Subgroups — #576×D₉₇ #27156484

Order 12416.

1512 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×D₉₇	C₂	1	2	yes	maximal, complemented
2	#20261540	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261539	C₂	1	6	yes	maximal, complemented
4	#20261538	C₂	1	2	yes	maximal, complemented
5	#20261537	C₂	1	3	yes	maximal, complemented
6	#20261536	C₂	1	3	yes	maximal, complemented
7	AGL(1,ℤ/8)×D₉₇	C₂	1	6	yes	maximal, complemented
8	#135×D₉₇	C₂	1	3	yes	maximal, complemented
9	C₂×D₈×D₉₇	C₂	1	3	yes	maximal, complemented
10	#131×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#576×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#15690856	—	2	2	no	—
13	#15690856	C₂²	1	2	yes	complemented
14	#15690856	C₂²	1	6	yes	complemented
15	#15690855	—	2	6	no	—
16	#15690855	C₂²	1	6	yes	complemented
17	#41×D₉₇	—	2	2	no	—
18	#41×D₉₇	C₂²	1	1	yes	characteristic
19	#41×D₉₇	C₂²	1	6	yes	complemented
20	C₂×D₄×D₉₇	—	2	6	no	—
21	C₂×D₄×D₉₇	C₂²	1	6	yes	complemented
22	#15690838	C₂²	1	6	yes	complemented
23	#15690837	C₂²	1	12	yes	complemented
24	#15690836	C₂²	1	6	yes	complemented
25	#15690835	C₂²	1	6	yes	complemented
26	#15690834	C₂²	1	12	yes	complemented
27	#15690833	C₂²	1	3	yes	complemented
28	#15690832	C₂²	1	3	yes	complemented
29	#15690831	C₂²	1	6	yes	complemented
30	#15690830	C₂²	1	6	yes	complemented
31	#15690829	C₂²	1	3	yes	complemented
32	Q₁₆×D₉₇	C₂²	1	1	yes	characteristic, complemented
33	SD₁₆×D₉₇	C₂²	1	6	yes	complemented
34	C₂×D₇₇₆	C₂²	1	3	yes	complemented
35	C₂×#13095496	C₂²	1	6	yes	complemented
36	D₈×D₉₇	C₂²	1	3	yes	complemented
37	D₈×D₉₇	C₂²	1	6	yes	complemented
38	D₈×D₉₇	C₂²	1	6	yes	complemented
39	D₈×D₉₇	C₂²	1	6	yes	complemented
40	#15690818	C₂²	1	3	yes	complemented
41	#15690817	C₂²	1	1	yes	characteristic, complemented
42	#15690816	C₂²	1	3	yes	complemented
43	M₁₆×D₉₇	C₂²	1	3	yes	complemented
44	C₂×C₈×D₉₇	C₂²	1	3	yes	complemented
45	H(5,𝔽₂)×C₉₇	C₂²	1	2	yes	complemented
46	AGL(1,ℤ/8)×C₉₇	C₂²	1	6	yes	complemented
47	#135×C₉₇	C₂²	1	3	yes	complemented
48	C₂×D₈×C₉₇	C₂²	1	3	yes	complemented
49	#131×C₉₇	C₂²	1	1	yes	characteristic, complemented
50	C₂³×D₉₇	—	2	6	no	—
