Subgroups — #27150568

Order 12416.

506 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	C₂×#15690836	C₂	1	1	yes	maximal, characteristic
2	#20261215	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260845	C₂	1	2	yes	maximal, complemented
4	#20260835	C₂	1	2	yes	maximal, complemented
5	C₄×#13095499	C₂	1	2	yes	maximal, complemented
6	C₄×#13095496	C₂	1	2	yes	maximal, complemented
7	#20260778	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260729	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15690731	C₂	1	1	yes	maximal, characteristic, complemented
10	#20260691	C₂	1	1	yes	maximal, characteristic, complemented
11	#518×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095511	C₂²	1	1	yes	characteristic
13	#15690836	C₄	1	8	yes	complemented
14	C₂×#13095499	C₂²	1	2	yes	—
15	C₂×#13095496	C₂²	1	2	yes	—
16	C₂×#13095495	C₂²	1	1	yes	characteristic
17	C₂²×#11715895	C₂²	1	1	yes	characteristic
18	C₄×Dic₁₉₄	C₂²	1	2	yes	complemented
19	C₄×D₃₈₈	C₂²	1	2	yes	complemented
20	#15690770	—	2	2	no	—
21	#15690768	—	2	1	no	—
22	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₉₇	—	2	1	no	—
24	#15690743	C₂²	1	2	yes	complemented
25	#15690741	C₂²	1	2	yes	complemented
26	#15690733	C₂²	1	1	yes	characteristic, complemented
27	#15690733	C₂²	1	1	yes	characteristic, complemented
28	#15690731	C₂²	1	2	yes	—
29	#15690731	C₂²	1	2	yes	—
30	#15690730	C₂²	1	1	yes	characteristic, complemented
31	#15690730	C₂²	1	1	yes	characteristic, complemented
32	#15690721	C₂²	1	2	yes	complemented
33	C₄×#11715895	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₉₇	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₉₇	—	2	1	no	—
36	C₄×D₄×C₉₇	—	2	1	no	—
37	C₄×D₄×C₉₇	C₂²	1	2	yes	—
38	C₄×D₄×C₉₇	C₂²	1	2	yes	—
39	#117×C₉₇	C₂²	1	1	yes	characteristic
40	#117×C₉₇	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₉₇	—	2	2	no	—
42	#13095511	—	2	2	no	—
43	#13095511	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₁₉₄	C₂³	1	1	yes	characteristic
45	C₂×D₃₈₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₉₇	—	2	1	no	—
47	C₂²×Dic₉₇	—	2	1	no	—
48	C₂×C₄×D₉₇	—	2	1	no	—
49	C₂×C₄×D₉₇	—	2	2	no	—
50	#13095499	C₂×C₄	1	8	yes	complemented
51	#13095496	C₂×C₄	1	8	yes	complemented
52	#13095495	C₂×C₄	1	4	yes	complemented
53	C₂×#11715895	—	2	2	no	—
54	C₂×#11715895	C₂³	1	1	yes	characteristic
55	C₂×#11715895	C₂³	1	1	yes	characteristic
56	C₂×#11715895	C₂³	1	2	yes	—
57	C₄⋊Dic₉₇	—	2	1	no	—
58	C₄⋊Dic₉₇	C₂³	1	2	yes	—
59	#13095490	—	2	2	no	—
60	#13095488	—	2	2	no	—
61	C₄×Dic₉₇	—	2	2	no	—
62	C₂⁴×C₉₇	—	2	1	no	—
63	C₂×D₄×C₉₇	—	2	1	no	—
64	C₂×D₄×C₉₇	—	2	2	no	—
65	C₂×D₄×C₉₇	C₂×C₄	1	4	yes	complemented
66	C₂×D₄×C₉₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₉₇	—	2	2	no	—
68	C₂²×C₄×C₉₇	—	2	2	no	—
69	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₉₇	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₉₇	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₉₇	—	2	1	no	—
75	C₂²⋊C₄×C₉₇	—	2	2	no	—
76	C₂²⋊C₄×C₉₇	—	2	2	no	—
77	C₂²⋊C₄×C₉₇	D₄	1	2	yes	—
78	C₄²×C₉₇	—	2	1	no	—
79	C₄²×C₉₇	C₂³	1	2	yes	—
80	C₂³×C₉₇	—	2	2	no	—
81	C₂³×C₉₇	—	2	4	no	—
82	C₂³×C₉₇	—	8	1	no	—
83	C₂³×C₉₇	—	8	1	no	—
84	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₉₇	—	2	1	no	—
86	D₄×C₉₇	—	2	4	no	—
87	D₄×C₉₇	—	2	2	no	—
88	D₄×C₉₇	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₉₇	—	2	1	no	—
90	C₂×C₄×C₉₇	—	2	1	no	—
91	C₂×C₄×C₉₇	—	8	2	no	—
92	C₂×C₄×C₉₇	—	8	2	no	—
93	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉₇	#41	1	2	yes	—
98	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₉₇	—	2	4	no	—
100	C₂×Dic₉₇	—	2	1	no	—
101	C₂×Dic₉₇	—	2	2	no	—
102	C₂×Dic₉₇	—	8	1	no	—
103	D₃₈₈	—	2	1	no	—
104	D₃₈₈	C₂²×C₄	1	2	yes	—
105	C₄×D₉₇	—	2	4	no	—
106	C₄×D₉₇	—	8	2	no	—
107	Dic₁₉₄	—	2	1	no	—
108	Dic₁₉₄	C₂²×C₄	1	2	yes	—
109	#11715895	—	2	1	no	—
110	#11715895	—	2	1	no	—
111	#11715895	C₂²×C₄	1	4	yes	—
112	C₂²×C₉₇	—	2	1	no	—
113	C₂²×C₉₇	—	2	1	no	—
114	C₂²×C₉₇	—	2	4	no	—
115	C₂²×C₉₇	—	4	2	no	—
116	C₂²×C₉₇	—	4	2	no	—
117	C₂²×C₉₇	—	8	4	no	—
118	C₂²×C₉₇	C₄×D₄	1	2	yes	—
119	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₉₇	—	2	2	no	—
