Subgroups — #555×D₉₇ #27155272

Order 12416.

1240 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261451	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261450	C₂	1	2	yes	maximal, complemented
3	#20261449	C₂	1	4	yes	maximal, complemented
4	#20261448	C₂	1	4	yes	maximal, complemented
5	#20261447	C₂	1	4	yes	maximal, complemented
6	C₄⋊Q₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#124×D₉₇	C₂	1	4	yes	maximal, complemented
8	#121×D₉₇	C₂	1	4	yes	maximal, complemented
9	C₄×D₄×D₉₇	C₂	1	2	yes	maximal, complemented
10	#117×D₉₇	C₂	1	4	yes	maximal, complemented
11	#555×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₉₇	C₂²	1	2	yes	—
13	C₂×D₄×D₉₇	C₂²	1	2	yes	—
14	C₂×D₄×D₉₇	C₂²	1	4	yes	—
15	C₂²×C₄×D₉₇	C₂²	1	4	yes	—
16	#15690810	C₂²	1	1	yes	characteristic, complemented
17	#15690809	C₂²	1	4	yes	complemented
18	#15690808	C₂²	1	2	yes	complemented
19	#15690799	C₂²	1	4	yes	complemented
20	#15690798	C₂²	1	4	yes	complemented
21	#15690797	C₂²	1	4	yes	complemented
22	#15690796	C₂²	1	8	yes	complemented
23	#15690796	C₂²	1	8	yes	complemented
24	#15690785	C₂²	1	4	yes	complemented
25	#15690784	C₂²	1	4	yes	complemented
26	#15690783	C₂²	1	8	yes	complemented
27	#15690782	C₂²	1	8	yes	complemented
28	#15690781	C₂²	1	4	yes	complemented
29	C₄×D₃₈₈	C₂²	1	2	yes	complemented
30	D₄×Dic₉₇	C₂²	1	2	yes	complemented
31	#15690772	C₂²	1	2	yes	complemented
32	C₄×C₂²⋊D₉₇	C₂²	1	4	yes	complemented
33	#15690770	C₂²	1	4	yes	complemented
34	#15690769	C₂²	1	4	yes	complemented
35	#15690768	C₂²	1	4	yes	complemented
36	#15690768	C₂²	1	4	yes	complemented
37	#15690767	C₂²	1	8	yes	complemented
38	#15690766	C₂²	1	8	yes	complemented
39	C₄⋊C₄×D₉₇	C₂²	1	2	yes	—
40	C₄⋊C₄×D₉₇	C₂²	1	4	yes	—
41	C₂²⋊C₄×D₉₇	C₂²	1	8	yes	—
42	C₂²⋊C₄×D₉₇	C₂²	1	4	yes	—
43	C₄²×D₉₇	C₂²	1	4	yes	—
44	C₄²×D₉₇	C₂²	1	1	yes	characteristic
45	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
46	#124×C₉₇	C₂²	1	4	yes	complemented
47	#121×C₉₇	C₂²	1	4	yes	complemented
48	C₄×D₄×C₉₇	C₂²	1	2	yes	complemented
49	#117×C₉₇	C₂²	1	4	yes	complemented
50	C₂³×D₉₇	C₂³	1	4	yes	—
51	C₂×Dic₁₉₄	C₂³	1	2	yes	—
52	C₂×Dic₁₉₄	C₂³	1	4	yes	—
53	Q₈×D₉₇	—	4	4	no	—
54	C₂×D₃₈₈	C₂³	1	2	yes	—
55	C₂×D₃₈₈	C₂³	1	4	yes	—
56	C₂×C₂²⋊D₉₇	C₂³	1	4	yes	—
57	C₂×C₂²⋊D₉₇	C₂³	1	8	yes	—
58	D₄×D₉₇	—	2	8	no	—
59	D₄×D₉₇	—	4	8	no	—
60	C₂²×Dic₉₇	C₂³	1	4	yes	—
61	C₂×C₄×D₉₇	—	2	16	no	—
62	C₂×C₄×D₉₇	C₂³	1	4	yes	—
63	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₉₇	C₂³	1	4	yes	—
65	C₂×C₄×D₉₇	C₂³	1	4	yes	—
66	C₂×C₄×D₉₇	C₂³	1	4	yes	—
67	C₂×C₄×D₉₇	C₂³	1	2	yes	—
68	C₄⋊Dic₉₇	C₂³	1	2	yes	—
69	C₄⋊Dic₉₇	C₂³	1	4	yes	—
