Subgroups — #27150644

Order 12416.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261532	C₂	1	1	yes	maximal, characteristic
2	#20261500	C₂	1	2	yes	maximal, complemented
3	#20260858	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260854	C₂	1	2	yes	maximal, complemented
5	#20260852	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260851	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260850	C₂	1	2	yes	maximal, complemented
8	#20260782	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₉₇	C₂	1	2	yes	maximal, complemented
10	#20260687	C₂	1	1	yes	maximal, characteristic, complemented
11	#444×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₉₇	C₂²	1	2	yes	—
13	#15690842	C₄	1	2	yes	complemented
14	#15690839	C₄	1	2	yes	complemented
15	#15690837	C₄	1	2	yes	complemented
16	#15690834	C₄	1	2	yes	complemented
17	#15690832	C₂²	1	1	yes	characteristic
18	#15690830	C₂²	1	2	yes	—
19	#15690818	—	2	2	no	—
20	#15690818	C₂²	1	2	yes	complemented
21	#15690817	C₂²	1	2	yes	complemented
22	#15690816	—	2	2	no	—
23	#15690816	C₂²	1	2	yes	complemented
24	C₂×#13095494	—	2	2	no	—
25	C₂×#13095494	C₂²	1	1	yes	characteristic
26	C₂×#13095494	C₂²	1	1	yes	characteristic
27	M₁₆×D₉₇	—	2	2	no	—
28	M₁₆×D₉₇	C₂²	1	2	yes	—
29	C₄²×D₉₇	C₂²	1	1	yes	characteristic
30	#15690746	C₂²	1	1	yes	characteristic, complemented
31	#15690745	C₂²	1	2	yes	complemented
32	#15690738	C₂²	1	2	yes	complemented
33	#15690737	C₂²	1	2	yes	complemented
34	#15690736	C₂²	1	2	yes	complemented
35	#15690721	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
38	#135×C₉₇	C₂²	1	1	yes	characteristic
39	#131×C₉₇	C₂²	1	2	yes	complemented
40	#108×C₉₇	C₂²	1	1	yes	characteristic, complemented
41	#104×C₉₇	C₂²	1	2	yes	complemented
42	#97×C₉₇	C₂²	1	1	yes	characteristic, complemented
43	#13095512	C₂×C₄	1	2	yes	—
44	#13095511	—	2	2	no	—
45	#13095511	C₂³	1	2	yes	—
46	#13095510	—	2	2	no	—
47	#13095510	C₂×C₄	1	2	yes	—
48	Q₈×D₉₇	C₂×C₄	1	2	yes	—
49	D₄×D₉₇	—	2	2	no	—
50	D₄×D₉₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₉₇	—	2	2	no	—
52	C₂×C₄×D₉₇	—	2	1	no	—
53	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
54	Dic₃₈₈	C₂×C₄	1	1	yes	characteristic, complemented
55	#13095501	C₂×C₄	1	2	yes	complemented
56	#13095500	C₂×C₄	1	2	yes	complemented
57	#13095499	C₂×C₄	1	2	yes	complemented
58	#13095498	C₂×C₄	1	2	yes	complemented
59	D₇₇₆	C₂×C₄	1	1	yes	characteristic, complemented
60	#13095496	C₂×C₄	1	2	yes	complemented
61	#13095495	—	2	2	no	—
62	#13095495	C₂³	1	2	yes	—