121	C₂×D₉₇	—	4	1	no	—
122	C₂×D₉₇	—	4	1	no	—
123	C₄×C₉₇	—	2	2	no	—
124	C₄×C₉₇	—	2	2	no	—
125	C₄×C₉₇	—	4	1	no	—
126	C₄×C₉₇	—	4	1	no	—
127	C₄×C₉₇	C₄×D₄	1	2	yes	—
128	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
130	Dic₉₇	—	2	2	no	—
131	Dic₉₇	—	4	2	no	—
132	C₂×C₉₇	—	2	2	no	—
133	C₂×C₉₇	—	4	4	no	—
134	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₉₇	#576	1	1	yes	characteristic
136	C₂×C₉₇	#577	1	1	yes	characteristic
137	D₉₇	—	4	2	no	—
138	#2946	—	97	1	no	maximal, Sylow
139	C₉₇	#2946	1	1	yes	characteristic, complemented, Sylow
140	C₂×AGL(1,ℤ/8)	—	97	1	no	—
141	#520	—	97	1	no	—
142	#518	D₉₇	1	1	yes	characteristic, complemented
143	#442	—	97	2	no	—
144	#440	—	97	2	no	—
145	C₄×SD₁₆	—	97	2	no	—
146	C₄×D₈	—	97	2	no	—
147	#428	—	97	1	no	—
148	#416	—	97	1	no	—
149	C₂×#102	—	97	1	no	—
150	#405	—	97	1	no	—
151	C₂×#41	—	97	1	no	—
152	C₂²×D₄	C₂×D₉₇	1	1	yes	characteristic
153	AGL(1,ℤ/8)	—	97	8	no	—
154	C₂×SD₁₆	—	97	2	no	—
155	C₂×D₈	—	97	2	no	—
156	C₂×M₁₆	—	97	1	no	—
157	C₂²×C₈	—	97	1	no	—
158	C₄×Q₈	—	97	2	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	97	2	no	—
162	C₄×D₄	—	194	2	no	—
163	C₄×D₄	C₂×D₉₇	1	2	yes	—
164	C₄×D₄	C₂×D₉₇	1	2	yes	—
165	#117	—	194	1	no	—
166	#117	C₂×D₉₇	1	1	yes	characteristic
167	#117	C₂×D₉₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	97	1	no	—
169	C₂×C₄⋊C₄	—	194	1	no	—
170	C₂×C₂²⋊C₄	—	2	2	no	—
171	#107	—	97	2	no	—
172	#106	—	97	2	no	—
173	#103	—	97	1	no	—
174	#103	—	97	1	no	—
175	#102	—	97	2	no	—
176	#102	—	97	2	no	—
177	#102	—	97	1	no	—
178	#102	—	97	1	no	—
179	#97	—	97	2	no	—
180	C₄×C₈	—	97	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	97	4	no	—
183	#41	—	194	2	no	—
184	C₂×Q₈	—	97	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	97	1	no	—
188	C₂×D₄	—	194	1	no	—
189	C₂×D₄	C₄×D₉₇	1	4	yes	complemented
190	C₂×D₄	C₂²×D₉₇	1	2	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	194	1	no	—
194	C₂²×C₄	—	194	2	no	—
195	C₂²×C₄	—	194	1	no	—
196	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
197	SD₁₆	—	97	8	no	—
198	D₈	—	97	8	no	—
199	M₁₆	—	97	4	no	—
200	C₂×C₈	—	97	2	no	—
201	C₂×C₈	—	97	1	no	—
202	C₂×C₈	—	97	1	no	—
203	C₂×C₈	—	194	2	no	—
204	C₄⋊C₄	—	97	2	no	—
205	C₄⋊C₄	—	194	2	no	—
206	C₄⋊C₄	—	194	1	no	—
207	C₄⋊C₄	C₂²⋊D₉₇	1	1	yes	characteristic
208	C₄⋊C₄	C₂²⋊D₉₇	1	1	yes	characteristic
209	C₄⋊C₄	C₂²×D₉₇	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₉₇	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	194	2	no	—
215	C₂²⋊C₄	C₂²⋊D₉₇	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	194	2	no	—
218	C₄²	C₂²×D₉₇	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	194	1	no	—
224	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
225	Q₈	—	97	2	no	—
226	Q₈	—	194	1	no	—
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	—	97	2	no	—
230	D₄	—	194	4	no	—
231	D₄	—	194	1	no	—
232	D₄	C₂×C₄×D₉₇	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	776	2	no	—
238	C₂×C₄	—	776	1	no	—
239	C₂×C₄	—	194	2	no	—
240	C₂×C₄	—	194	1	no	—
241	C₂×C₄	—	194	4	no	—
242	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
246	C₂×C₄	#13095511	1	2	yes	—
247	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
248	C₈	—	97	4	no	—
249	C₈	—	194	1	no	—
250	C₈	—	194	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	388	1	no	—
258	C₂²	—	388	1	no	—
259	C₂²	—	194	2	no	—
260	C₂²	C₄×C₂²⋊D₉₇	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	388	2	no	—
267	C₄	—	194	2	no	—
268	C₄	C₄×C₂²⋊D₉₇	1	2	yes	—
269	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
270	C₄	C₂×#13095511	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	388	2	no	—
274	C₂	C₂×C₄×C₂²⋊D₉₇	1	1	yes	characteristic
275	C₂	#20261538	1	1	yes	characteristic
276	C₂	#20261546	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₉₇
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#518×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂×#15690836	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#518	D₉₇
hypocenter	C₉₇	#2946
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #518

