Subgroups — #27150566

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×#15690838	C₂	1	1	yes	maximal, characteristic
2	#20261216	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261209	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260846	C₂	1	2	yes	maximal, complemented
5	#20260834	C₂	1	2	yes	maximal, complemented
6	#20260822	C₂	1	2	yes	maximal, complemented
7	#20260817	C₂	1	2	yes	maximal, complemented
8	#20260729	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15690730	C₂	1	1	yes	maximal, characteristic, complemented
10	#20260693	C₂	1	1	yes	maximal, characteristic, complemented
11	#428×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095511	C₂²	1	1	yes	characteristic
13	C₂²×D₃₈₈	C₂²	1	1	yes	characteristic
14	#15690838	C₄	1	8	yes	complemented
15	C₂×#13095500	C₂²	1	2	yes	—
16	C₂×D₇₇₆	C₂²	1	2	yes	—
17	C₂²×#11715895	C₂²	1	1	yes	characteristic
18	#15690775	C₂²	1	2	yes	complemented
19	C₄×D₃₈₈	—	2	1	no	—
20	C₄×D₃₈₈	—	2	1	no	—
21	#15690772	C₂²	1	2	yes	—
22	#15690772	C₂²	1	2	yes	—
23	#15690772	C₂²	1	2	yes	complemented
24	C₄×C₂²⋊D₉₇	—	2	2	no	—
25	#15690769	C₂²	1	1	yes	characteristic
26	#15690768	—	2	1	no	—
27	C₄⋊C₄×D₉₇	—	2	1	no	—
28	C₂²⋊C₄×D₉₇	—	2	2	no	—
29	#15690733	C₂²	1	1	yes	characteristic, complemented
30	#15690733	C₂²	1	1	yes	characteristic, complemented
31	#15690730	C₂²	1	2	yes	—
32	#15690730	C₂²	1	2	yes	—
33	#15690730	C₂²	1	1	yes	characteristic, complemented
34	#15690730	C₂²	1	1	yes	characteristic, complemented
35	#15690720	C₂²	1	2	yes	complemented
36	C₈×Dic₉₇	C₂²	1	2	yes	complemented
37	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
38	#117×C₉₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
40	#107×C₉₇	C₂²	1	2	yes	complemented
41	#106×C₉₇	C₂²	1	2	yes	complemented
42	C₂³×D₉₇	—	2	1	no	—
43	#13095511	—	2	2	no	—
44	#13095511	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₁₉₄	C₂³	1	1	yes	characteristic
46	C₂×D₃₈₈	—	2	1	no	—
47	C₂×D₃₈₈	—	2	2	no	—
48	C₂×D₃₈₈	C₂×C₄	1	4	yes	complemented
49	C₂×D₃₈₈	C₂³	1	2	yes	—
50	C₂×D₃₈₈	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₉₇	—	2	1	no	—
52	C₂×C₄×D₉₇	—	2	1	no	—
53	C₂×C₄×D₉₇	—	2	2	no	—
54	C₂×C₄×D₉₇	—	2	2	no	—
55	C₂×C₄×D₉₇	—	2	2	no	—
56	#13095500	C₂×C₄	1	8	yes	complemented
57	D₇₇₆	C₂×C₄	1	8	yes	complemented
58	C₂×#11715895	—	2	2	no	—
59	C₂×#11715895	C₂³	1	1	yes	characteristic
60	C₂×#11715895	C₂³	1	1	yes	characteristic
61	C₄⋊Dic₉₇	D₄	1	1	yes	characteristic
62	C₄⋊Dic₉₇	D₄	1	1	yes	characteristic
63	#13095490	—	2	2	no	—
64	C₂²⋊Dic₉₇	D₄	1	2	yes	—
65	#13095488	—	2	2	no	—
66	#13095488	—	2	2	no	—
67	#13095488	—	2	2	no	—
68	C₄×Dic₉₇	—	2	2	no	—
69	C₄×Dic₉₇	C₂³	1	2	yes	—
70	C₂²×C₄×C₉₇	—	2	1	no	—
71	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
72	M₁₆×C₉₇	C₂×C₄	1	4	yes	complemented
73	C₂×C₈×C₉₇	C₂³	1	2	yes	—
74	C₄⋊C₄×C₉₇	—	2	1	no	—
75	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₉₇	—	2	1	no	—
79	C₄²×C₉₇	—	2	1	no	—
80	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₉₇	—	2	1	no	—
82	C₂²×D₉₇	—	2	2	no	—
