Subgroups — C₉₇×#3272 #27139230

Order 12416.

546 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/8)×C₉₇	C₂	1	2	yes	maximal, complemented
2	#547×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄²×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#492×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#468×C₉₇	C₂	1	2	yes	maximal, complemented
6	#461×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#459×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#450×C₉₇	C₂	1	2	yes	maximal, complemented
9	#449×C₉₇	C₂	1	2	yes	maximal, complemented
10	C₄×D₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₉₇	C₂²	1	2	yes	complemented
13	C₂²×D₄×C₉₇	—	2	1	no	—
14	C₂²×D₄×C₉₇	C₂²	1	2	yes	—
15	AGL(1,ℤ/8)×C₉₇	—	2	8	no	—
16	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	complemented
18	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	complemented
19	C₂×D₈×C₉₇	C₂²	1	1	yes	characteristic
20	C₂×D₈×C₉₇	C₂²	1	2	yes	complemented
21	C₂×M₁₆×C₉₇	C₂²	1	2	yes	complemented
22	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
23	#127×C₉₇	C₂²	1	1	yes	characteristic
24	#123×C₉₇	—	2	2	no	—
25	C₂²⋊Q₈×C₉₇	C₂²	1	2	yes	complemented
26	#121×C₉₇	—	2	1	no	—
27	#121×C₉₇	—	2	1	no	—
28	#121×C₉₇	C₂²	1	2	yes	—
29	#120×C₉₇	—	2	2	no	—
30	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
34	#107×C₉₇	C₂²	1	1	yes	characteristic, complemented
35	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
36	#103×C₉₇	C₂²	1	2	yes	complemented
37	#102×C₉₇	C₂²	1	1	yes	characteristic
38	#102×C₉₇	C₂²	1	1	yes	characteristic, complemented
39	#102×C₉₇	C₂²	1	2	yes	complemented
40	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	complemented
41	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
42	C₂⁴×C₉₇	—	2	2	no	—
43	#41×C₉₇	—	2	4	no	—
44	#41×C₉₇	—	4	2	no	—
45	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
46	C₂×D₄×C₉₇	—	2	4	no	—
47	C₂×D₄×C₉₇	—	2	2	no	—
48	C₂×D₄×C₉₇	—	2	2	no	—
49	C₂×D₄×C₉₇	—	2	2	no	—
50	C₂×D₄×C₉₇	—	2	1	no	—
51	C₂×D₄×C₉₇	—	2	1	no	—
52	C₂×D₄×C₉₇	—	4	2	no	—
53	C₂×D₄×C₉₇	—	4	4	no	—
54	C₂×D₄×C₉₇	D₄	1	1	yes	characteristic
55	C₂×D₄×C₉₇	C₂³	1	2	yes	—
56	C₂×D₄×C₉₇	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₉₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₉₇	—	2	1	no	—
59	C₂²×C₄×C₉₇	—	2	2	no	—
60	C₂²×C₄×C₉₇	—	2	1	no	—
61	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
62	SD₁₆×C₉₇	—	2	4	no	—
63	SD₁₆×C₉₇	—	2	4	no	—
64	SD₁₆×C₉₇	—	4	2	no	—
65	D₈×C₉₇	—	2	4	no	—
66	D₈×C₉₇	D₄	1	4	yes	complemented
67	M₁₆×C₉₇	—	2	4	no	—
68	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₉₇	C₂³	1	2	yes	—
70	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₉₇	—	2	1	no	—
