Subgroups — #27150573

Order 12416.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690841	C₂	1	1	yes	maximal, characteristic
2	#20261216	C₂	1	1	yes	maximal, characteristic
3	#20261213	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260839	C₂	1	2	yes	maximal, complemented
5	#20260838	C₂	1	2	yes	maximal, complemented
6	#20260827	C₂	1	2	yes	maximal, complemented
7	#20260820	C₂	1	2	yes	maximal, complemented
8	#20260775	C₂	1	1	yes	maximal, characteristic, complemented
9	#20260732	C₂	1	1	yes	maximal, characteristic
10	#20260690	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#103×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095511	C₂²	1	1	yes	characteristic
13	#15690841	C₄	1	8	yes	complemented
14	C₂×#13095501	C₂²	1	2	yes	—
15	C₂×#13095498	C₂²	1	2	yes	—
16	C₂×#13095495	C₂²	1	1	yes	characteristic
17	Q₈×Dic₉₇	—	2	1	no	—
18	Q₈×Dic₉₇	—	2	1	no	—
19	Q₈×Dic₉₇	C₂²	1	2	yes	—
20	Q₈×Dic₉₇	C₂²	1	2	yes	—
21	#15690775	C₂²	1	2	yes	—
22	#15690772	C₂²	1	2	yes	—
23	C₄×C₂²⋊D₉₇	—	2	2	no	—
24	#15690769	—	2	2	no	—
25	#15690769	C₂²	1	1	yes	characteristic
26	#15690769	C₂²	1	1	yes	characteristic
27	#15690768	—	2	1	no	—
28	C₄⋊C₄×D₉₇	—	2	1	no	—
29	#15690743	C₂²	1	2	yes	—
30	#15690741	C₂²	1	2	yes	—
31	#15690735	C₂²	1	1	yes	characteristic
32	#15690735	C₂²	1	1	yes	characteristic
33	#15690732	C₂²	1	1	yes	characteristic
34	#15690732	C₂²	1	1	yes	characteristic
35	#15690720	C₂²	1	2	yes	—
36	C₈×Dic₉₇	C₂²	1	2	yes	—
37	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
40	#103×C₉₇	C₂²	1	2	yes	—
41	#103×C₉₇	C₂²	1	2	yes	—
42	#13095511	—	2	2	no	—
43	#13095511	C₂×C₄	1	4	yes	—
44	C₂×Dic₁₉₄	C₂³	1	1	yes	characteristic
45	C₂×D₃₈₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₉₇	—	2	1	no	—
47	C₂×C₄×D₉₇	—	2	1	no	—
48	C₂×C₄×D₉₇	—	2	2	no	—
49	#13095501	C₂×C₄	1	8	yes	complemented
50	#13095498	C₂×C₄	1	8	yes	complemented
51	#13095495	C₂×C₄	1	4	yes	complemented
52	C₂×#11715895	C₂³	1	2	yes	—
53	C₄⋊Dic₉₇	—	2	2	no	—
54	C₄⋊Dic₉₇	—	2	2	no	—
55	C₄⋊Dic₉₇	D₄	1	1	yes	characteristic
56	C₄⋊Dic₉₇	D₄	1	1	yes	characteristic
57	C₄⋊Dic₉₇	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₉₇	C₂³	1	1	yes	characteristic
59	#13095490	—	2	2	no	—
60	C₂²⋊Dic₉₇	—	2	1	no	—
61	C₂²⋊Dic₉₇	D₄	1	2	yes	—
62	#13095488	—	2	2	no	—
63	C₄×Dic₉₇	—	2	1	no	—
64	C₄×Dic₉₇	—	2	2	no	—
65	C₄×Dic₉₇	—	2	2	no	—
66	C₄×Dic₉₇	—	2	2	no	—
67	C₄×Dic₉₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₉₇	—	2	1	no	—
69	C₂×Q₈×C₉₇	—	2	2	no	—
70	C₂×Q₈×C₉₇	C₂×C₄	1	4	yes	complemented
71	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
72	C₂²×C₄×C₉₇	—	2	1	no	—
73	C₂²×C₄×C₉₇	—	2	1	no	—
74	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₉₇	—	2	2	no	—
76	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₉₇	—	2	1	no	—
79	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
