Subgroups — #467×D₉₇ #27151306

Order 12416.

512 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Q₈×D₉₇	C₂	1	2	yes	maximal
4	#20260977	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260976	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260975	C₂	1	2	yes	maximal, complemented
7	#20260974	C₂	1	2	yes	maximal, complemented
8	#20260973	C₂	1	1	yes	maximal, characteristic, complemented
9	#107×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#103×D₉₇	C₂	1	2	yes	maximal, complemented
11	#467×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₉₇	C₂²	1	2	yes	—
13	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₈₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×#13095501	C₂²	1	2	yes	complemented
16	Q₁₆×D₉₇	—	2	4	no	—
17	C₂²×#11715895	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₉₇	—	2	4	no	—
19	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
23	#15690790	C₂²	1	2	yes	—
24	#15690789	C₂²	1	2	yes	—
25	#15690788	—	2	2	no	—
26	#15690787	—	2	2	no	—
27	#15690786	C₂²	1	2	yes	—
28	C₄⋊C₄×D₉₇	—	2	2	no	—
29	C₄⋊C₄×D₉₇	C₂²	1	2	yes	—
30	C₂²⋊C₄×D₉₇	—	2	2	no	—
31	#15690744	C₂²	1	1	yes	characteristic, complemented
32	#15690743	C₂²	1	2	yes	complemented
33	#15690735	C₂²	1	2	yes	complemented
34	#15690734	C₂²	1	2	yes	complemented
35	#15690733	C₂²	1	2	yes	complemented
36	C₂×Q₁₆×C₉₇	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
38	C₂²⋊Q₈×C₉₇	C₂²	1	2	yes	—
39	#107×C₉₇	C₂²	1	1	yes	characteristic, complemented
40	#103×C₉₇	C₂²	1	2	yes	complemented
41	C₂³×D₉₇	D₄	1	1	yes	characteristic
42	C₂×Dic₁₉₄	—	2	2	no	—
43	C₂×Dic₁₉₄	C₂³	1	2	yes	—
44	Q₈×D₉₇	—	2	4	no	—
45	Q₈×D₉₇	—	2	4	no	—
46	C₂²×Dic₉₇	D₄	1	1	yes	characteristic
47	C₂×C₄×D₉₇	—	2	2	no	—
48	C₂×C₄×D₉₇	—	2	2	no	—
49	C₂×C₄×D₉₇	—	2	2	no	—
50	C₂×C₄×D₉₇	—	2	2	no	—
51	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
52	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
53	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
55	Dic₃₈₈	—	2	2	no	—
56	#13095501	—	2	4	no	—
57	C₂×#11715895	—	2	2	no	—
58	C₂×#11715895	C₂³	1	1	yes	characteristic
59	C₂×#11715895	C₂³	1	1	yes	characteristic
60	C₈×D₉₇	—	2	2	no	—
61	C₈×D₉₇	—	2	2	no	—
62	C₈×D₉₇	—	2	2	no	—
63	C₈×D₉₇	D₄	1	4	yes	—
64	C₄⋊Dic₉₇	—	2	2	no	—
65	C₄⋊Dic₉₇	C₂³	1	2	yes	—
66	#13095490	—	2	2	no	—
67	#13095490	—	2	2	no	—
68	#13095490	—	2	2	no	—
69	C₂²⋊Dic₉₇	—	2	2	no	—
70	#13095488	—	2	2	no	—
71	#13095488	—	2	2	no	—
72	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
73	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
74	Q₁₆×C₉₇	—	2	2	no	—
75	C₂×C₈×C₉₇	—	2	2	no	—
76	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₉₇	—	2	2	no	—
79	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₉₇	—	2	2	no	—
81	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₉₇	—	2	2	no	—
83	C₂²×D₉₇	—	2	2	no	—
84	C₂²×D₉₇	Q₁₆	1	4	yes	complemented
85	C₂²×D₉₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₉₇	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₉₇	—	2	2	no	—
88	Q₈×C₉₇	—	2	2	no	—
89	C₂×C₄×C₉₇	—	2	1	no	—
90	C₂×C₄×C₉₇	—	2	1	no	—