83	C₂²×D₉₇	—	2	4	no	—
84	C₂²×D₉₇	—	8	1	no	—
85	C₂²×D₉₇	—	8	1	no	—
86	C₂×C₄×C₉₇	—	2	1	no	—
87	C₂×C₄×C₉₇	—	2	1	no	—
88	C₂×C₄×C₉₇	—	2	2	no	—
89	C₂×C₄×C₉₇	—	8	1	no	—
90	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₉₇	#41	1	2	yes	—
93	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₉₇	—	2	4	no	—
95	C₂×Dic₉₇	—	2	1	no	—
96	C₂×Dic₉₇	C₂×D₄	1	2	yes	—
97	C₂×Dic₉₇	C₂×D₄	1	2	yes	—
98	D₃₈₈	—	2	4	no	—
99	D₃₈₈	—	2	2	no	—
100	D₃₈₈	—	2	1	no	—
101	D₃₈₈	C₂²×C₄	1	2	yes	—
102	D₃₈₈	C₂²×C₄	1	4	yes	—
103	C₄×D₉₇	—	2	4	no	—
104	C₄×D₉₇	—	8	2	no	—
105	C₄×D₉₇	—	8	2	no	—
106	C₄×D₉₇	—	8	2	no	—
107	Dic₁₉₄	—	2	1	no	—
108	Dic₁₉₄	C₂²×C₄	1	2	yes	—
109	C₈×C₉₇	C₂²×C₄	1	4	yes	—
110	#11715895	—	2	1	no	—
111	#11715895	—	2	1	no	—
112	C₂²×C₉₇	—	2	1	no	—
113	C₂²×C₉₇	—	2	1	no	—
114	C₂²×C₉₇	C₄×D₄	1	2	yes	—
115	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₉₇	—	2	2	no	—
117	C₂×D₉₇	—	2	4	no	—
118	C₂×D₉₇	—	4	1	no	—
119	C₂×D₉₇	—	4	2	no	—
120	C₂×D₉₇	—	4	2	no	—
121	C₂×D₉₇	—	4	1	no	—
122	C₂×D₉₇	—	8	4	no	—
123	C₄×C₉₇	—	4	1	no	—
124	C₄×C₉₇	—	4	2	no	—
125	C₄×C₉₇	—	4	1	no	—
126	C₄×C₉₇	C₄×D₄	1	2	yes	—
127	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
129	Dic₉₇	—	2	2	no	—
130	Dic₉₇	—	2	2	no	—
131	Dic₉₇	—	2	2	no	—
132	C₂×C₉₇	—	2	2	no	—
133	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₉₇	#576	1	1	yes	characteristic
135	C₂×C₉₇	#577	1	1	yes	characteristic
136	D₉₇	—	4	2	no	—
137	D₉₇	—	4	4	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	—	97	1	no	—
143	#442	—	97	2	no	—
144	#440	—	97	2	no	—
145	C₄×SD₁₆	—	97	2	no	—
146	C₄×D₈	—	97	2	no	—
147	#428	D₉₇	1	1	yes	characteristic, complemented
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	—	97	1	no	—
153	AGL(1,ℤ/8)	—	97	8	no	—
154	C₂×SD₁₆	—	97	2	no	—
155	C₂×D₈	—	97	2	no	—
156	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
157	C₂²×C₈	—	97	1	no	—
158	C₄×Q₈	—	97	2	no	—
159	C₄×D₄	—	97	2	no	—
160	C₄×D₄	—	97	2	no	—
161	C₄×D₄	—	97	2	no	—
162	C₄×D₄	—	194	1	no	—
163	C₄×D₄	—	194	1	no	—
164	C₄×D₄	—	194	2	no	—
165	#117	—	97	1	no	—
166	#117	—	194	1	no	—
167	#117	C₂×D₉₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	194	1	no	—
169	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	194	2	no	—
171	#107	C₂×D₉₇	1	2	yes	complemented
172	#106	C₂×D₉₇	1	2	yes	complemented
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	—	194	1	no	—
182	#41	—	97	4	no	—
183	#41	—	194	2	no	—
184	C₂×Q₈	—	97	1	no	—
185	C₂×D₄	—	97	4	no	—
186	C₂×D₄	—	97	1	no	—
187	C₂×D₄	—	97	2	no	—
188	C₂×D₄	—	194	1	no	—
189	C₂×D₄	—	194	1	no	—
190	C₂×D₄	—	194	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	194	2	no	—
193	C₂²×C₄	—	194	2	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	C₄×D₉₇	1	4	yes	complemented
200	C₂×C₈	—	97	1	no	—
201	C₂×C₈	—	97	1	no	—
202	C₂×C₈	—	194	2	no	—
203	C₂×C₈	C₂²×D₉₇	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	97	1	no	—
206	C₄⋊C₄	—	97	1	no	—
207	C₄⋊C₄	—	194	2	no	—
208	C₄⋊C₄	C₂²×D₉₇	1	2	yes	—
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	—	97	2	no	—