70	#13095490	C₂³	1	4	yes	—
71	#13095490	C₂³	1	8	yes	—
72	C₂²⋊Dic₉₇	C₂³	1	8	yes	—
73	C₂²⋊Dic₉₇	C₂³	1	4	yes	—
74	#13095488	C₂³	1	8	yes	—
75	#13095488	C₂³	1	8	yes	—
76	#13095488	C₂³	1	4	yes	—
77	#13095488	C₂³	1	4	yes	—
78	C₄×Dic₉₇	C₂³	1	4	yes	—
79	C₄×Dic₉₇	C₂³	1	8	yes	—
80	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
81	C₄×Dic₉₇	C₂³	1	2	yes	—
82	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
83	C₂×D₄×C₉₇	C₂³	1	2	yes	—
84	C₂×D₄×C₉₇	C₂³	1	4	yes	—
85	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
86	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
87	C₄⋊C₄×C₉₇	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₉₇	C₂³	1	8	yes	—
89	C₂²⋊C₄×C₉₇	C₂³	1	4	yes	—
90	C₄²×C₉₇	C₂³	1	4	yes	—
91	C₄²×C₉₇	C₂³	1	1	yes	characteristic
92	C₂³×C₉₇	C₂⁴	1	4	yes	—
93	C₂²×D₉₇	—	2	8	no	—
94	C₂²×D₉₇	—	2	8	no	—
95	C₂²×D₉₇	—	2	8	no	—
96	C₂²×D₉₇	C₂⁴	1	4	yes	—
97	C₂²×D₉₇	C₂⁴	1	1	yes	characteristic
98	Q₈×C₉₇	—	4	2	no	—
99	D₄×C₉₇	—	2	4	no	—
100	D₄×C₉₇	—	4	4	no	—
101	C₂×C₄×C₉₇	—	2	8	no	—
102	C₂×C₄×C₉₇	C₂⁴	1	4	yes	—
103	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₉₇	C₂⁴	1	4	yes	—
105	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
106	C₂²⋊D₉₇	—	2	8	no	—
107	C₂²⋊D₉₇	—	4	8	no	—
108	C₂×Dic₉₇	—	2	8	no	—
109	C₂×Dic₉₇	C₂⁴	1	4	yes	—
110	C₂×Dic₉₇	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₉₇	C₂⁴	1	4	yes	—
112	C₂×Dic₉₇	C₂⁴	1	2	yes	—
113	D₃₈₈	—	2	4	no	—
114	D₃₈₈	—	4	4	no	—
115	C₄×D₉₇	—	2	8	no	—
116	C₄×D₉₇	—	2	8	no	—
117	C₄×D₉₇	—	2	2	no	—
118	C₄×D₉₇	—	2	8	no	—
119	C₄×D₉₇	—	2	8	no	—
120	C₄×D₉₇	#41	1	8	yes	—
121	Dic₁₉₄	—	4	4	no	—
122	Dic₁₉₄	—	4	2	no	—
123	C₂²×C₉₇	—	2	4	no	—
124	C₂²×C₉₇	—	2	4	no	—
125	C₂²×C₉₇	—	2	4	no	—
126	C₂²×C₉₇	C₂⁵	1	1	yes	characteristic
127	C₂×D₉₇	—	2	4	no	—
128	C₂×D₉₇	—	2	4	no	—
129	C₂×D₉₇	—	2	4	no	—
130	C₂×D₉₇	—	4	16	no	—
131	C₂×D₉₇	C₂×#41	1	4	yes	—
132	C₂×D₉₇	H(5,𝔽₂)	1	2	yes	—
133	C₄×C₉₇	—	2	4	no	—
134	C₄×C₉₇	—	2	1	no	—
135	C₄×C₉₇	—	2	4	no	—
136	C₄×C₉₇	C₂×#41	1	4	yes	—
137	Dic₉₇	—	2	4	no	—
138	Dic₉₇	—	2	1	no	—
139	Dic₉₇	—	2	4	no	—
140	Dic₉₇	C₂×#41	1	4	yes	—
141	C₂×C₉₇	—	4	4	no	—
142	C₂×C₉₇	C₂²×#41	1	2	yes	—
143	C₂×C₉₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
144	D₉₇	—	4	4	no	—
145	D₉₇	#555	1	4	yes	direct product factor
146	C₂×#555	—	97	1	no	maximal, Sylow
147	C₉₇	C₂×#555	1	1	yes	characteristic, complemented, Sylow
148	#555	—	97	1	no	—
149	#555	—	97	4	no	—
150	#555	—	97	4	no	—
151	#555	—	97	2	no	—
152	#555	—	97	4	no	—
153	#555	D₉₇	1	1	yes	characteristic, direct product factor
154	C₂×C₄⋊Q₈	—	97	1	no	—
155	C₂×#124	—	97	4	no	—
156	C₂×#121	—	97	4	no	—
157	C₂×C₄×D₄	—	97	2	no	—