91	C₂×C₄×C₉₇	—	2	2	no	—
92	C₂×C₄×C₉₇	—	2	2	no	—
93	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₉₇	—	2	2	no	—
96	C₂×Dic₉₇	—	2	2	no	—
97	C₂×Dic₉₇	—	2	2	no	—
98	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
100	C₄×D₉₇	—	2	2	no	—
101	C₄×D₉₇	—	2	1	no	—
102	C₄×D₉₇	—	2	1	no	—
103	C₄×D₉₇	—	2	2	no	—
104	C₄×D₉₇	—	4	4	no	—
105	C₄×D₉₇	—	4	4	no	—
106	C₄×D₉₇	C₂×D₄	1	2	yes	—
107	C₄×D₉₇	#41	1	2	yes	—
108	Dic₁₉₄	—	2	2	no	—
109	Dic₁₉₄	—	2	2	no	—
110	Dic₁₉₄	—	4	4	no	—
111	C₈×C₉₇	—	2	1	no	—
112	C₈×C₉₇	C₂×D₄	1	2	yes	—
113	#11715895	—	2	1	no	—
114	#11715895	C₂×D₄	1	2	yes	—
115	C₂²×C₉₇	—	2	1	no	—
116	C₂²×C₉₇	—	2	1	no	—
117	C₂²×C₉₇	C₂×Q₁₆	1	2	yes	complemented
118	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₉₇	—	2	1	no	—
120	C₂×D₉₇	—	2	1	no	—
121	C₂×D₉₇	—	2	8	no	—
122	C₂×D₉₇	#121	1	2	yes	—
123	C₂×D₉₇	C₂×Q₁₆	1	2	yes	—
124	C₂×D₉₇	C₂×Q₁₆	1	2	yes	complemented
125	C₂×D₉₇	#135	1	2	yes	—
126	C₄×C₉₇	—	2	1	no	—
127	C₄×C₉₇	—	4	2	no	—
128	C₄×C₉₇	—	4	2	no	—
129	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
131	Dic₉₇	—	2	1	no	—
132	Dic₉₇	—	4	2	no	—
133	Dic₉₇	—	4	2	no	—
134	Dic₉₇	#121	1	2	yes	—
135	C₂×C₉₇	—	2	2	no	—
136	C₂×C₉₇	C₂×#121	1	1	yes	characteristic
137	C₂×C₉₇	C₂²×Q₁₆	1	1	yes	characteristic
138	C₂×C₉₇	C₂×#135	1	1	yes	characteristic
139	D₉₇	—	2	2	no	—
140	D₉₇	#467	1	4	yes	direct product factor
141	C₂×#467	—	97	1	no	maximal, Sylow
142	C₉₇	C₂×#467	1	1	yes	characteristic, complemented, Sylow
143	C₂²×Q₁₆	—	97	1	no	—
144	C₂³×C₈	—	97	1	no	—
145	C₂×C₂²⋊Q₈	—	97	2	no	—
146	#467	—	97	1	no	—
147	#467	—	97	1	no	—
148	#467	—	97	2	no	—
149	#467	—	97	1	no	—
150	#467	—	97	2	no	—
151	#467	D₉₇	1	1	yes	characteristic, direct product factor
152	C₂×#107	—	97	1	no	—
153	C₂×#103	—	97	2	no	—
154	C₂²×Q₈	—	97	2	no	—
155	C₂³×C₄	—	97	1	no	—
156	C₂×Q₁₆	—	97	2	no	—
157	C₂×Q₁₆	—	97	1	no	—
158	C₂×Q₁₆	—	194	4	no	—
159	C₂×Q₁₆	C₂×D₉₇	1	1	yes	characteristic, complemented
160	C₂²×C₈	—	97	1	no	—
161	C₂²×C₈	—	97	1	no	—
162	C₂²×C₈	—	97	1	no	—
163	C₂²×C₈	—	97	1	no	—
164	C₂²×C₈	—	97	1	no	—
165	C₂²×C₈	—	194	4	no	—
166	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
167	C₂²⋊Q₈	—	97	2	no	—
168	C₂²⋊Q₈	—	97	2	no	—
169	C₂²⋊Q₈	—	97	2	no	—
170	C₂²⋊Q₈	—	194	2	no	—
171	C₂²⋊Q₈	—	194	2	no	—
172	C₂²⋊Q₈	C₂×D₉₇	1	2	yes	—
173	C₂×C₄⋊C₄	—	97	2	no	—
174	C₂×C₄⋊C₄	—	194	2	no	—
175	C₂×C₂²⋊C₄	—	194	2	no	—
176	#107	—	97	1	no	—
177	#107	—	97	2	no	—
178	#107	C₂×D₉₇	1	1	yes	characteristic, complemented
179	#103	—	97	2	no	—
180	#103	—	97	2	no	—
181	#103	—	97	2	no	—
182	#103	C₂×D₉₇	1	2	yes	complemented
183	C₂⁴	—	97	1	no	—
184	C₂×Q₈	—	97	2	no	—
185	C₂×Q₈	—	194	2	no	—
186	C₂×Q₈	—	194	4	no	—
187	C₂×Q₈	—	194	4	no	—
188	C₂×Q₈	C₂²×D₉₇	1	2	yes	—
189	C₂²×C₄	—	97	1	no	—
190	C₂²×C₄	—	97	1	no	—
191	C₂²×C₄	—	97	1	no	—
192	C₂²×C₄	—	97	1	no	—
193	C₂²×C₄	—	97	1	no	—
194	C₂²×C₄	—	194	2	no	—
195	C₂²×C₄	—	194	2	no	—
196	C₂²×C₄	—	194	2	no	—
197	C₂²×C₄	—	194	2	no	—
198	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	194	4	no	—
201	Q₁₆	—	194	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	97	1	no	—
204	C₂×C₈	—	97	4	no	—
