	1	no	—
232	C₂³	D₄×D₁₉₇	1	1	yes	characteristic
233	Q₈	—	394	4	no	—
234	Q₈	—	394	2	no	—
235	Q₈	—	394	1	no	—
236	Q₈	—	197	4	no	—
237	Q₈	—	197	2	no	—
238	D₄	—	394	4	no	—
239	D₄	—	394	1	no	—
240	D₄	—	197	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	788	2	no	—
246	C₂×C₄	—	788	2	no	—
247	C₂×C₄	—	788	1	no	—
248	C₂×C₄	—	394	4	no	—
249	C₂×C₄	—	394	1	no	—
250	C₂×C₄	—	394	2	no	—
251	C₂×C₄	—	394	2	no	—
252	C₂×C₄	—	394	4	no	—
253	C₂×C₄	—	197	1	no	—
254	C₂×C₄	—	197	1	no	—
255	C₂×C₄	—	197	1	no	—
256	C₂×C₄	—	197	1	no	—
257	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
258	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
259	C₂×C₄	#15691359	1	2	yes	—
260	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
261	C₈	—	394	2	no	—
262	C₈	C₂×C₄×D₁₉₇	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	788	1	no	—
266	C₂²	—	788	1	no	—
267	C₂²	—	394	2	no	—
268	C₂²	#20263288	1	2	yes	—
269	C₂²	C₂×D₄×D₁₉₇	1	1	yes	characteristic
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	394	1	no	—
274	C₄	—	394	1	no	—
275	C₄	—	394	1	no	—
276	C₄	—	394	2	no	—
277	C₄	—	394	4	no	—
278	C₄	—	197	2	no	—
279	C₄	#20263288	1	2	yes	—
280	C₄	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
281	C₄	C₂×#15691359	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	788	2	no	—
284	C₂	C₂×#20263288	1	1	yes	characteristic
285	C₂	#137×D₁₉₇	1	1	yes	characteristic
286	C₂	#27161875	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₇
commutator	C₄×C₁₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#428×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
prime radical	C₂×#15691358	C₂²
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	#428	D₁₉₇
hypocenter	C₁₉₇	C₄×#137
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
3. C₂×#20263291	C₂	semidirect
4. #27161185	C₂	semidirect
5. #27161178	C₂	semidirect
6. #27161173	C₂	semidirect
7. #27161166	C₂	semidirect
8. #27161087	C₂	semidirect
10. M₁₆×Dic₁₉₇	C₂	semidirect
11. #428×C₁₉₇	C₂	semidirect
14. #20263359	C₄	semidirect
48. C₂×Dic₃₉₄	C₂×C₄	semidirect
59. Dic₇₈₈	C₂×C₄	semidirect
60. #15691347	C₂×C₄	semidirect
79. M₁₆×C₁₉₇	C₂×C₄	semidirect
137. Dic₁₉₇	#137	semidirect
143. C₁₉₇	C₄×#137	semidirect
152. #428	D₁₉₇	semidirect
208. M₁₆	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×#20263359	2
#27161547	2
C₂×#20263291	2
#27161185	2
#27161178	2
#27161173	2
#27161166	2
#27161087	2
#27161078	2
M₁₆×Dic₁₉₇	2
#428×C₁₉₇	2
C₄×#137	197