Subgroups — #27150701

Order 12416.

412 conjugacy classes of subgroups.

The subgroup lattice has 252 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	#20260814	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260812	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260808	C₂	1	2	yes	maximal, complemented
6	#20260805	C₂	1	2	yes	maximal, complemented
7	#20260787	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260787	C₂	1	2	yes	maximal, complemented
9	#20260687	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic, 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×#13095494	C₂²	1	2	yes	—
15	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
17	Q₈×Dic₉₇	C₄	1	1	yes	characteristic
18	C₄×D₃₈₈	C₂²	1	2	yes	—
19	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
20	#15690772	C₄	1	2	yes	—
21	#15690772	C₄	1	1	yes	characteristic
22	#15690768	C₄	1	2	yes	—
23	#15690768	C₄	1	1	yes	characteristic
24	C₄²×D₉₇	C₂²	1	1	yes	characteristic
25	C₄²×D₉₇	C₂²	1	2	yes	—
26	C₄⋊#11715895	C₂²	1	2	yes	—
27	C₄⋊#11715895	C₂²	1	1	yes	characteristic
28	#15690722	C₂²	1	2	yes	—
29	#15690722	C₂²	1	2	yes	—
30	#15690722	C₂²	1	2	yes	—
31	#15690721	C₂²	1	2	yes	—
32	C₄×#11715895	C₂²	1	1	yes	characteristic
33	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
34	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
35	C₈×Dic₉₇	C₂²	1	2	yes	—
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	2	yes	—
46	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₉₇	C₂³	1	2	yes	—
50	#13095494	—	2	4	no	—
51	#13095494	—	2	4	no	—
52	C₂×#11715895	C₂³	1	1	yes	characteristic
53	C₂×#11715895	C₂³	1	2	yes	—
54	C₂×#11715895	C₂³	1	1	yes	characteristic
55	C₈×D₉₇	—	2	2	no	—
56	C₈×D₉₇	—	2	2	no	—
57	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
59	#13095488	C₂×C₄	1	2	yes	—
60	#13095488	C₂×C₄	1	2	yes	—
61	#13095488	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	2	yes	—
65	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
67	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄	1	2	yes	—
72	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
73	C₄²×C₉₇	C₂³	1	1	yes	characteristic
74	C₄²×C₉₇	C₂³	1	2	yes	—
75	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
76	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
77	Q₈×C₉₇	—	2	2	no	—
78	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
85	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
87	D₃₈₈	—	2	4	no	—
88	D₃₈₈	—	2	2	no	—
89	C₄×D₉₇	—	2	1	no	—
90	C₄×D₉₇	—	2	1	no	—
91	C₄×D₉₇	—	2	2	no	—
92	C₄×D₉₇	—	2	2	no	—
93	C₄×D₉₇	—	2	4	no	—
94	C₄×D₉₇	—	2	2	no	—
95	C₄×D₉₇	—	2	4	no	—
96	C₄×D₉₇	M₁₆	1	4	yes	complemented
97	C₈×C₉₇	—	2	1	no	—
98	C₈×C₉₇	—	2	2	no	—
99	C₈×C₉₇	#41	1	2	yes	complemented
100	#11715895	—	2	1	no	—
101	#11715895	—	2	2	no	—
102	#11715895	#41	1	2	yes	complemented
103	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₉₇	—	2	1	no	—
105	C₂×D₉₇	—	2	1	no	—
106	C₂×D₉₇	—	2	2	no	—
107	C₂×D₉₇	—	2	2	no	—
108	C₂×D₉₇	—	2	2	no	—
109	C₂×D₉₇	C₂×M₁₆	1	2	yes	—
110	C₄×C₉₇	—	2	2	no	—
111	C₄×C₉₇	—	2	1	no	—
112	C₄×C₉₇	—	2	2	no	—
113	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
114	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
115	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
116	Dic₉₇	—	2	1	no	—
117	Dic₉₇	—	2	2	no	—
118	Dic₉₇	C₂×M₁₆	1	2	yes	—
119	Dic₉₇	#131	1	2	yes	—
120	C₂×C₉₇	C₄×#41	1	1	yes	characteristic
121	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
123	D₉₇	—	2	2	no	—
124	D₉₇	—	4	2	no	—
125	#2965	—	97	1	no	maximal, Sylow
126	C₉₇	#2965	1	1	yes	characteristic, complemented, Sylow
127	C₄×#41	—	97	1	no	—