205	C₂×C₈	—	97	1	no	—
206	C₂×C₈	—	194	2	no	—
207	C₂×C₈	—	194	2	no	—
208	C₂×C₈	—	194	2	no	—
209	C₂×C₈	—	194	2	no	—
210	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	97	2	no	—
214	C₄⋊C₄	—	194	2	no	—
215	C₄⋊C₄	—	194	2	no	—
216	C₄⋊C₄	—	194	2	no	—
217	C₄⋊C₄	—	194	2	no	—
218	C₄⋊C₄	C₂²×D₉₇	1	2	yes	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	194	2	no	—
221	C₂²⋊C₄	—	194	2	no	—
222	C₂²⋊C₄	—	194	2	no	—
223	C₂³	—	97	1	no	—
224	C₂³	—	97	4	no	—
225	C₂³	—	97	1	no	—
226	C₂³	—	194	2	no	—
227	C₂³	—	194	2	no	—
228	C₂³	D₄×D₉₇	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	2	2	no	—
231	Q₈	—	388	4	no	—
232	Q₈	—	194	2	no	—
233	Q₈	—	194	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	97	1	no	—
239	C₂×C₄	—	97	2	no	—
240	C₂×C₄	—	97	2	no	—
241	C₂×C₄	—	97	1	no	—
242	C₂×C₄	—	388	4	no	—
243	C₂×C₄	—	388	4	no	—
244	C₂×C₄	—	194	2	no	—
245	C₂×C₄	—	194	2	no	—
246	C₂×C₄	—	194	2	no	—
247	C₂×C₄	—	194	2	no	—
248	C₂×C₄	—	194	2	no	—
249	C₂×C₄	—	194	1	no	—
250	C₂×C₄	—	194	1	no	—
251	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	97	2	no	—
255	C₈	—	194	1	no	—
256	C₈	D₄×D₉₇	1	2	yes	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	97	2	no	—
260	C₂²	—	97	2	no	—
261	C₂²	—	97	2	no	—
262	C₂²	—	97	2	no	—
263	C₂²	—	194	8	no	—
264	C₂²	—	194	1	no	—
265	C₂²	—	194	1	no	—
266	C₂²	Q₁₆×D₉₇	1	2	yes	complemented
267	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	97	2	no	—
272	C₄	—	388	2	no	—
273	C₄	—	388	2	no	—
274	C₄	—	194	1	no	—
275	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
276	C₄	#41×D₉₇	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	97	4	no	—
279	C₂	—	194	2	no	—
280	C₂	#121×D₉₇	1	1	yes	characteristic
281	C₂	C₂×Q₁₆×D₉₇	1	1	yes	characteristic
282	C₂	#135×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#467×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂³×D₉₇	D₄
hypercenter	#467	D₉₇
hypocenter	C₉₇	C₂×#467
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×D₉₇	C₂	semidirect
4. #20260977	C₂	semidirect
5. #20260976	C₂	semidirect
6. #20260975	C₂	semidirect
7. #20260974	C₂	semidirect
8. #20260973	C₂	semidirect
9. #107×D₉₇	C₂	semidirect
10. #103×D₉₇	C₂	semidirect
11. #467×C₉₇	C₂	semidirect
14. C₂×Dic₃₈₈	C₂²	semidirect
15. C₂×#13095501	C₂²	semidirect
31. #15690744	C₂²	semidirect
32. #15690743	C₂²	semidirect
33. #15690735	C₂²	semidirect
34. #15690734	C₂²	semidirect
35. #15690733	C₂²	semidirect
36. C₂×Q₁₆×C₉₇	C₂²	semidirect
39. #107×C₉₇	C₂²	semidirect
40. #103×C₉₇	C₂²	semidirect
84. C₂²×D₉₇	Q₁₆	semidirect
117. C₂²×C₉₇	C₂×Q₁₆	semidirect
124. C₂×D₉₇	C₂×Q₁₆	semidirect
140. D₉₇	#467	direct (N × H)
142. C₉₇	C₂×#467	semidirect
151. #467	D₉₇	direct (N × H)
159. C₂×Q₁₆	C₂×D₉₇	semidirect
178. #107	C₂×D₉₇	semidirect
182. #103	C₂×D₉₇	semidirect
266. C₂²	Q₁₆×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×D₉₇	2
C₂²×C₈×D₉₇	2
C₂²⋊Q₈×D₉₇	2
#20260977	2
#20260976	2
#20260975	2
#20260974	2
#20260973	2
#107×D₉₇	2
#103×D₉₇	2
#467×C₉₇	2
C₂×#467	97

Subgroups — #467×D97 #27151306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #467×D₉₇ #27151306