80	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₉₇	—	2	1	no	—
82	Q₈×C₉₇	—	2	4	no	—
83	Q₈×C₉₇	—	2	2	no	—
84	Q₈×C₉₇	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₉₇	—	2	4	no	—
86	C₂×C₄×C₉₇	—	2	2	no	—
87	C₂×C₄×C₉₇	—	2	2	no	—
88	C₂×C₄×C₉₇	—	8	1	no	—
89	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉₇	#41	1	2	yes	—
92	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₉₇	—	2	4	no	—
94	C₂×Dic₉₇	—	2	1	no	—
95	C₂×Dic₉₇	—	2	1	no	—
96	C₂×Dic₉₇	—	2	1	no	—
97	C₂×Dic₉₇	C₂×D₄	1	2	yes	—
98	C₂×Dic₉₇	C₂×D₄	1	2	yes	—
99	D₃₈₈	—	2	1	no	—
100	D₃₈₈	C₂²×C₄	1	2	yes	—
101	C₄×D₉₇	—	2	4	no	—
102	C₄×D₉₇	—	8	2	no	—
103	Dic₁₉₄	—	2	1	no	—
104	Dic₁₉₄	C₂²×C₄	1	2	yes	—
105	C₈×C₉₇	—	2	1	no	—
106	C₈×C₉₇	—	2	1	no	—
107	#11715895	C₂²×C₄	1	4	yes	—
108	C₂²×C₉₇	—	2	1	no	—
109	C₂²×C₉₇	—	2	1	no	—
110	C₂²×C₉₇	C₄×D₄	1	2	yes	—
111	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₉₇	—	2	2	no	—
113	C₂×D₉₇	—	4	1	no	—
114	C₂×D₉₇	—	4	1	no	—
115	C₄×C₉₇	—	2	4	no	—
116	C₄×C₉₇	—	4	2	no	—
117	C₄×C₉₇	C₄×D₄	1	2	yes	—
118	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
120	Dic₉₇	—	2	2	no	—
121	Dic₉₇	—	2	2	no	—
122	Dic₉₇	—	2	2	no	—
123	Dic₉₇	—	4	1	no	—
124	Dic₉₇	—	4	1	no	—
125	C₂×C₉₇	—	2	2	no	—
126	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×C₉₇	#577	1	1	yes	characteristic
128	C₂×C₉₇	#578	1	1	yes	characteristic
129	D₉₇	—	4	2	no	—
130	#2947	—	97	1	no	maximal, Sylow
131	C₉₇	#2947	1	1	yes	characteristic, complemented, Sylow
132	C₂×#137	—	97	1	no	—
133	#520	—	97	1	no	—
134	#519	—	97	1	no	—
135	#441	—	97	2	no	—
136	#440	—	97	2	no	—
137	C₄×Q₁₆	—	97	2	no	—
138	C₄×SD₁₆	—	97	2	no	—
139	#428	—	97	1	no	—
140	#416	—	97	1	no	—
141	C₂×#103	D₉₇	1	1	yes	characteristic, complemented
142	#405	—	97	1	no	—
143	C₂×#41	—	97	1	no	—
144	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
145	#137	—	97	8	no	—
146	C₂×Q₁₆	—	97	2	no	—
147	C₂×SD₁₆	—	97	2	no	—
148	C₂×M₁₆	—	97	1	no	—
149	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
150	C₄×Q₈	—	97	2	no	—
151	C₄×Q₈	—	97	2	no	—
152	C₄×Q₈	—	97	2	no	—
153	C₄×Q₈	—	194	1	no	—
154	C₄×Q₈	—	194	1	no	—
155	C₄×D₄	—	97	2	no	—
156	C₄×D₄	—	194	2	no	—
157	#117	—	97	1	no	—
158	#117	—	97	1	no	—
159	#117	—	194	2	no	—
160	#117	—	194	1	no	—
161	C₂×C₄⋊C₄	—	194	1	no	—
162	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
163	#107	—	97	2	no	—
164	#106	—	97	2	no	—
165	#103	—	97	1	no	—
166	#103	—	97	1	no	—
167	#103	C₂×D₉₇	1	2	yes	—
168	#103	C₂×D₉₇	1	2	yes	—
169	#102	—	97	1	no	—
170	#102	—	97	1	no	—
171	#97	—	97	2	no	—
172	C₄×C₈	—	97	2	no	—
173	#41	—	97	4	no	—
174	#41	—	194	2	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	97	1	no	—
178	C₂×Q₈	C₄×D₉₇	1	4	yes	complemented
179	C₂×Q₈	C₂²×D₉₇	1	2	yes	—
180	C₂×D₄	—	97	1	no	—
181	C₂×D₄	—	194	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	194	2	no	—