158	C₂×#117	—	97	4	no	—
159	C₂²×Q₈	—	97	2	no	—
160	C₂²×D₄	—	97	4	no	—
161	C₂²×D₄	—	97	2	no	—
162	C₂³×C₄	—	97	4	no	—
163	C₄⋊Q₈	—	97	2	no	—
164	C₄⋊Q₈	—	97	4	no	—
165	C₄⋊Q₈	—	97	1	no	—
166	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic, complemented
167	#124	—	97	4	no	—
168	#124	—	97	4	no	—
169	#124	—	97	8	no	—
170	#124	—	97	8	no	—
171	#124	—	97	4	no	—
172	#124	C₂×D₉₇	1	4	yes	complemented
173	#121	—	97	4	no	—
174	#121	—	97	8	no	—
175	#121	—	97	4	no	—
176	#121	—	97	4	no	—
177	#121	—	97	8	no	—
178	#121	C₂×D₉₇	1	4	yes	complemented
179	C₄×D₄	—	97	2	no	—
180	C₄×D₄	—	97	4	no	—
181	C₄×D₄	—	97	2	no	—
182	C₄×D₄	—	97	4	no	—
183	C₄×D₄	—	97	2	no	—
184	C₄×D₄	C₂×D₉₇	1	2	yes	complemented
185	#117	—	97	4	no	—
186	#117	—	97	4	no	—
187	#117	—	97	4	no	—
188	#117	—	97	8	no	—
189	#117	—	97	8	no	—
190	#117	C₂×D₉₇	1	4	yes	complemented
191	C₂×C₄⋊C₄	—	97	2	no	—
192	C₂×C₄⋊C₄	—	97	4	no	—
193	C₂×C₂²⋊C₄	—	97	8	no	—
194	C₂×C₂²⋊C₄	—	97	4	no	—
195	C₂×C₄²	—	97	4	no	—
196	C₂×C₄²	—	97	1	no	—
197	C₂⁴	—	97	4	no	—
198	C₂×Q₈	—	97	4	no	—
199	C₂×Q₈	—	97	2	no	—
200	C₂×Q₈	—	388	4	no	—
201	C₂×Q₈	C₂²×D₉₇	1	2	yes	—
202	C₂×D₄	—	97	8	no	—
203	C₂×D₄	—	97	4	no	—
204	C₂×D₄	—	97	4	no	—
205	C₂×D₄	—	97	2	no	—
206	C₂×D₄	—	388	8	no	—
207	C₂×D₄	—	194	8	no	—
208	C₂×D₄	C₂²×D₉₇	1	2	yes	—
209	C₂×D₄	C₂²×D₉₇	1	4	yes	—
210	C₂²×C₄	—	97	2	no	—
211	C₂²×C₄	—	97	4	no	—
212	C₂²×C₄	—	97	4	no	—
213	C₂²×C₄	—	97	1	no	—
214	C₂²×C₄	—	97	4	no	—
215	C₂²×C₄	—	97	4	no	—
216	C₂²×C₄	—	97	4	no	—
217	C₂²×C₄	—	194	16	no	—
218	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
219	C₄⋊C₄	—	97	4	no	—
220	C₄⋊C₄	—	97	8	no	—
221	C₄⋊C₄	—	97	2	no	—
222	C₄⋊C₄	—	97	4	no	—
223	C₄⋊C₄	C₂²×D₉₇	1	2	yes	—
224	C₄⋊C₄	C₂²×D₉₇	1	4	yes	—
225	C₂²⋊C₄	—	97	8	no	—
226	C₂²⋊C₄	—	97	4	no	—
227	C₂²⋊C₄	—	97	8	no	—
228	C₂²⋊C₄	—	97	8	no	—
229	C₂²⋊C₄	—	97	4	no	—
230	C₂²⋊C₄	—	97	4	no	—
231	C₂²⋊C₄	C₂²×D₉₇	1	8	yes	—
232	C₂²⋊C₄	C₂²×D₉₇	1	4	yes	—
233	C₄²	—	97	4	no	—
234	C₄²	—	97	8	no	—
235	C₄²	—	97	1	no	—
236	C₄²	—	97	2	no	—
237	C₄²	C₂²×D₉₇	1	4	yes	—
238	C₄²	C₂²×D₉₇	1	1	yes	characteristic
239	C₂³	—	97	1	no	—
240	C₂³	—	97	4	no	—
241	C₂³	—	194	8	no	—
242	C₂³	—	194	8	no	—
243	C₂³	—	194	8	no	—
244	C₂³	C₂³×D₉₇	1	4	yes	—
245	Q₈	—	4	2	no	—
246	Q₈	—	388	4	no	—
247	Q₈	—	388	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	388	4	no	—
251	D₄	—	388	8	no	—
252	D₄	—	194	8	no	—
253	D₄	—	194	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	97	2	no	—
256	C₂×C₄	—	97	8	no	—
257	C₂×C₄	—	97	4	no	—
258	C₂×C₄	—	97	1	no	—
259	C₂×C₄	—	97	4	no	—
260	C₂×C₄	—	194	8	no	—
261	C₂×C₄	—	194	8	no	—
262	C₂×C₄	—	194	8	no	—
263	C₂×C₄	—	194	2	no	—