51	#13095512	—	2	2	no	—
52	#13095512	D₄	1	1	yes	characteristic
53	#13095512	C₂³	1	6	yes	complemented
54	#13095511	—	2	6	no	—
55	#13095511	—	2	6	no	—
56	#13095511	C₂³	1	3	yes	—
57	#13095511	C₂³	1	6	yes	complemented
58	#13095510	—	2	6	no	—
59	#13095510	—	2	6	no	—
60	#13095510	D₄	1	3	yes	—
61	#13095510	C₂³	1	6	yes	complemented
62	Q₈×D₉₇	D₄	1	1	yes	characteristic
63	Q₈×D₉₇	C₂³	1	2	yes	—
64	C₂×D₃₈₈	—	2	6	no	—
65	C₂×D₃₈₈	C₂³	1	6	yes	complemented
66	C₂×C₂²⋊D₉₇	—	2	12	no	—
67	C₂×C₂²⋊D₉₇	—	2	6	no	—
68	D₄×D₉₇	—	2	6	no	—
69	D₄×D₉₇	—	2	6	no	—
70	D₄×D₉₇	—	2	12	no	—
71	D₄×D₉₇	—	2	6	no	—
72	D₄×D₉₇	—	2	6	no	—
73	D₄×D₉₇	—	2	6	no	—
74	D₄×D₉₇	D₄	1	3	yes	—
75	D₄×D₉₇	C₂³	1	6	yes	—
76	D₄×D₉₇	C₂³	1	12	yes	complemented
77	C₂×C₄×D₉₇	—	2	6	no	—
78	C₂×C₄×D₉₇	C₂³	1	3	yes	—
79	Dic₃₈₈	C₂³	1	1	yes	characteristic, complemented
80	#13095501	C₂³	1	2	yes	complemented
81	#13095500	C₂³	1	6	yes	complemented
82	#13095499	C₂³	1	6	yes	complemented
83	#13095498	C₂³	1	6	yes	complemented
84	D₇₇₆	C₂³	1	3	yes	complemented
85	D₇₇₆	C₂³	1	6	yes	complemented
86	#13095496	C₂³	1	6	yes	complemented
87	#13095496	C₂³	1	12	yes	complemented
88	#13095495	C₂³	1	3	yes	complemented
89	#13095494	C₂³	1	6	yes	complemented
90	C₂×#11715895	C₂³	1	3	yes	complemented
91	C₈×D₉₇	C₂³	1	1	yes	characteristic
92	C₈×D₉₇	C₂³	1	3	yes	—
93	C₈×D₉₇	C₂³	1	3	yes	complemented
94	C₈×D₉₇	C₂³	1	3	yes	complemented
95	#41×C₉₇	—	2	2	no	—
96	#41×C₉₇	C₂³	1	1	yes	characteristic
97	#41×C₉₇	C₂³	1	6	yes	complemented
98	C₂×D₄×C₉₇	—	2	6	no	—
99	C₂×D₄×C₉₇	C₂³	1	6	yes	complemented
100	Q₁₆×C₉₇	C₂³	1	1	yes	characteristic, complemented
101	SD₁₆×C₉₇	C₂³	1	6	yes	complemented
102	D₈×C₉₇	C₂³	1	3	yes	complemented
103	D₈×C₉₇	C₂³	1	6	yes	complemented
104	M₁₆×C₉₇	C₂³	1	3	yes	complemented
105	C₂×C₈×C₉₇	C₂³	1	3	yes	complemented
106	C₂³×C₉₇	—	2	6	no	—
107	C₂²×D₉₇	—	2	6	no	—
108	C₂²×D₉₇	—	2	6	no	—
109	C₂²×D₉₇	—	2	12	no	—
110	C₂²×D₉₇	—	8	12	no	—
111	C₂²×D₉₇	C₂×D₄	1	3	yes	—
112	Q₈×C₉₇	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₉₇	C₂⁴	1	2	yes	—
114	D₄×C₉₇	—	2	6	no	—
115	D₄×C₉₇	—	2	6	no	—
116	D₄×C₉₇	C₂×D₄	1	3	yes	—
117	D₄×C₉₇	C₂⁴	1	6	yes	—
118	C₂×C₄×C₉₇	—	2	6	no	—