72	C₄⋊C₄×C₉₇	—	2	2	no	—
73	C₄⋊C₄×C₉₇	—	2	1	no	—
74	C₄⋊C₄×C₉₇	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₉₇	—	2	2	no	—
78	C₂²⋊C₄×C₉₇	—	2	1	no	—
79	C₂²⋊C₄×C₉₇	—	2	1	no	—
80	C₂²⋊C₄×C₉₇	—	2	2	no	—
81	C₂²⋊C₄×C₉₇	D₄	1	2	yes	—
82	C₄²×C₉₇	C₂³	1	1	yes	characteristic
83	C₂³×C₉₇	—	2	1	no	—
84	C₂³×C₉₇	—	2	1	no	—
85	C₂³×C₉₇	—	2	2	no	—
86	C₂³×C₉₇	—	4	4	no	—
87	C₂³×C₉₇	—	4	4	no	—
88	C₂³×C₉₇	—	8	2	no	—
89	C₂³×C₉₇	C₂×D₄	1	2	yes	—
90	Q₈×C₉₇	—	2	2	no	—
91	Q₈×C₉₇	—	2	2	no	—
92	D₄×C₉₇	—	2	2	no	—
93	D₄×C₉₇	—	2	2	no	—
94	D₄×C₉₇	—	2	1	no	—
95	D₄×C₉₇	—	2	1	no	—
96	D₄×C₉₇	—	2	2	no	—
97	D₄×C₉₇	—	4	4	no	—
98	D₄×C₉₇	—	4	4	no	—
99	D₄×C₉₇	—	4	4	no	—
100	D₄×C₉₇	—	4	2	no	—
101	D₄×C₉₇	—	8	1	no	—
102	D₄×C₉₇	C₂×D₄	1	2	yes	—
103	D₄×C₉₇	C₂×D₄	1	2	yes	complemented
104	C₂×C₄×C₉₇	—	2	2	no	—
105	C₂×C₄×C₉₇	—	2	2	no	—
106	C₂×C₄×C₉₇	—	2	1	no	—
107	C₂×C₄×C₉₇	—	2	1	no	—
108	C₂×C₄×C₉₇	—	2	2	no	—
109	C₂×C₄×C₉₇	—	4	4	no	—
110	C₂×C₄×C₉₇	—	4	2	no	—
111	C₂×C₄×C₉₇	—	4	2	no	—
112	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
116	C₈×C₉₇	—	2	2	no	—
117	C₈×C₉₇	—	2	1	no	—
118	C₈×C₉₇	C₂×D₄	1	2	yes	complemented
119	C₂²×C₉₇	—	2	2	no	—
120	C₂²×C₉₇	—	2	2	no	—
121	C₂²×C₉₇	—	2	2	no	—
122	C₂²×C₉₇	—	2	2	no	—
123	C₂²×C₉₇	—	2	2	no	—
124	C₂²×C₉₇	—	4	1	no	—
125	C₂²×C₉₇	—	4	1	no	—
126	C₂²×C₉₇	—	4	1	no	—
127	C₂²×C₉₇	—	4	1	no	—
128	C₂²×C₉₇	—	4	2	no	—
129	C₂²×C₉₇	—	4	2	no	—
130	C₂²×C₉₇	—	4	2	no	—
131	C₂²×C₉₇	—	8	8	no	—
132	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₉₇	—	2	2	no	—
134	C₄×C₉₇	—	2	1	no	—
135	C₄×C₉₇	—	4	2	no	—
136	C₄×C₉₇	—	4	1	no	—
137	C₄×C₉₇	—	4	1	no	—
138	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	complemented
139	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₉₇	—	4	2	no	—
142	C₂×C₉₇	—	4	2	no	—
143	C₂×C₉₇	—	4	2	no	—
144	C₂×C₉₇	—	8	2	no	—
145	C₂×C₉₇	D₄²	1	1	yes	characteristic
146	C₂×C₉₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₉₇	#577	1	1	yes	characteristic
148	#3272	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₉₇	#3272	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×AGL(1,ℤ/8)	C₂×C₉₇	1	2	yes	complemented
151	#547	C₂×C₉₇	1	1	yes	characteristic, complemented
152	D₄²	C₂×C₉₇	1	1	yes	characteristic, complemented
153	#492	C₂×C₉₇	1	1	yes	characteristic, complemented
154	#468	C₂×C₉₇	1	2	yes	complemented
155	#461	C₂×C₉₇	1	1	yes	characteristic, complemented
156	#459	C₂×C₉₇	1	1	yes	characteristic, complemented
157	#450	C₂×C₉₇	1	2	yes	complemented
158	#449	C₂×C₉₇	1	2	yes	complemented
159	C₄×D₈	C₂×C₉₇	1	1	yes	characteristic, complemented
160	#436	C₂×C₉₇	1	1	yes	characteristic, complemented
161	C₂×#41	C₂²×C₉₇	1	2	yes	complemented
162	C₂²×D₄	—	2	1	no	—
163	C₂²×D₄	C₂²×C₉₇	1	2	yes	—
164	AGL(1,ℤ/8)	—	2	8	no	—