Complements

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

N	H ≅ G/N	type
2. #20261215	C₂	semidirect
3. #20260845	C₂	semidirect
4. #20260835	C₂	semidirect
5. C₄×#13095499	C₂	semidirect
6. C₄×#13095496	C₂	semidirect
7. #20260778	C₂	semidirect
8. #20260729	C₂	semidirect
9. C₂×#15690731	C₂	semidirect
10. #20260691	C₂	semidirect
11. #518×C₉₇	C₂	semidirect
13. #15690836	C₄	semidirect
18. C₄×Dic₁₉₄	C₂²	semidirect
19. C₄×D₃₈₈	C₂²	semidirect
24. #15690743	C₂²	semidirect
25. #15690741	C₂²	semidirect
26. #15690733	C₂²	semidirect
27. #15690733	C₂²	semidirect
30. #15690730	C₂²	semidirect
31. #15690730	C₂²	semidirect
32. #15690721	C₂²	semidirect
33. C₄×#11715895	C₂²	semidirect
43. #13095511	C₂×C₄	semidirect
50. #13095499	C₂×C₄	semidirect
51. #13095496	C₂×C₄	semidirect
52. #13095495	C₂×C₄	semidirect
65. C₂×D₄×C₉₇	C₂×C₄	semidirect
139. C₉₇	#2946	semidirect
142. #518	D₉₇	semidirect
189. C₂×D₄	C₄×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690836	2
#20261215	2
#20260845	2
#20260835	2
C₄×#13095499	2
C₄×#13095496	2
#20260778	2
#20260729	2
C₂×#15690731	2
#20260691	2
#518×C₉₇	2
#2946	97