63	#13095494	—	2	2	no	—
64	#13095494	—	2	2	no	—
65	#13095494	—	2	2	no	—
66	#13095494	C₂×C₄	1	2	yes	—
67	#13095494	C₂×C₄	1	2	yes	—
68	#13095494	D₄	1	2	yes	—
69	#13095494	D₄	1	2	yes	—
70	C₂×#11715895	—	2	2	no	—
71	C₂×#11715895	C₂³	1	1	yes	characteristic
72	C₂×#11715895	C₂³	1	1	yes	characteristic
73	C₈×D₉₇	—	2	2	no	—
74	C₈×D₉₇	—	2	2	no	—
75	C₈×D₉₇	—	2	2	no	—
76	C₄×Dic₉₇	—	2	1	no	—
77	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
78	#41×C₉₇	C₂³	1	2	yes	—
79	Q₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
80	SD₁₆×C₉₇	C₂×C₄	1	2	yes	complemented
81	D₈×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
82	M₁₆×C₉₇	—	2	2	no	—
83	M₁₆×C₉₇	C₂³	1	2	yes	—
84	C₂×C₈×C₉₇	—	2	2	no	—
85	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
87	C₄²×C₉₇	C₂³	1	1	yes	characteristic
88	C₂²×D₉₇	—	2	2	no	—
89	C₂²×D₉₇	#41	1	1	yes	characteristic
90	Q₈×C₉₇	C₂²×C₄	1	2	yes	—
91	D₄×C₉₇	—	2	2	no	—
92	D₄×C₉₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₉₇	—	2	2	no	—
94	C₂×C₄×C₉₇	—	2	1	no	—
95	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₉₇	—	2	2	no	—
97	C₂²⋊D₉₇	—	2	2	no	—
98	C₂²⋊D₉₇	—	2	2	no	—
99	C₂×Dic₉₇	—	2	1	no	—
100	C₂×Dic₉₇	—	2	2	no	—
101	C₂×Dic₉₇	#41	1	1	yes	characteristic
102	D₃₈₈	—	2	2	no	—
103	D₃₈₈	C₂²×C₄	1	2	yes	—
104	C₄×D₉₇	—	2	2	no	—
105	C₄×D₉₇	—	2	2	no	—
106	C₄×D₉₇	—	2	2	no	—
107	C₄×D₉₇	—	4	2	no	—
108	C₄×D₉₇	C₂²×C₄	1	2	yes	—
109	C₄×D₉₇	C₂×D₄	1	2	yes	—
110	Dic₁₉₄	—	2	2	no	—
111	Dic₁₉₄	C₂²×C₄	1	2	yes	—
112	C₈×C₉₇	—	2	2	no	—
113	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
115	C₈×C₉₇	C₂×D₄	1	2	yes	—
116	#11715895	—	2	2	no	—
117	#11715895	C₂²×C₄	1	2	yes	—
118	#11715895	C₂×D₄	1	1	yes	characteristic
119	#11715895	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₉₇	—	2	2	no	—
121	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
122	C₂×D₉₇	—	2	2	no	—
123	C₂×D₉₇	—	4	1	no	—
124	C₂×D₉₇	—	8	2	no	—
125	C₂×D₉₇	C₄×D₄	1	2	yes	—
126	C₄×C₉₇	—	2	2	no	—
127	C₄×C₉₇	—	4	1	no	—
128	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
130	Dic₉₇	—	2	2	no	—
131	Dic₉₇	—	2	2	no	—
132	Dic₉₇	C₄×D₄	1	2	yes	—
133	C₂×C₉₇	—	2	1	no	—
134	C₂×C₉₇	—	4	2	no	—
135	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
136	D₉₇	—	2	2	no	—
137	D₉₇	—	4	2	no	—
138	#2956	—	97	1	no	maximal, Sylow
139	C₉₇	#2956	1	1	yes	characteristic, complemented, Sylow
140	#575	—	97	1	no	—
141	#568	—	97	2	no	—
142	#444	—	97	2	no	—
143	#444	—	97	1	no	—
144	#444	D₉₇	1	1	yes	characteristic, complemented
145	#443	—	97	1	no	—
146	#443	—	97	2	no	—
147	#443	—	97	1	no	—