213	C₂²⋊C₄	—	194	2	no	—
214	C₂²⋊C₄	—	194	2	no	—
215	C₂²⋊C₄	—	194	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	97	2	no	—
218	C₄²	—	194	2	no	—
219	C₂³	—	776	1	no	—
220	C₂³	—	776	1	no	—
221	C₂³	—	194	4	no	—
222	C₂³	—	194	1	no	—
223	C₂³	—	194	2	no	—
224	C₂³	D₄×D₉₇	1	1	yes	characteristic
225	Q₈	—	97	2	no	—
226	Q₈	—	194	1	no	—
227	D₄	—	97	2	no	—
228	D₄	—	97	4	no	—
229	D₄	—	194	4	no	—
230	D₄	—	194	4	no	—
231	D₄	—	194	2	no	—
232	D₄	—	194	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	—	776	2	no	—
238	C₂×C₄	—	776	2	no	—
239	C₂×C₄	—	776	2	no	—
240	C₂×C₄	—	97	2	no	—
241	C₂×C₄	—	97	2	no	—
242	C₂×C₄	—	194	1	no	—
243	C₂×C₄	—	194	4	no	—
244	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
245	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
246	C₂×C₄	#13095512	1	2	yes	—
247	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
248	C₈	—	194	1	no	—
249	C₈	—	194	1	no	—
250	C₈	C₂×C₄×D₉₇	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	776	4	no	—
254	C₂²	—	388	1	no	—
255	C₂²	—	388	2	no	—
256	C₂²	—	388	2	no	—
257	C₂²	—	388	1	no	—
258	C₂²	—	194	2	no	—
259	C₂²	—	194	4	no	—
260	C₂²	#15690772	1	2	yes	—
261	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
262	C₄	—	4	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	194	2	no	—
266	C₄	—	194	2	no	—
267	C₄	—	194	2	no	—
268	C₄	#15690772	1	2	yes	—
269	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
270	C₄	C₂×#13095512	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	388	4	no	—
273	C₂	—	388	2	no	—
274	C₂	C₂×#15690772	1	1	yes	characteristic
275	C₂	#20261539	1	1	yes	characteristic
276	C₂	#20261544	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₉₇
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#428×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂×#15690838	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#428	D₉₇
hypocenter	C₉₇	#2946
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
2. #20261216	C₂	semidirect
3. #20261209	C₂	semidirect
4. #20260846	C₂	semidirect
5. #20260834	C₂	semidirect
6. #20260822	C₂	semidirect
7. #20260817	C₂	semidirect
8. #20260729	C₂	semidirect
9. C₂×#15690730	C₂	semidirect
10. #20260693	C₂	semidirect
11. #428×C₉₇	C₂	semidirect
14. #15690838	C₄	semidirect
18. #15690775	C₂²	semidirect
23. #15690772	C₂²	semidirect
29. #15690733	C₂²	semidirect
30. #15690733	C₂²	semidirect
33. #15690730	C₂²	semidirect
34. #15690730	C₂²	semidirect
35. #15690720	C₂²	semidirect
36. C₈×Dic₉₇	C₂²	semidirect
40. #107×C₉₇	C₂²	semidirect
41. #106×C₉₇	C₂²	semidirect
44. #13095511	C₂×C₄	semidirect
48. C₂×D₃₈₈	C₂×C₄	semidirect
56. #13095500	C₂×C₄	semidirect
57. D₇₇₆	C₂×C₄	semidirect
72. M₁₆×C₉₇	C₂×C₄	semidirect
139. C₉₇	#2946	semidirect
147. #428	D₉₇	semidirect
171. #107	C₂×D₉₇	semidirect
172. #106	C₂×D₉₇	semidirect
199. M₁₆	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×#15690838	2
#20261216	2
#20261209	2
#20260846	2
#20260834	2
#20260822	2
#20260817	2
#20260729	2
C₂×#15690730	2
#20260693	2
#428×C₉₇	2
#2946	97