Subgroups — #27150860

Order 12416.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13095512	C₂	1	1	yes	maximal, characteristic
2	#20260866	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260808	C₂	1	2	yes	maximal, complemented
4	#20260805	C₂	1	2	yes	maximal, complemented
5	C₈×D₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260800	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260790	C₂	1	2	yes	maximal, complemented
8	#20260786	C₂	1	1	yes	maximal, characteristic, complemented
9	#20260757	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₉₇	C₂	1	2	yes	maximal, complemented
11	#446×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095512	C₄	1	1	yes	characteristic
13	C₂×#13095494	C₂²	1	2	yes	—
14	C₂×C₈×D₉₇	C₂²	1	2	yes	—
15	C₂×C₈×D₉₇	C₂²	1	2	yes	—
16	Q₈×Dic₉₇	C₄	1	1	yes	characteristic
17	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
18	C₄×D₃₈₈	C₂²	1	2	yes	—
19	#15690772	C₄	1	2	yes	—
20	#15690772	C₄	1	1	yes	characteristic
21	#15690768	C₄	1	2	yes	—
22	#15690768	C₄	1	1	yes	characteristic
23	C₄²×D₉₇	C₂²	1	2	yes	—
24	C₄²×D₉₇	C₂²	1	1	yes	characteristic
25	C₄⋊#11715895	C₂²	1	2	yes	—
26	C₄⋊#11715895	C₂²	1	1	yes	characteristic
27	#15690739	C₂²	1	1	yes	characteristic
28	#15690739	C₂²	1	2	yes	—
29	#15690739	C₂²	1	1	yes	characteristic
30	#15690722	C₂²	1	1	yes	characteristic
31	#15690722	C₂²	1	1	yes	characteristic
32	#15690722	C₂²	1	2	yes	—
33	#15690722	C₂²	1	2	yes	—
34	#15690721	C₂²	1	2	yes	—
35	C₄×#11715895	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
38	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
39	#97×C₉₇	C₂²	1	2	yes	—
40	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
41	#13095512	—	2	4	no	—
42	C₂×D₃₈₈	C₂×C₄	1	2	yes	—
43	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₉₇	C₂³	1	2	yes	—
49	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
50	#13095494	—	4	2	no	—
51	C₂×#11715895	C₂³	1	1	yes	characteristic
52	C₂×#11715895	C₂³	1	2	yes	—
53	C₂×#11715895	C₂³	1	1	yes	characteristic
54	C₈×D₉₇	—	4	2	no	—
55	C₈×D₉₇	—	4	2	no	—
56	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
58	#13095488	C₂×C₄	1	2	yes	—
59	#13095488	C₂×C₄	1	2	yes	—
60	#13095488	C₂×C₄	1	2	yes	—
61	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
62	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
64	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₉₇	C₂³	1	2	yes	—
69	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₉₇	C₂³	1	1	yes	characteristic
73	C₄²×C₉₇	C₂³	1	2	yes	—
74	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
76	Q₈×C₉₇	—	2	2	no	—
77	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
86	D₃₈₈	—	2	4	no	—
87	D₃₈₈	M₁₆	1	4	yes	complemented
88	C₄×D₉₇	—	2	4	no	—
89	C₄×D₉₇	—	2	2	no	—
90	C₄×D₉₇	—	2	2	no	—
91	C₄×D₉₇	—	2	1	no	—
92	C₄×D₉₇	—	2	2	no	—
93	C₄×D₉₇	—	2	1	no	—
94	C₄×D₉₇	—	2	4	no	—
95	C₄×D₉₇	—	2	2	no	—
96	C₈×C₉₇	—	2	1	no	—
97	C₈×C₉₇	—	2	2	no	—
98	C₈×C₉₇	—	2	1	no	—
99	#11715895	—	2	1	no	—
100	#11715895	—	2	2	no	—
101	#11715895	—	2	1	no	—
102	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₉₇	—	2	2	no	—
104	C₂×D₉₇	—	2	2	no	—
105	C₂×D₉₇	—	2	1	no	—
106	C₂×D₉₇	—	2	1	no	—
107	C₂×D₉₇	—	2	1	no	—
108	C₂×D₉₇	C₂×M₁₆	1	4	yes	—
109	C₄×C₉₇	—	2	2	no	—
110	C₄×C₉₇	—	2	2	no	—
111	C₄×C₉₇	—	2	1	no	—
112	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
113	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
114	C₄×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
115	Dic₉₇	—	2	2	no	—
116	Dic₉₇	—	2	1	no	—
117	Dic₉₇	—	2	1	no	—
118	Dic₉₇	#131	1	2	yes	—
119	C₂×C₉₇	#520	1	1	yes	characteristic
120	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
121	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
122	D₉₇	—	2	4	no	—
123	D₉₇	—	4	1	no	—
124	#2990	—	97	1	no	maximal, Sylow