148	#430	—	97	1	no	—
149	C₂×#104	—	97	2	no	—
150	C₄×M₁₆	—	97	1	no	—
151	C₂×#41	—	97	2	no	—
152	#137	—	97	2	no	—
153	#137	—	97	2	no	—
154	AGL(1,ℤ/8)	—	97	2	no	—
155	AGL(1,ℤ/8)	—	97	2	no	—
156	#135	—	97	2	no	—
157	#135	—	97	1	no	—
158	#135	C₂×D₉₇	1	1	yes	characteristic
159	#131	—	97	2	no	—
160	#131	—	97	2	no	—
161	#131	—	97	2	no	—
162	#131	—	194	2	no	—
163	#131	—	194	2	no	—
164	#131	C₂×D₉₇	1	2	yes	complemented
165	C₂×M₁₆	—	97	1	no	—
166	C₂×M₁₆	—	97	1	no	—
167	C₂×M₁₆	—	97	2	no	—
168	C₂×M₁₆	—	194	2	no	—
169	C₂×M₁₆	—	194	2	no	—
170	C₂×C₄²	—	97	1	no	—
171	#108	—	97	2	no	—
172	#108	—	97	1	no	—
173	#108	C₂×D₉₇	1	1	yes	characteristic, complemented
174	#104	—	97	2	no	—
175	#104	—	97	2	no	—
176	#104	—	97	2	no	—
177	#104	C₂×D₉₇	1	2	yes	complemented
178	#97	—	97	1	no	—
179	#97	C₂×D₉₇	1	1	yes	characteristic, complemented
180	C₄×C₈	—	97	1	no	—
181	C₄×C₈	—	97	1	no	—
182	#41	—	97	2	no	—
183	#41	—	97	2	no	—
184	#41	—	97	2	no	—
185	#41	—	194	2	no	—
186	#41	—	194	2	no	—
187	#41	C₂²×D₉₇	1	2	yes	—
188	C₂×Q₈	—	97	2	no	—
189	C₂×D₄	—	97	2	no	—
190	C₂×D₄	—	194	2	no	—
191	C₂²×C₄	—	97	1	no	—
192	C₂²×C₄	—	194	2	no	—
193	C₂²×C₄	—	194	1	no	—
194	Q₁₆	—	97	2	no	—
195	Q₁₆	—	97	1	no	—
196	Q₁₆	C₄×D₉₇	1	1	yes	characteristic, complemented
197	SD₁₆	—	97	2	no	—
198	SD₁₆	—	97	2	no	—
199	SD₁₆	—	97	2	no	—
200	SD₁₆	C₄×D₉₇	1	2	yes	complemented
201	D₈	—	97	2	no	—
202	D₈	—	97	1	no	—
203	D₈	C₄×D₉₇	1	1	yes	characteristic, complemented
204	M₁₆	—	2	2	no	—
205	M₁₆	—	97	2	no	—
206	M₁₆	—	97	2	no	—
207	M₁₆	—	97	2	no	—
208	M₁₆	—	97	2	no	—
209	M₁₆	—	97	2	no	—
210	M₁₆	—	194	2	no	—
211	M₁₆	—	194	2	no	—
212	M₁₆	—	194	2	no	—
213	M₁₆	—	194	2	no	—
214	M₁₆	C₂²×D₉₇	1	2	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	97	1	no	—
217	C₂×C₈	—	97	1	no	—
218	C₂×C₈	—	194	2	no	—
219	C₂×C₈	—	194	2	no	—
220	C₂×C₈	—	194	2	no	—
221	C₂×C₈	—	194	2	no	—
222	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
224	C₄²	—	97	1	no	—
225	C₄²	—	194	1	no	—
226	C₄²	C₂²×D₉₇	1	1	yes	characteristic
227	C₂³	—	97	1	no	—
228	C₂³	—	194	2	no	—
229	Q₈	—	97	2	no	—
230	Q₈	—	194	2	no	—
231	Q₈	C₂×C₄×D₉₇	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	97	2	no	—
234	D₄	—	194	2	no	—
235	D₄	—	194	2	no	—
236	D₄	—	194	2	no	—
237	D₄	—	194	2	no	—
238	D₄	C₂×C₄×D₉₇	1	2	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	97	1	no	—
242	C₂×C₄	—	97	2	no	—
243	C₂×C₄	—	97	2	no	—
244	C₂×C₄	—	388	2	no	—
245	C₂×C₄	—	194	2	no	—
246	C₂×C₄	—	194	2	no	—
247	C₂×C₄	—	194	2	no	—
248	C₂×C₄	—	194	1	no	—
249	C₂×C₄	—	194	2	no	—