119	C₂×C₄×C₉₇	C₂⁴	1	3	yes	—
120	C₂²⋊D₉₇	—	2	6	no	—
121	C₂²⋊D₉₇	—	2	6	no	—
122	C₂²⋊D₉₇	—	2	6	no	—
123	C₂²⋊D₉₇	—	2	12	no	—
124	C₂²⋊D₉₇	C₂×D₄	1	6	yes	—
125	C₂×Dic₉₇	—	2	6	no	—
126	C₂×Dic₉₇	C₂×D₄	1	3	yes	—
127	D₃₈₈	—	2	6	no	—
128	D₃₈₈	—	2	6	no	—
129	D₃₈₈	C₂×D₄	1	3	yes	—
130	D₃₈₈	C₂⁴	1	6	yes	—
131	C₄×D₉₇	—	2	2	no	—
132	C₄×D₉₇	—	2	6	no	—
133	C₄×D₉₇	—	2	6	no	—
134	C₄×D₉₇	C₂×D₄	1	3	yes	—
135	C₄×D₉₇	C₂×D₄	1	3	yes	—
136	C₄×D₉₇	C₂⁴	1	1	yes	characteristic
137	C₄×D₉₇	C₂⁴	1	3	yes	—
138	Dic₁₉₄	—	2	2	no	—
139	Dic₁₉₄	C₂×D₄	1	3	yes	—
140	Dic₁₉₄	C₂⁴	1	2	yes	—
141	C₈×C₉₇	C₂⁴	1	1	yes	characteristic
142	C₈×C₉₇	C₂⁴	1	3	yes	—
143	#11715895	C₂⁴	1	1	yes	characteristic
144	#11715895	C₂⁴	1	3	yes	—
145	C₂²×C₉₇	—	2	6	no	—
146	C₂²×C₉₇	—	8	6	no	—
147	C₂²×C₉₇	C₂²×D₄	1	3	yes	—
148	C₂×D₉₇	—	2	6	no	—
149	C₂×D₉₇	—	2	6	no	—
150	C₂×D₉₇	—	4	12	no	—
151	C₂×D₉₇	—	8	12	no	—
152	C₂×D₉₇	—	8	6	no	—
153	C₂×D₉₇	C₂²×D₄	1	3	yes	—
154	C₂×D₉₇	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₉₇	—	2	2	no	—
156	C₄×C₉₇	C₂²×D₄	1	3	yes	—
157	C₄×C₉₇	C₂⁵	1	1	yes	characteristic
158	Dic₉₇	—	2	2	no	—
159	Dic₉₇	C₂²×D₄	1	1	yes	characteristic
160	Dic₉₇	C₂²×D₄	1	3	yes	—
161	C₂×C₉₇	—	2	3	no	—
162	C₂×C₉₇	—	4	6	no	—
163	C₂×C₉₇	C₂³×D₄	1	1	yes	characteristic
164	D₉₇	—	2	3	no	—
165	D₉₇	—	4	6	no	—
166	D₉₇	#576	1	2	yes	direct product factor
167	C₂×#576	—	97	1	no	maximal, Sylow
168	C₉₇	C₂×#576	1	1	yes	characteristic, complemented, Sylow
169	C₂×H(5,𝔽₂)	—	97	2	no	—
170	#576	—	97	2	no	—
171	#576	—	97	6	no	—
172	#576	—	97	3	no	—
173	#576	—	97	3	no	—
174	#576	—	97	1	no	—
175	#576	D₉₇	1	1	yes	characteristic, direct product factor
176	C₂×AGL(1,ℤ/8)	—	97	6	no	—
177	C₂×#135	—	97	3	no	—
178	C₂²×D₈	—	97	3	no	—
179	C₂×#131	—	97	1	no	—
180	H(5,𝔽₂)	—	97	2	no	—
181	H(5,𝔽₂)	—	97	6	no	—
182	H(5,𝔽₂)	—	97	6	no	—
183	H(5,𝔽₂)	—	194	6	no	—
184	H(5,𝔽₂)	—	194	2	no	—
185	H(5,𝔽₂)	C₂×D₉₇	1	2	yes	complemented
186	C₂×#41	—	97	1	no	—
187	C₂×#41	—	97	6	no	—
188	C₂×#41	—	194	2	no	—
189	C₂²×D₄	—	97	6	no	—
190	C₂²×D₄	—	194	6	no	—
191	AGL(1,ℤ/8)	—	97	6	no	—
192	AGL(1,ℤ/8)	—	97	6	no	—
193	AGL(1,ℤ/8)	—	97	12	no	—
194	AGL(1,ℤ/8)	—	97	12	no	—