185	C₂²×C₄	—	194	1	no	—
186	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
187	Q₁₆	—	97	8	no	—
188	SD₁₆	—	97	8	no	—
189	M₁₆	—	97	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	97	2	no	—
192	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	97	1	no	—
196	C₄⋊C₄	—	97	1	no	—
197	C₄⋊C₄	—	97	1	no	—
198	C₄⋊C₄	—	97	1	no	—
199	C₄⋊C₄	—	194	2	no	—
200	C₄⋊C₄	—	194	2	no	—
201	C₄⋊C₄	—	194	2	no	—
202	C₄⋊C₄	C₂²×D₉₇	1	2	yes	—
203	C₂²⋊C₄	—	97	2	no	—
204	C₂²⋊C₄	—	194	2	no	—
205	C₂²⋊C₄	—	194	1	no	—
206	C₄²	—	97	2	no	—
207	C₄²	—	194	2	no	—
208	C₄²	—	194	2	no	—
209	C₄²	—	194	2	no	—
210	C₄²	—	194	1	no	—
211	C₂³	—	194	1	no	—
212	C₂³	D₄×D₉₇	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	97	2	no	—
216	Q₈	—	194	1	no	—
217	Q₈	C₂×C₄×D₉₇	1	4	yes	—
218	D₄	—	97	2	no	—
219	D₄	—	194	4	no	—
220	D₄	—	194	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	776	2	no	—
226	C₂×C₄	—	97	2	no	—
227	C₂×C₄	—	97	2	no	—
228	C₂×C₄	—	194	1	no	—
229	C₂×C₄	—	194	1	no	—
230	C₂×C₄	—	194	1	no	—
231	C₂×C₄	—	194	4	no	—
232	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
233	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
234	C₂×C₄	#13095510	1	2	yes	—
235	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	97	4	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	388	1	no	—
242	C₂²	—	388	1	no	—
243	C₂²	—	194	2	no	—
244	C₂²	#15690770	1	2	yes	—
245	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	—	388	1	no	—
249	C₄	—	388	1	no	—
250	C₄	—	194	2	no	—
251	C₄	—	194	2	no	—
252	C₄	—	194	2	no	—
253	C₄	#15690770	1	2	yes	—
254	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
255	C₄	C₂×#13095510	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	388	2	no	—
258	C₂	C₂×#15690770	1	1	yes	characteristic
259	C₂	#20261541	1	1	yes	characteristic
260	C₂	#20261548	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₉₇
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×#103×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂×#13095511	C₂²
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×#103	D₉₇
hypocenter	C₉₇	#2947
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
3. #20261213	C₂	semidirect
4. #20260839	C₂	semidirect
5. #20260838	C₂	semidirect
6. #20260827	C₂	semidirect
7. #20260820	C₂	semidirect
8. #20260775	C₂	semidirect
10. #20260690	C₂	semidirect
11. C₂×#103×C₉₇	C₂	semidirect
13. #15690841	C₄	semidirect
49. #13095501	C₂×C₄	semidirect
50. #13095498	C₂×C₄	semidirect
51. #13095495	C₂×C₄	semidirect
70. C₂×Q₈×C₉₇	C₂×C₄	semidirect
131. C₉₇	#2947	semidirect
141. C₂×#103	D₉₇	semidirect
178. C₂×Q₈	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×#15690841	2
#20261216	2
#20261213	2
#20260839	2
#20260838	2
#20260827	2
#20260820	2
#20260775	2
#20260732	2
#20260690	2
C₂×#103×C₉₇	2
#2947	97