250	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	97	2	no	—
253	C₈	—	97	1	no	—
254	C₈	—	97	1	no	—
255	C₈	—	194	2	no	—
256	C₈	C₂×C₄×D₉₇	1	1	yes	characteristic
257	C₈	C₂×C₄×D₉₇	1	1	yes	characteristic
258	C₈	D₄×D₉₇	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	776	2	no	—
261	C₂²	—	97	2	no	—
262	C₂²	—	388	1	no	—
263	C₂²	—	194	2	no	—
264	C₂²	#41×D₉₇	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	97	2	no	—
268	C₄	—	194	2	no	—
269	C₄	—	194	2	no	—
270	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
271	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	388	2	no	—
275	C₂	—	194	2	no	—
276	C₂	C₄×D₄×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₉₇
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#444×C₉₇	C₂
socle	C₂×C₉₇	C₂×C₄×D₄
prime radical	#20261532	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#444	D₉₇
hypocenter	C₉₇	#2956
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #20261500	C₂	semidirect
3. #20260858	C₂	semidirect
4. #20260854	C₂	semidirect
5. #20260852	C₂	semidirect
6. #20260851	C₂	semidirect
7. #20260850	C₂	semidirect
8. #20260782	C₂	semidirect
9. #104×D₉₇	C₂	semidirect
10. #20260687	C₂	semidirect
11. #444×C₉₇	C₂	semidirect
13. #15690842	C₄	semidirect
14. #15690839	C₄	semidirect
15. #15690837	C₄	semidirect
16. #15690834	C₄	semidirect
20. #15690818	C₂²	semidirect
21. #15690817	C₂²	semidirect
23. #15690816	C₂²	semidirect
30. #15690746	C₂²	semidirect
31. #15690745	C₂²	semidirect
32. #15690738	C₂²	semidirect
33. #15690737	C₂²	semidirect
34. #15690736	C₂²	semidirect
35. #15690721	C₂²	semidirect
36. C₈×Dic₉₇	C₂²	semidirect
37. C₈×Dic₉₇	C₂²	semidirect
39. #131×C₉₇	C₂²	semidirect
40. #108×C₉₇	C₂²	semidirect
41. #104×C₉₇	C₂²	semidirect
42. #97×C₉₇	C₂²	semidirect
54. Dic₃₈₈	C₂×C₄	semidirect
55. #13095501	C₂×C₄	semidirect
56. #13095500	C₂×C₄	semidirect
57. #13095499	C₂×C₄	semidirect
58. #13095498	C₂×C₄	semidirect
59. D₇₇₆	C₂×C₄	semidirect
60. #13095496	C₂×C₄	semidirect
79. Q₁₆×C₉₇	C₂×C₄	semidirect
80. SD₁₆×C₉₇	C₂×C₄	semidirect
81. D₈×C₉₇	C₂×C₄	semidirect
139. C₉₇	#2956	semidirect
144. #444	D₉₇	semidirect
164. #131	C₂×D₉₇	semidirect
173. #108	C₂×D₉₇	semidirect
177. #104	C₂×D₉₇	semidirect
179. #97	C₂×D₉₇	semidirect
196. Q₁₆	C₄×D₉₇	semidirect
200. SD₁₆	C₄×D₉₇	semidirect
203. D₈	C₄×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261532	2
#20261500	2
#20260858	2
#20260854	2
#20260852	2
#20260851	2
#20260850	2
#20260782	2
#104×D₉₇	2
#20260687	2
#444×C₉₇	2
#2956	97