195	AGL(1,ℤ/8)	—	97	6	no	—
196	AGL(1,ℤ/8)	C₂×D₉₇	1	6	yes	complemented
197	#135	—	97	6	no	—
198	#135	—	97	3	no	—
199	#135	—	97	6	no	—
200	#135	—	97	3	no	—
201	#135	—	97	3	no	—
202	#135	C₂×D₉₇	1	3	yes	complemented
203	C₂×Q₁₆	—	97	1	no	—
204	C₂×SD₁₆	—	97	6	no	—
205	C₂×D₈	—	97	6	no	—
206	C₂×D₈	—	97	3	no	—
207	C₂×D₈	—	97	6	no	—
208	C₂×D₈	—	97	6	no	—
209	C₂×D₈	—	97	6	no	—
210	C₂×D₈	—	97	3	no	—
211	C₂×D₈	C₂×D₉₇	1	3	yes	complemented
212	#131	—	97	1	no	—
213	#131	—	97	3	no	—
214	#131	—	97	3	no	—
215	#131	C₂×D₉₇	1	1	yes	characteristic, complemented
216	C₂×M₁₆	—	97	3	no	—
217	C₂²×C₈	—	97	3	no	—
218	C₂⁴	—	194	6	no	—
219	#41	—	2	2	no	—
220	#41	—	97	1	no	—
221	#41	—	97	3	no	—
222	#41	—	97	3	no	—
223	#41	—	97	6	no	—
224	#41	—	97	6	no	—
225	#41	—	97	6	no	—
226	#41	—	194	6	no	—
227	#41	—	194	6	no	—
228	#41	—	194	6	no	—
229	#41	—	194	6	no	—
230	#41	—	194	2	no	—
231	#41	C₂²×D₉₇	1	1	yes	characteristic
232	#41	C₂²×D₉₇	1	6	yes	complemented
233	C₂×Q₈	—	97	1	no	—
234	C₂×Q₈	—	97	2	no	—
235	C₂×D₄	—	2	6	no	—
236	C₂×D₄	—	97	3	no	—
237	C₂×D₄	—	97	6	no	—
238	C₂×D₄	—	97	12	no	—
239	C₂×D₄	—	97	6	no	—
240	C₂×D₄	—	194	6	no	—
241	C₂×D₄	—	194	6	no	—
242	C₂×D₄	—	194	6	no	—
243	C₂×D₄	—	194	6	no	—
244	C₂×D₄	—	194	12	no	—
245	C₂×D₄	—	194	6	no	—
246	C₂×D₄	—	194	6	no	—
247	C₂×D₄	—	194	12	no	—
248	C₂×D₄	—	194	6	no	—
249	C₂×D₄	C₂²×D₉₇	1	6	yes	complemented
250	C₂²×C₄	—	97	3	no	—
251	C₂²×C₄	—	194	6	no	—
252	Q₁₆	—	97	2	no	—
253	Q₁₆	—	97	1	no	—
254	Q₁₆	C₂²×D₉₇	1	1	yes	characteristic, complemented
255	SD₁₆	—	97	6	no	—
256	SD₁₆	—	97	6	no	—
257	SD₁₆	—	97	6	no	—
258	SD₁₆	C₂²×D₉₇	1	6	yes	complemented
259	D₈	—	97	6	no	—
260	D₈	—	97	12	no	—
261	D₈	—	97	3	no	—
262	D₈	—	97	6	no	—
263	D₈	C₂²×D₉₇	1	3	yes	complemented
264	D₈	C₂²×D₉₇	1	6	yes	complemented
265	M₁₆	—	97	3	no	—
266	M₁₆	—	97	6	no	—
267	M₁₆	C₂²×D₉₇	1	3	yes	complemented
268	C₂×C₈	—	97	3	no	—
269	C₂×C₈	—	97	1	no	—
270	C₂×C₈	—	97	3	no	—
271	C₂×C₈	—	97	3	no	—
272	C₂×C₈	—	97	3	no	—
273	C₂×C₈	C₂²×D₉₇	1	3	yes	complemented
274	C₂³	—	2	6	no	—
275	C₂³	—	776	12	no	—
276	C₂³	—	97	3	no	—
277	C₂³	—	194	12	no	—
278	C₂³	—	194	6	no	—
279	C₂³	—	194	6	no	—
280	Q₈	—	97	3	no	—