264	C₂×C₄	—	194	8	no	—
265	C₂×C₄	—	194	8	no	—
266	C₂×C₄	C₂³×D₉₇	1	4	yes	—
267	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
268	C₂×C₄	C₂³×D₉₇	1	4	yes	—
269	C₂×C₄	C₂³×D₉₇	1	2	yes	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	97	4	no	—
274	C₂²	—	97	2	no	—
275	C₂²	—	388	16	no	—
276	C₂²	—	194	4	no	—
277	C₂²	—	194	4	no	—
278	C₂²	—	194	4	no	—
279	C₂²	C₂⁴×D₉₇	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	1	no	—
283	C₄	—	97	4	no	—
284	C₄	—	194	4	no	—
285	C₄	—	194	4	no	—
286	C₄	—	194	1	no	—
287	C₄	#41×D₉₇	1	4	yes	—
288	C₂	—	4	4	no	—
289	C₂	—	97	4	no	—
290	C₂	—	388	4	no	—
291	C₂	C₂×#41×D₉₇	1	2	yes	—
292	C₂	H(5,𝔽₂)×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₉₇
commutator, socle, cosocle	C₂²×C₉₇	C₂⁵
Fitting	#555×C₉₇	C₂
hypercenter	#555	D₉₇
hypocenter	C₉₇	C₂×#555
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #555

Complements

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

N	H ≅ G/N	type
1. #20261451	C₂	semidirect
2. #20261450	C₂	semidirect
3. #20261449	C₂	semidirect
4. #20261448	C₂	semidirect
5. #20261447	C₂	semidirect
6. C₄⋊Q₈×D₉₇	C₂	semidirect
7. #124×D₉₇	C₂	semidirect
8. #121×D₉₇	C₂	semidirect
9. C₄×D₄×D₉₇	C₂	semidirect
10. #117×D₉₇	C₂	semidirect
11. #555×C₉₇	C₂	semidirect
16. #15690810	C₂²	semidirect
17. #15690809	C₂²	semidirect
18. #15690808	C₂²	semidirect
19. #15690799	C₂²	semidirect
20. #15690798	C₂²	semidirect
21. #15690797	C₂²	semidirect
22. #15690796	C₂²	semidirect
23. #15690796	C₂²	semidirect
24. #15690785	C₂²	semidirect
25. #15690784	C₂²	semidirect
26. #15690783	C₂²	semidirect
27. #15690782	C₂²	semidirect
28. #15690781	C₂²	semidirect
29. C₄×D₃₈₈	C₂²	semidirect
30. D₄×Dic₉₇	C₂²	semidirect
31. #15690772	C₂²	semidirect
32. C₄×C₂²⋊D₉₇	C₂²	semidirect
33. #15690770	C₂²	semidirect
34. #15690769	C₂²	semidirect
35. #15690768	C₂²	semidirect
36. #15690768	C₂²	semidirect
37. #15690767	C₂²	semidirect
38. #15690766	C₂²	semidirect
45. C₄⋊Q₈×C₉₇	C₂²	semidirect
46. #124×C₉₇	C₂²	semidirect
47. #121×C₉₇	C₂²	semidirect
48. C₄×D₄×C₉₇	C₂²	semidirect
49. #117×C₉₇	C₂²	semidirect
145. D₉₇	#555	direct (N × H)
147. C₉₇	C₂×#555	semidirect
153. #555	D₉₇	direct (N × H)
166. C₄⋊Q₈	C₂×D₉₇	semidirect
172. #124	C₂×D₉₇	semidirect
178. #121	C₂×D₉₇	semidirect
184. C₄×D₄	C₂×D₉₇	semidirect
190. #117	C₂×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261451	2
#20261450	2
#20261449	2
#20261448	2
#20261447	2
C₄⋊Q₈×D₉₇	2
#124×D₉₇	2
#121×D₉₇	2
C₄×D₄×D₉₇	2
#117×D₉₇	2
#555×C₉₇	2
C₂×#555	97

Subgroups — #555×D97 #27155272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #555×D₉₇ #27155272