Subgroups — #27154942

Order 12416.

1660 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690856	C₂	1	1	yes	maximal, characteristic
2	#20261217	C₂	1	2	yes	maximal, complemented
3	#20261216	C₂	1	3	yes	maximal, complemented
4	#20261209	C₂	1	3	yes	maximal, complemented
5	#20261207	C₂	1	6	yes	maximal, complemented
6	#41×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#13095512	C₂	1	2	yes	maximal, complemented
8	#20261203	C₂	1	3	yes	maximal, complemented
9	C₂×#15690772	C₂	1	3	yes	maximal, complemented
10	C₄×D₄×D₉₇	C₂	1	6	yes	maximal, complemented
11	#520×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#15690856	C₄	1	16	yes	complemented
13	C₂×#13095512	C₂²	1	2	yes	—
14	C₂×#13095511	C₂²	1	3	yes	—
15	C₂²×D₃₈₈	C₂²	1	3	yes	—
16	C₂×D₄×D₉₇	C₂²	1	6	yes	—
17	C₂²×C₄×D₉₇	C₂²	1	6	yes	complemented
18	Q₈×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
19	Q₈×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
20	#15690775	C₂²	1	1	yes	characteristic, complemented
21	#15690775	C₂²	1	3	yes	complemented
22	C₄×D₃₈₈	C₂²	1	6	yes	complemented
23	C₄×D₃₈₈	C₂²	1	6	yes	complemented
24	D₄×Dic₉₇	C₂²	1	3	yes	complemented
25	D₄×Dic₉₇	C₂²	1	3	yes	complemented
26	#15690772	C₂²	1	6	yes	complemented
27	#15690772	C₂²	1	3	yes	complemented
28	#15690772	C₂²	1	6	yes	complemented
29	#15690772	C₂²	1	6	yes	complemented
30	#15690772	C₂²	1	3	yes	complemented
31	C₄×C₂²⋊D₉₇	C₂²	1	12	yes	complemented
32	#15690770	C₂²	1	6	yes	complemented
33	#15690770	C₂²	1	6	yes	complemented
34	#15690769	C₂²	1	3	yes	complemented
35	#15690768	C₂²	1	6	yes	complemented
36	#15690767	C₂²	1	6	yes	complemented
37	C₄⋊C₄×D₉₇	C₂²	1	6	yes	complemented
38	C₂×#13095488	C₂²	1	6	yes	complemented
39	C₂²⋊C₄×D₉₇	C₂²	1	12	yes	complemented
40	C₂×C₄×Dic₉₇	C₂²	1	3	yes	complemented
41	C₄²×D₉₇	C₂²	1	6	yes	complemented
42	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₉₇	C₂²	1	2	yes	complemented
44	C₄×D₄×C₉₇	C₂²	1	6	yes	complemented
45	#117×C₉₇	C₂²	1	3	yes	complemented
46	C₂×C₄⋊C₄×C₉₇	C₂²	1	3	yes	complemented
47	C₂³×D₉₇	C₂³	1	6	yes	—
48	#13095512	C₂×C₄	1	16	yes	complemented
49	#13095511	C₂×C₄	1	24	yes	complemented
50	C₂×Dic₁₉₄	C₂³	1	1	yes	characteristic