128	#446	—	97	1	no	—
129	#446	D₉₇	1	1	yes	characteristic, complemented
130	#436	—	97	1	no	—
131	#436	—	97	1	no	—
132	#435	—	97	2	no	—
133	#435	—	97	2	no	—
134	#431	—	97	2	no	—
135	#431	—	97	1	no	—
136	C₄×M₁₆	—	97	2	no	—
137	C₂×C₄×C₈	—	97	1	no	—
138	C₂×#41	—	97	1	no	—
139	C₂×M₁₆	—	97	2	no	—
140	C₂×M₁₆	—	97	2	no	—
141	C₂²×C₈	—	97	1	no	—
142	C₂²×C₈	—	97	1	no	—
143	C₄×Q₈	—	97	1	no	—
144	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic
145	C₄×D₄	—	97	2	no	—
146	C₄×D₄	—	97	1	no	—
147	C₄×D₄	—	97	2	no	—
148	C₄×D₄	—	97	1	no	—
149	#117	—	97	1	no	—
150	#117	—	97	2	no	—
151	C₂×C₄²	—	97	1	no	—
152	C₂×C₄²	—	97	2	no	—
153	C₄⋊C₈	—	97	2	no	—
154	C₄⋊C₈	—	97	1	no	—
155	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
156	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
157	C₂²⋊C₈	—	97	2	no	—
158	C₂²⋊C₈	—	97	2	no	—
159	C₂²⋊C₈	—	97	2	no	—
160	#97	—	97	2	no	—
161	#97	C₂×D₉₇	1	2	yes	—
162	C₄×C₈	—	97	1	no	—
163	C₄×C₈	—	97	1	no	—
164	C₄×C₈	—	97	2	no	—
165	C₄×C₈	—	97	1	no	—
166	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
167	#41	—	194	4	no	—
168	C₂×Q₈	C₄×D₉₇	1	1	yes	characteristic
169	C₂×D₄	—	97	2	no	—
170	C₂×D₄	—	97	1	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	C₂²×C₄	—	97	2	no	—
177	M₁₆	—	194	4	no	—
178	M₁₆	—	194	4	no	—
179	C₂×C₈	—	97	1	no	—
180	C₂×C₈	—	97	1	no	—
181	C₂×C₈	—	97	2	no	—
182	C₂×C₈	—	194	2	no	—
183	C₂×C₈	—	194	2	no	—
184	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
185	C₂×C₈	C₂²×D₉₇	1	2	yes	—
186	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
187	C₄⋊C₄	—	97	2	no	—
188	C₄⋊C₄	—	97	1	no	—
189	C₄⋊C₄	C₄×D₉₇	1	2	yes	—
190	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
191	C₂²⋊C₄	—	97	2	no	—
192	C₂²⋊C₄	—	97	2	no	—
193	C₂²⋊C₄	—	97	2	no	—
194	C₄²	—	97	1	no	—
195	C₄²	—	97	2	no	—
196	C₄²	—	97	2	no	—
197	C₄²	—	97	1	no	—
198	C₄²	—	97	1	no	—
199	C₄²	C₂²×D₉₇	1	1	yes	characteristic
200	C₄²	C₂²×D₉₇	1	2	yes	—
201	C₂³	—	97	1	no	—
202	C₂³	—	97	2	no	—
203	Q₈	—	2	2	no	—
204	D₄	—	194	4	no	—
205	D₄	—	194	2	no	—
206	C₂×C₄	—	97	1	no	—
207	C₂×C₄	—	97	1	no	—
208	C₂×C₄	—	97	4	no	—
209	C₂×C₄	—	97	1	no	—
210	C₂×C₄	—	97	2	no	—
211	C₂×C₄	—	194	4	no	—
212	C₂×C₄	—	194	4	no	—
213	C₂×C₄	—	194	2	no	—
214	C₂×C₄	—	194	1	no	—
215	C₂×C₄	—	194	2	no	—
216	C₂×C₄	—	194	1	no	—
217	C₂×C₄	—	194	2	no	—
218	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	97	2	no	—
226	C₈	—	194	1	no	—
227	C₈	—	194	2	no	—
228	C₈	#13095512	1	2	yes	complemented
229	C₂²	—	97	2	no	—
230	C₂²	—	194	2	no	—
231	C₂²	—	194	2	no	—
232	C₂²	—	194	2	no	—
233	C₂²	—	194	1	no	—
234	C₂²	—	194	1	no	—
235	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	1	no	—
239	C₄	—	97	2	no	—
240	C₄	—	97	2	no	—
241	C₄	—	194	1	no	—
242	C₄	—	194	2	no	—
243	C₄	M₁₆×D₉₇	1	2	yes	—
244	C₄	#41×D₉₇	1	1	yes	characteristic
245	C₄	C₂×#13095512	1	1	yes	characteristic
246	C₂	—	388	2	no	—
247	C₂	—	194	2	no	—
248	C₂	C₄×#13095512	1	1	yes	characteristic
249	C₂	C₂×M₁₆×D₉₇	1	1	yes	characteristic
250	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	#2965
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. #20260814	C₂	semidirect
4. #20260812	C₂	semidirect
5. #20260808	C₂	semidirect
6. #20260805	C₂	semidirect
7. #20260787	C₂	semidirect
8. #20260787	C₂	semidirect
9. #20260687	C₂	semidirect
10. C₄×C₈×D₉₇	C₂	semidirect
11. #446×C₉₇	C₂	semidirect
96. C₄×D₉₇	M₁₆	semidirect
99. C₈×C₉₇	#41	semidirect
102. #11715895	#41	semidirect
126. C₉₇	#2965	semidirect
129. #446	D₉₇	semidirect
228. C₈	#13095512	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
#20260814	2
#20260812	2
#20260808	2
#20260805	2
#20260787	2
#20260687	2
C₄×C₈×D₉₇	2
#446×C₉₇	2
#2965	97