281	Q₈	—	97	2	no	—
282	Q₈	—	194	2	no	—
283	Q₈	D₄×D₉₇	1	1	yes	characteristic
284	Q₈	C₂³×D₉₇	1	2	yes	—
285	D₄	—	2	6	no	—
286	D₄	—	2	6	no	—
287	D₄	—	97	6	no	—
288	D₄	—	97	3	no	—
289	D₄	—	97	6	no	—
290	D₄	—	194	6	no	—
291	D₄	—	194	6	no	—
292	D₄	—	194	12	no	—
293	D₄	—	194	6	no	—
294	D₄	—	194	6	no	—
295	D₄	—	194	6	no	—
296	D₄	D₄×D₉₇	1	3	yes	—
297	D₄	C₂³×D₉₇	1	6	yes	—
298	C₂×C₄	—	2	6	no	—
299	C₂×C₄	—	97	3	no	—
300	C₂×C₄	—	97	3	no	—
301	C₂×C₄	—	97	3	no	—
302	C₂×C₄	—	97	1	no	—
303	C₂×C₄	—	97	3	no	—
304	C₂×C₄	—	194	6	no	—
305	C₂×C₄	—	194	2	no	—
306	C₂×C₄	—	194	6	no	—
307	C₂×C₄	—	194	6	no	—
308	C₂×C₄	C₂³×D₉₇	1	3	yes	—
309	C₈	—	97	1	no	—
310	C₈	—	97	3	no	—
311	C₈	C₂³×D₉₇	1	1	yes	characteristic
312	C₈	C₂³×D₉₇	1	3	yes	—
313	C₂²	—	2	6	no	—
314	C₂²	—	8	6	no	—
315	C₂²	—	776	6	no	—
316	C₂²	—	776	12	no	—
317	C₂²	—	97	1	no	—
318	C₂²	—	97	3	no	—
319	C₂²	—	388	12	no	—
320	C₂²	—	194	6	no	—
321	C₂²	—	194	6	no	—
322	C₂²	C₂×D₄×D₉₇	1	3	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	97	1	no	—
325	C₄	—	97	3	no	—
326	C₄	—	194	2	no	—
327	C₄	C₂×D₄×D₉₇	1	3	yes	—
328	C₄	C₂⁴×D₉₇	1	1	yes	characteristic
329	C₂	—	2	3	no	—
330	C₂	—	4	6	no	—
331	C₂	—	97	2	no	—
332	C₂	—	388	6	no	—
333	C₂	—	194	3	no	—
334	C₂	C₂²×D₄×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₉₇
commutator, cosocle	C₄×C₉₇	C₂⁵
Frattini	C₄	C₂⁴×D₉₇
Fitting	#576×C₉₇	C₂
socle	C₂×C₉₇	C₂³×D₄
hypercenter	#576	D₉₇
hypocenter	C₉₇	C₂×#576
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×D₉₇	C₂	semidirect
2. #20261540	C₂	semidirect
3. #20261539	C₂	semidirect
4. #20261538	C₂	semidirect
5. #20261537	C₂	semidirect
6. #20261536	C₂	semidirect
7. AGL(1,ℤ/8)×D₉₇	C₂	semidirect
8. #135×D₉₇	C₂	semidirect
9. C₂×D₈×D₉₇	C₂	semidirect
10. #131×D₉₇	C₂	semidirect
11. #576×C₉₇	C₂	semidirect
13. #15690856	C₂²	semidirect
14. #15690856	C₂²	semidirect
16. #15690855	C₂²	semidirect
19. #41×D₉₇	C₂²	semidirect
21. C₂×D₄×D₉₇	C₂²	semidirect
22. #15690838	C₂²	semidirect
23. #15690837	C₂²	semidirect
24. #15690836	C₂²	semidirect
25. #15690835	C₂²	semidirect
26. #15690834	C₂²	semidirect
27. #15690833	C₂²	semidirect
28. #15690832	C₂²	semidirect