51	C₂×D₃₈₈	C₂×C₄	1	24	yes	complemented
52	C₂×D₃₈₈	C₂³	1	3	yes	—
53	C₂×D₃₈₈	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₉₇	C₂³	1	6	yes	—
55	D₄×D₉₇	C₂×C₄	1	48	yes	complemented
56	C₂²×Dic₉₇	C₂³	1	3	yes	—
57	C₂²×Dic₉₇	C₂³	1	3	yes	—
58	C₂×C₄×D₉₇	—	4	12	no	—
59	C₂×C₄×D₉₇	C₂³	1	6	yes	—
60	C₂×C₄×D₉₇	C₂³	1	6	yes	—
61	C₂×C₄×D₉₇	C₂³	1	12	yes	—
62	C₂×C₄×D₉₇	C₂³	1	6	yes	—
63	C₄⋊Dic₉₇	C₂³	1	3	yes	—
64	C₄⋊Dic₉₇	C₂³	1	3	yes	—
65	#13095490	C₂³	1	6	yes	—
66	#13095490	C₂³	1	2	yes	—
67	C₂²⋊Dic₉₇	C₂³	1	6	yes	—
68	#13095488	C₂³	1	6	yes	—
69	#13095488	C₂³	1	12	yes	—
70	#13095488	C₂³	1	6	yes	—
71	C₄×Dic₉₇	C₂³	1	2	yes	—
72	C₄×Dic₉₇	C₂³	1	3	yes	—
73	C₄×Dic₉₇	C₂³	1	6	yes	—
74	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
75	C₄×Dic₉₇	C₂³	1	3	yes	—
76	C₄×Dic₉₇	C₂³	1	3	yes	—
77	#41×C₉₇	C₂×C₄	1	8	yes	complemented
78	C₂×Q₈×C₉₇	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₉₇	C₂³	1	3	yes	—
80	C₂²×C₄×C₉₇	C₂³	1	6	yes	—
81	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
82	C₄⋊C₄×C₉₇	C₂³	1	6	yes	—
83	C₄⋊C₄×C₉₇	C₂³	1	3	yes	—
84	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₉₇	C₂³	1	6	yes	—
86	C₄²×C₉₇	C₂³	1	6	yes	—
87	C₂³×C₉₇	C₂⁴	1	3	yes	—
88	C₂²×D₉₇	—	4	6	no	—
89	C₂²×D₉₇	—	4	6	no	—
90	C₂²×D₉₇	C₂²×C₄	1	24	yes	complemented
91	C₂²×D₉₇	C₂⁴	1	6	yes	—
92	Q₈×C₉₇	C₂²×C₄	1	4	yes	complemented
93	D₄×C₉₇	C₂²×C₄	1	12	yes	complemented
94	C₂×C₄×C₉₇	—	4	6	no	—
95	C₂×C₄×C₉₇	C₂²×C₄	1	12	yes	complemented
96	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
97	C₂×C₄×C₉₇	C₂⁴	1	6	yes	—
98	C₂×C₄×C₉₇	C₂⁴	1	3	yes	—
99	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
100	C₂²⋊D₉₇	C₂²×C₄	1	24	yes	complemented
101	C₂×Dic₉₇	—	2	6	no	—
102	C₂×Dic₉₇	—	4	3	no	—
103	C₂×Dic₉₇	C₂⁴	1	1	yes	characteristic
104	C₂×Dic₉₇	C₂⁴	1	3	yes	—
105	C₂×Dic₉₇	C₂⁴	1	3	yes	—
106	C₂×Dic₉₇	C₂⁴	1	2	yes	—
107	C₂×Dic₉₇	C₂⁴	1	1	yes	characteristic
108	D₃₈₈	C₂²×C₄	1	24	yes	complemented
109	D₃₈₈	C₂²×C₄	1	12	yes	complemented
110	C₄×D₉₇	—	2	12	no	—
111	C₄×D₉₇	—	4	12	no	—
112	C₄×D₉₇	—	4	6	no	—
113	C₄×D₉₇	C₂²×C₄	1	24	yes	complemented
114	Dic₁₉₄	C₂²×C₄	1	4	yes	complemented
115	C₂²×C₉₇	—	2	3	no	—
116	C₂²×C₉₇	—	2	3	no	—
117	C₂²×C₉₇	C₂³×C₄	1	6	yes	—