165	C₂×SD₁₆	C₂²×C₉₇	1	2	yes	—
166	C₂×SD₁₆	C₂²×C₉₇	1	2	yes	complemented
167	C₂×SD₁₆	C₂²×C₉₇	1	2	yes	complemented
168	C₂×D₈	C₂²×C₉₇	1	1	yes	characteristic
169	C₂×D₈	C₂²×C₉₇	1	2	yes	complemented
170	C₂×M₁₆	C₂²×C₉₇	1	2	yes	complemented
171	C₄⋊Q₈	C₂²×C₉₇	1	1	yes	characteristic, complemented
172	#127	C₂²×C₉₇	1	1	yes	characteristic
173	#123	—	2	2	no	—
174	C₂²⋊Q₈	C₂²×C₉₇	1	2	yes	complemented
175	#121	—	2	1	no	—
176	#121	—	2	1	no	—
177	#121	C₂²×C₉₇	1	2	yes	—
178	#120	—	2	2	no	—
179	C₄×D₄	C₂²×C₉₇	1	1	yes	characteristic
180	C₄×D₄	C₂²×C₉₇	1	1	yes	characteristic
181	C₄×D₄	C₂²×C₉₇	1	1	yes	characteristic, complemented
182	C₂×C₄⋊C₄	—	2	1	no	—
183	#107	C₂²×C₉₇	1	1	yes	characteristic, complemented
184	C₄⋊C₈	C₂²×C₉₇	1	1	yes	characteristic, complemented
185	#103	C₂²×C₉₇	1	2	yes	complemented
186	#102	C₂²×C₉₇	1	1	yes	characteristic
187	#102	C₂²×C₉₇	1	1	yes	characteristic, complemented
188	#102	C₂²×C₉₇	1	2	yes	complemented
189	C₂²⋊C₈	C₂²×C₉₇	1	2	yes	complemented
190	C₄×C₈	C₂²×C₉₇	1	1	yes	characteristic, complemented
191	C₂⁴	—	2	2	no	—
192	#41	—	2	4	no	—
193	#41	—	4	2	no	—
194	C₂×Q₈	C₂³×C₉₇	1	2	yes	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	D₄×C₉₇	1	1	yes	characteristic
204	C₂×D₄	C₂³×C₉₇	1	2	yes	—
205	C₂×D₄	C₂³×C₉₇	1	1	yes	characteristic
206	C₂×D₄	C₂³×C₉₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	C₂³×C₉₇	1	2	yes	—
211	SD₁₆	—	2	4	no	—
212	SD₁₆	—	2	4	no	—
213	SD₁₆	—	4	2	no	—
214	D₈	—	2	4	no	—
215	D₈	D₄×C₉₇	1	4	yes	complemented
216	M₁₆	—	2	4	no	—
217	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
218	C₂×C₈	C₂³×C₉₇	1	2	yes	—
219	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	D₄×C₉₇	1	1	yes	characteristic
224	C₄⋊C₄	C₂³×C₉₇	1	1	yes	characteristic
225	C₄⋊C₄	C₂³×C₉₇	1	1	yes	characteristic
226	C₂²⋊C₄	—	2	2	no	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	D₄×C₉₇	1	2	yes	—
231	C₄²	C₂³×C₉₇	1	1	yes	characteristic
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	4	4	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	8	2	no	—
238	C₂³	C₂×D₄×C₉₇	1	2	yes	—
239	Q₈	—	2	2	no	—
240	Q₈	—	2	2	no	—
241	D₄	—	2	2	no	—
242	D₄	—	2	2	no	—
243	D₄	—	2	1	no	—
244	D₄	—	2	1	no	—
245	D₄	—	2	2	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	8	1	no	—
251	D₄	C₂×D₄×C₉₇	1	2	yes	—
252	D₄	C₂×D₄×C₉₇	1	2	yes	complemented
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
263	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
264	C₂×C₄	C₂⁴×C₉₇	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	1	no	—
267	C₈	C₂×D₄×C₉₇	1	2	yes	complemented
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	8	8	no	—