29. #15690831	C₂²	semidirect
30. #15690830	C₂²	semidirect
31. #15690829	C₂²	semidirect
32. Q₁₆×D₉₇	C₂²	semidirect
33. SD₁₆×D₉₇	C₂²	semidirect
34. C₂×D₇₇₆	C₂²	semidirect
35. C₂×#13095496	C₂²	semidirect
36. D₈×D₉₇	C₂²	semidirect
37. D₈×D₉₇	C₂²	semidirect
38. D₈×D₉₇	C₂²	semidirect
39. D₈×D₉₇	C₂²	semidirect
40. #15690818	C₂²	semidirect
41. #15690817	C₂²	semidirect
42. #15690816	C₂²	semidirect
43. M₁₆×D₉₇	C₂²	semidirect
44. C₂×C₈×D₉₇	C₂²	semidirect
45. H(5,𝔽₂)×C₉₇	C₂²	semidirect
46. AGL(1,ℤ/8)×C₉₇	C₂²	semidirect
47. #135×C₉₇	C₂²	semidirect
48. C₂×D₈×C₉₇	C₂²	semidirect
49. #131×C₉₇	C₂²	semidirect
53. #13095512	C₂³	semidirect
57. #13095511	C₂³	semidirect
61. #13095510	C₂³	semidirect
65. C₂×D₃₈₈	C₂³	semidirect
76. D₄×D₉₇	C₂³	semidirect
79. Dic₃₈₈	C₂³	semidirect
80. #13095501	C₂³	semidirect
81. #13095500	C₂³	semidirect
82. #13095499	C₂³	semidirect
83. #13095498	C₂³	semidirect
84. D₇₇₆	C₂³	semidirect
85. D₇₇₆	C₂³	semidirect
86. #13095496	C₂³	semidirect
87. #13095496	C₂³	semidirect
88. #13095495	C₂³	semidirect
89. #13095494	C₂³	semidirect
90. C₂×#11715895	C₂³	semidirect
93. C₈×D₉₇	C₂³	semidirect
94. C₈×D₉₇	C₂³	semidirect
97. #41×C₉₇	C₂³	semidirect
99. C₂×D₄×C₉₇	C₂³	semidirect
100. Q₁₆×C₉₇	C₂³	semidirect
101. SD₁₆×C₉₇	C₂³	semidirect
102. D₈×C₉₇	C₂³	semidirect
103. D₈×C₉₇	C₂³	semidirect
104. M₁₆×C₉₇	C₂³	semidirect
105. C₂×C₈×C₉₇	C₂³	semidirect
166. D₉₇	#576	direct (N × H)
168. C₉₇	C₂×#576	semidirect
175. #576	D₉₇	direct (N × H)
185. H(5,𝔽₂)	C₂×D₉₇	semidirect
196. AGL(1,ℤ/8)	C₂×D₉₇	semidirect
202. #135	C₂×D₉₇	semidirect
211. C₂×D₈	C₂×D₉₇	semidirect
215. #131	C₂×D₉₇	semidirect
232. #41	C₂²×D₉₇	semidirect
249. C₂×D₄	C₂²×D₉₇	semidirect
254. Q₁₆	C₂²×D₉₇	semidirect
258. SD₁₆	C₂²×D₉₇	semidirect
263. D₈	C₂²×D₉₇	semidirect
264. D₈	C₂²×D₉₇	semidirect
267. M₁₆	C₂²×D₉₇	semidirect
273. C₂×C₈	C₂²×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×D₉₇	2
#20261540	2
#20261539	2
#20261538	2
#20261537	2
#20261536	2
AGL(1,ℤ/8)×D₉₇	2
#135×D₉₇	2
C₂×D₈×D₉₇	2
#131×D₉₇	2
#576×C₉₇	2
C₂×#576	97

Subgroups — #576×D97 #27156484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #576×D₉₇ #27156484