118	C₂²×C₉₇	C₂⁵	1	1	yes	characteristic
119	C₂×D₉₇	—	2	6	no	—
120	C₂×D₉₇	—	2	6	no	—
121	C₂×D₉₇	—	4	24	no	—
122	C₂×D₉₇	C₂³×C₄	1	12	yes	—
123	C₄×C₉₇	—	2	6	no	—
124	C₄×C₉₇	—	2	2	no	—
125	C₄×C₉₇	C₂³×C₄	1	2	yes	—
126	C₄×C₉₇	C₂³×C₄	1	6	yes	—
127	Dic₉₇	—	2	3	no	—
128	Dic₉₇	—	2	3	no	—
129	Dic₉₇	—	2	1	no	—
130	Dic₉₇	C₂³×C₄	1	4	yes	—
131	Dic₉₇	H(5,𝔽₂)	1	2	yes	complemented
132	C₂×C₉₇	—	2	6	no	—
133	C₂×C₉₇	C₂⁴×C₄	1	1	yes	characteristic
134	C₂×C₉₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₉₇	#585	1	1	yes	characteristic
136	D₉₇	—	2	12	no	—
137	C₄×H(5,𝔽₂)	—	97	1	no	maximal, Sylow
138	C₉₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
139	C₂×H(5,𝔽₂)	—	97	1	no	—
140	#520	—	97	2	no	—
141	#520	—	97	3	no	—
142	#520	D₉₇	1	1	yes	characteristic, complemented
143	#518	—	97	6	no	—
144	#518	—	97	3	no	—
145	C₄×#41	—	97	1	no	—
146	C₄×#41	—	97	2	no	—
147	C₄×#41	—	97	3	no	—
148	C₂×C₄×D₄	—	97	6	no	—
149	C₂×C₄×D₄	—	97	3	no	—
150	H(5,𝔽₂)	—	97	16	no	—
151	C₂×#41	—	97	2	no	—
152	C₂×#41	—	97	3	no	—
153	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
154	C₂²×D₄	—	97	6	no	—
155	C₂²×D₄	—	97	3	no	—
156	C₂³×C₄	—	97	6	no	—
157	C₄×Q₈	—	97	1	no	—
158	C₄×Q₈	—	97	1	no	—
159	C₄×Q₈	—	97	1	no	—
160	C₄×Q₈	—	97	3	no	—
161	C₄×Q₈	C₂×D₉₇	1	2	yes	complemented
162	C₄×D₄	—	97	3	no	—
163	C₄×D₄	—	97	3	no	—
164	C₄×D₄	—	97	12	no	—
165	C₄×D₄	—	97	6	no	—
166	C₄×D₄	—	97	3	no	—
167	C₄×D₄	—	97	6	no	—
168	C₄×D₄	—	97	6	no	—
169	C₄×D₄	—	97	3	no	—
170	C₄×D₄	—	97	6	no	—
171	C₄×D₄	—	97	6	no	—
172	C₄×D₄	—	97	6	no	—
173	C₄×D₄	—	97	6	no	—
174	C₄×D₄	C₂×D₉₇	1	6	yes	complemented
175	#117	—	97	3	no	—
176	#117	—	97	6	no	—
177	#117	—	97	6	no	—
178	#117	C₂×D₉₇	1	3	yes	complemented
179	C₂×C₄⋊C₄	—	97	6	no	—
180	C₂×C₄⋊C₄	C₂×D₉₇	1	3	yes	complemented
181	C₂×C₂²⋊C₄	—	97	6	no	—
182	C₂×C₂²⋊C₄	—	97	12	no	—
183	C₂×C₄²	—	97	3	no	—
184	C₂×C₄²	—	97	6	no	—
185	C₂⁴	—	97	6	no	—
186	#41	—	97	16	no	—
187	#41	—	97	24	no	—
188	#41	C₄×D₉₇	1	8	yes	complemented
189	C₂×Q₈	—	97	1	no	—
190	C₂×Q₈	C₂²×D₉₇	1	1	yes	characteristic
191	C₂×D₄	—	97	6	no	—
192	C₂×D₄	—	97	48	no	—
193	C₂×D₄	—	97	24	no	—
194	C₂×D₄	—	97	3	no	—
195	C₂×D₄	—	97	6	no	—
196	C₂×D₄	C₂²×D₉₇	1	3	yes	—
197	C₂²×C₄	—	97	6	no	—