281	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	1	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	AGL(1,ℤ/8)×C₉₇	1	2	yes	complemented
288	C₄	C₂²×D₄×C₉₇	1	1	yes	characteristic
289	C₄	H(5,𝔽₂)×C₉₇	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	—	4	2	no	—
292	C₂	—	4	2	no	—
293	C₂	—	8	2	no	—
294	C₂	D₄²×C₉₇	1	1	yes	characteristic
295	C₂	C₂×AGL(1,ℤ/8)×C₉₇	1	1	yes	characteristic
296	C₂	#577×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₉₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₉₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₄×D₄×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×AGL(1,ℤ/8)×C₉₇	C₂	semidirect
2. #547×C₉₇	C₂	semidirect
3. D₄²×C₉₇	C₂	semidirect
4. #492×C₉₇	C₂	semidirect
5. #468×C₉₇	C₂	semidirect
6. #461×C₉₇	C₂	semidirect
7. #459×C₉₇	C₂	semidirect
8. #450×C₉₇	C₂	semidirect
9. #449×C₉₇	C₂	semidirect
10. C₄×D₈×C₉₇	C₂	semidirect
11. #436×C₉₇	C₂	semidirect
12. C₂×#41×C₉₇	C₂²	semidirect
17. C₂×SD₁₆×C₉₇	C₂²	semidirect
18. C₂×SD₁₆×C₉₇	C₂²	semidirect
20. C₂×D₈×C₉₇	C₂²	semidirect
21. C₂×M₁₆×C₉₇	C₂²	semidirect
22. C₄⋊Q₈×C₉₇	C₂²	semidirect
25. C₂²⋊Q₈×C₉₇	C₂²	semidirect
32. C₄×D₄×C₉₇	C₂²	semidirect
34. #107×C₉₇	C₂²	semidirect
35. C₄⋊C₈×C₉₇	C₂²	semidirect
36. #103×C₉₇	C₂²	semidirect
38. #102×C₉₇	C₂²	semidirect
39. #102×C₉₇	C₂²	semidirect
40. C₂²⋊C₈×C₉₇	C₂²	semidirect
41. C₄×C₈×C₉₇	C₂²	semidirect
66. D₈×C₉₇	D₄	semidirect
103. D₄×C₉₇	C₂×D₄	semidirect
118. C₈×C₉₇	C₂×D₄	semidirect
138. C₄×C₉₇	AGL(1,ℤ/8)	semidirect
148. #3272	C₉₇	direct (N × H)
149. C₉₇	#3272	direct (N × H)
150. C₂×AGL(1,ℤ/8)	C₂×C₉₇	semidirect
151. #547	C₂×C₉₇	semidirect
152. D₄²	C₂×C₉₇	semidirect
153. #492	C₂×C₉₇	semidirect
154. #468	C₂×C₉₇	semidirect
155. #461	C₂×C₉₇	semidirect
156. #459	C₂×C₉₇	semidirect
157. #450	C₂×C₉₇	semidirect
158. #449	C₂×C₉₇	semidirect
159. C₄×D₈	C₂×C₉₇	semidirect
160. #436	C₂×C₉₇	semidirect
161. C₂×#41	C₂²×C₉₇	semidirect
166. C₂×SD₁₆	C₂²×C₉₇	semidirect
167. C₂×SD₁₆	C₂²×C₉₇	semidirect
169. C₂×D₈	C₂²×C₉₇	semidirect
170. C₂×M₁₆	C₂²×C₉₇	semidirect
171. C₄⋊Q₈	C₂²×C₉₇	semidirect
174. C₂²⋊Q₈	C₂²×C₉₇	semidirect
181. C₄×D₄	C₂²×C₉₇	semidirect
183. #107	C₂²×C₉₇	semidirect
184. C₄⋊C₈	C₂²×C₉₇	semidirect
185. #103	C₂²×C₉₇	semidirect
187. #102	C₂²×C₉₇	semidirect
188. #102	C₂²×C₉₇	semidirect
189. C₂²⋊C₈	C₂²×C₉₇	semidirect
190. C₄×C₈	C₂²×C₉₇	semidirect
215. D₈	D₄×C₉₇	semidirect
252. D₄	C₂×D₄×C₉₇	semidirect
267. C₈	C₂×D₄×C₉₇	semidirect
287. C₄	AGL(1,ℤ/8)×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×AGL(1,ℤ/8)×C₉₇	2
#547×C₉₇	2
D₄²×C₉₇	2
#492×C₉₇	2
#468×C₉₇	2
#461×C₉₇	2
#459×C₉₇	2
#450×C₉₇	2
#449×C₉₇	2
C₄×D₈×C₉₇	2
#436×C₉₇	2
#3272	97

Subgroups — C97×#3272 #27139230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#3272 #27139230