125	C₉₇	#2990	1	1	yes	characteristic, complemented, Sylow
126	C₄×#41	—	97	1	no	—
127	#446	—	97	1	no	—
128	#446	D₉₇	1	1	yes	characteristic, complemented
129	#435	—	97	2	no	—
130	#435	—	97	2	no	—
131	C₈×D₄	—	97	1	no	—
132	C₈×D₄	—	97	1	no	—
133	#432	—	97	2	no	—
134	#431	—	97	1	no	—
135	#423	—	97	1	no	—
136	C₂×C₄⋊C₈	—	97	2	no	—
137	C₂×#41	—	97	1	no	—
138	C₂×M₁₆	—	97	2	no	—
139	C₂²×C₈	—	97	2	no	—
140	C₂²×C₈	—	97	2	no	—
141	C₄×Q₈	—	97	1	no	—
142	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic
143	C₄×D₄	—	97	1	no	—
144	C₄×D₄	—	97	2	no	—
145	C₄×D₄	—	97	1	no	—
146	C₄×D₄	—	97	2	no	—
147	#117	—	97	1	no	—
148	#117	—	97	2	no	—
149	C₂×C₄²	—	97	2	no	—
150	C₂×C₄²	—	97	1	no	—
151	C₄⋊C₈	—	97	1	no	—
152	C₄⋊C₈	—	97	2	no	—
153	C₄⋊C₈	—	97	1	no	—
154	C₄⋊C₈	—	97	2	no	—
155	C₄⋊C₈	—	97	1	no	—
156	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
157	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
158	C₂²⋊C₈	—	97	1	no	—
159	C₂²⋊C₈	—	97	1	no	—
160	C₂²⋊C₈	—	97	2	no	—
161	C₂²⋊C₈	—	97	2	no	—
162	#97	—	97	2	no	—
163	#97	C₂×D₉₇	1	2	yes	—
164	C₄×C₈	—	97	1	no	—
165	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
166	#41	—	194	4	no	—
167	C₂×Q₈	C₄×D₉₇	1	1	yes	characteristic
168	C₂×D₄	—	97	1	no	—
169	C₂×D₄	—	97	2	no	—
170	C₂²×C₄	—	97	2	no	—
171	C₂²×C₄	—	97	1	no	—
172	C₂²×C₄	—	97	2	no	—
173	C₂²×C₄	—	97	1	no	—
174	C₂²×C₄	—	97	2	no	—
175	C₂²×C₄	—	97	1	no	—
176	M₁₆	—	388	2	no	—
177	C₂×C₈	—	97	1	no	—
178	C₂×C₈	—	97	1	no	—
179	C₂×C₈	—	97	2	no	—
180	C₂×C₈	—	388	2	no	—
181	C₂×C₈	—	388	2	no	—
182	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
184	C₂×C₈	C₂²×D₉₇	1	2	yes	—
185	C₄⋊C₄	—	97	2	no	—
186	C₄⋊C₄	—	97	1	no	—
187	C₄⋊C₄	C₄×D₉₇	1	2	yes	—
188	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
189	C₂²⋊C₄	—	97	2	no	—
190	C₂²⋊C₄	—	97	2	no	—
191	C₂²⋊C₄	—	97	2	no	—
192	C₄²	—	97	2	no	—
193	C₄²	—	97	1	no	—
194	C₄²	—	97	2	no	—
195	C₄²	—	97	1	no	—
196	C₄²	—	97	1	no	—
197	C₄²	C₂²×D₉₇	1	1	yes	characteristic
198	C₄²	C₂²×D₉₇	1	2	yes	—
199	C₂³	—	97	2	no	—
200	C₂³	—	97	1	no	—
201	Q₈	—	2	2	no	—
202	D₄	—	97	4	no	—
203	D₄	—	194	4	no	—
204	C₂×C₄	—	97	1	no	—
205	C₂×C₄	—	97	1	no	—
206	C₂×C₄	—	97	2	no	—
207	C₂×C₄	—	97	1	no	—
208	C₂×C₄	—	194	2	no	—
209	C₂×C₄	—	194	4	no	—
210	C₂×C₄	—	194	4	no	—
211	C₂×C₄	—	194	2	no	—
212	C₂×C₄	—	194	2	no	—
213	C₂×C₄	—	194	1	no	—
214	C₂×C₄	—	194	1	no	—
215	C₂×C₄	—	194	2	no	—
216	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	194	1	no	—
225	C₈	—	194	2	no	—
226	C₈	—	194	1	no	—
227	C₂²	—	97	4	no	—
228	C₂²	—	194	1	no	—
229	C₂²	—	194	1	no	—
230	C₂²	—	194	1	no	—
231	C₂²	—	194	2	no	—
232	C₂²	—	194	2	no	—
233	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	2	no	—
237	C₄	—	97	2	no	—
238	C₄	—	194	2	no	—
239	C₄	—	194	1	no	—
240	C₄	—	194	1	no	—
241	C₄	M₁₆×D₉₇	1	2	yes	—
242	C₄	#15690856	1	1	yes	characteristic
243	C₄	#15690858	1	1	yes	characteristic
244	C₂	—	388	1	no	—
245	C₂	—	194	4	no	—
246	C₂	#20261217	1	1	yes	characteristic
247	C₂	C₂×M₁₆×D₉₇	1	1	yes	characteristic
248	C₂	#131×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#446×C₉₇	C₂
prime radical	C₂×#13095512	C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#446	D₉₇
hypocenter	C₉₇	#2990
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #20260866	C₂	semidirect
3. #20260808	C₂	semidirect
4. #20260805	C₂	semidirect
5. C₈×D₃₈₈	C₂	semidirect
6. #20260800	C₂	semidirect
7. #20260790	C₂	semidirect
8. #20260786	C₂	semidirect
9. #20260757	C₂	semidirect
10. C₄⋊C₈×D₉₇	C₂	semidirect
11. #446×C₉₇	C₂	semidirect
87. D₃₈₈	M₁₆	semidirect
125. C₉₇	#2990	semidirect
128. #446	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13095512	2
#20260866	2
#20260808	2
#20260805	2
C₈×D₃₈₈	2
#20260800	2
#20260790	2
#20260786	2
#20260757	2
C₄⋊C₈×D₉₇	2
#446×C₉₇	2
#2990	97