198	C₂²×C₄	—	97	3	no	—
199	C₂²×C₄	—	97	3	no	—
200	C₂²×C₄	—	97	6	no	—
201	C₂²×C₄	—	97	12	no	—
202	C₂²×C₄	—	97	6	no	—
203	C₂²×C₄	—	388	12	no	—
204	C₂²×C₄	C₂²×D₉₇	1	6	yes	—
205	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
206	C₄⋊C₄	—	97	6	no	—
207	C₄⋊C₄	—	97	2	no	—
208	C₄⋊C₄	—	97	3	no	—
209	C₄⋊C₄	—	97	3	no	—
210	C₄⋊C₄	C₂²×D₉₇	1	6	yes	—
211	C₄⋊C₄	C₂²×D₉₇	1	3	yes	—
212	C₄⋊C₄	C₂²×D₉₇	1	1	yes	characteristic
213	C₂²⋊C₄	—	97	6	no	—
214	C₂²⋊C₄	—	97	12	no	—
215	C₂²⋊C₄	—	97	6	no	—
216	C₂²⋊C₄	—	97	6	no	—
217	C₂²⋊C₄	C₂²×D₉₇	1	6	yes	—
218	C₄²	—	97	2	no	—
219	C₄²	—	97	1	no	—
220	C₄²	—	97	3	no	—
221	C₄²	—	97	6	no	—
222	C₄²	—	97	3	no	—
223	C₄²	—	97	3	no	—
224	C₄²	C₂²×D₉₇	1	6	yes	—
225	C₂³	—	97	24	no	—
226	C₂³	—	97	6	no	—
227	C₂³	—	388	6	no	—
228	C₂³	—	388	6	no	—
229	C₂³	C₂³×D₉₇	1	3	yes	—
230	Q₈	—	97	4	no	—
231	Q₈	C₂×C₄×D₉₇	1	4	yes	complemented
232	D₄	—	97	24	no	—
233	D₄	—	97	24	no	—
234	D₄	—	97	12	no	—
235	D₄	C₂×C₄×D₉₇	1	12	yes	complemented
236	C₂×C₄	—	4	6	no	—
237	C₂×C₄	—	97	1	no	—
238	C₂×C₄	—	97	2	no	—
239	C₂×C₄	—	97	24	no	—
240	C₂×C₄	—	97	3	no	—
241	C₂×C₄	—	97	1	no	—
242	C₂×C₄	—	97	3	no	—
243	C₂×C₄	—	388	12	no	—
244	C₂×C₄	—	388	6	no	—
245	C₂×C₄	—	388	3	no	—
246	C₂×C₄	—	194	12	no	—
247	C₂×C₄	—	194	6	no	—
248	C₂×C₄	C₂×C₄×D₉₇	1	12	yes	complemented
249	C₂×C₄	C₂³×D₉₇	1	2	yes	—
250	C₂×C₄	C₂³×D₉₇	1	6	yes	—
251	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₉₇	1	3	yes	—
253	C₂²	—	2	3	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	97	12	no	—
256	C₂²	—	388	24	no	—
257	C₂²	—	194	6	no	—
258	C₂²	—	194	6	no	—
259	C₂²	C₂²×C₄×D₉₇	1	6	yes	—
260	C₂²	C₂⁴×D₉₇	1	1	yes	characteristic
261	C₄	—	2	6	no	—
262	C₄	—	2	2	no	—
263	C₄	—	97	4	no	—
264	C₄	—	97	2	no	—
265	C₄	—	194	3	no	—
266	C₄	—	194	3	no	—
267	C₄	—	194	1	no	—
268	C₄	C₂²×C₄×D₉₇	1	6	yes	—
269	C₄	C₂²×C₄×D₉₇	1	2	yes	—
270	C₂	—	2	6	no	—
271	C₂	—	194	12	no	—
272	C₂	C₂³×C₄×D₉₇	1	1	yes	characteristic
273	C₂	H(5,𝔽₂)×D₉₇	1	1	yes	characteristic
274	C₂	#20261572	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₉₇
commutator	C₂×C₉₇	C₂⁴×C₄
Fitting	#520×C₉₇	C₂
socle, cosocle	C₂²×C₉₇	C₂⁵
prime radical	C₂×#15690856	C₂
hypercenter	#520	D₉₇
hypocenter	C₉₇	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
2. #20261217	C₂	semidirect
3. #20261216	C₂	semidirect
4. #20261209	C₂	semidirect
5. #20261207	C₂	semidirect
6. #41×Dic₉₇	C₂	semidirect
7. C₄×#13095512	C₂	semidirect
8. #20261203	C₂	semidirect
9. C₂×#15690772	C₂	semidirect
10. C₄×D₄×D₉₇	C₂	semidirect
11. #520×C₉₇	C₂	semidirect
12. #15690856	C₄	semidirect
17. C₂²×C₄×D₉₇	C₂²	semidirect
18. Q₈×Dic₉₇	C₂²	semidirect
19. Q₈×Dic₉₇	C₂²	semidirect
20. #15690775	C₂²	semidirect
21. #15690775	C₂²	semidirect
22. C₄×D₃₈₈	C₂²	semidirect
23. C₄×D₃₈₈	C₂²	semidirect
24. D₄×Dic₉₇	C₂²	semidirect
25. D₄×Dic₉₇	C₂²	semidirect
26. #15690772	C₂²	semidirect
27. #15690772	C₂²	semidirect
28. #15690772	C₂²	semidirect
29. #15690772	C₂²	semidirect
30. #15690772	C₂²	semidirect
31. C₄×C₂²⋊D₉₇	C₂²	semidirect
32. #15690770	C₂²	semidirect
33. #15690770	C₂²	semidirect
34. #15690769	C₂²	semidirect
35. #15690768	C₂²	semidirect
36. #15690767	C₂²	semidirect
37. C₄⋊C₄×D₉₇	C₂²	semidirect
38. C₂×#13095488	C₂²	semidirect
39. C₂²⋊C₄×D₉₇	C₂²	semidirect
40. C₂×C₄×Dic₉₇	C₂²	semidirect
41. C₄²×D₉₇	C₂²	semidirect
43. C₄×Q₈×C₉₇	C₂²	semidirect
44. C₄×D₄×C₉₇	C₂²	semidirect
45. #117×C₉₇	C₂²	semidirect
46. C₂×C₄⋊C₄×C₉₇	C₂²	semidirect
48. #13095512	C₂×C₄	semidirect
49. #13095511	C₂×C₄	semidirect
51. C₂×D₃₈₈	C₂×C₄	semidirect
55. D₄×D₉₇	C₂×C₄	semidirect
77. #41×C₉₇	C₂×C₄	semidirect
90. C₂²×D₉₇	C₂²×C₄	semidirect
92. Q₈×C₉₇	C₂²×C₄	semidirect
93. D₄×C₉₇	C₂²×C₄	semidirect
95. C₂×C₄×C₉₇	C₂²×C₄	semidirect
100. C₂²⋊D₉₇	C₂²×C₄	semidirect
108. D₃₈₈	C₂²×C₄	semidirect
109. D₃₈₈	C₂²×C₄	semidirect
113. C₄×D₉₇	C₂²×C₄	semidirect
114. Dic₁₉₄	C₂²×C₄	semidirect
131. Dic₉₇	H(5,𝔽₂)	semidirect
138. C₉₇	C₄×H(5,𝔽₂)	semidirect
142. #520	D₉₇	semidirect
161. C₄×Q₈	C₂×D₉₇	semidirect
174. C₄×D₄	C₂×D₉₇	semidirect
178. #117	C₂×D₉₇	semidirect
180. C₂×C₄⋊C₄	C₂×D₉₇	semidirect
188. #41	C₄×D₉₇	semidirect
231. Q₈	C₂×C₄×D₉₇	semidirect
235. D₄	C₂×C₄×D₉₇	semidirect
248. C₂×C₄	C₂×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×#15690856	2
#20261217	2
#20261216	2
#20261209	2
#20261207	2
#41×Dic₉₇	2
C₄×#13095512	2
#20261203	2
C₂×#15690772	2
C₄×D₄×D₉₇	2
#520×C₉₇	2
C₄×H(5,𝔽₂)	97