Subgroups — #27150303

Order 12416.

484 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690818	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690816	C₂	1	1	yes	maximal, characteristic
3	#20261203	C₂	1	1	yes	maximal, characteristic
4	#20260762	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260760	C₂	1	2	yes	maximal, complemented
6	#20260760	C₂	1	2	yes	maximal, complemented
7	#20260758	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260757	C₂	1	2	yes	maximal, complemented
9	C₂×#15690739	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₉₇	C₂	1	2	yes	maximal, complemented
11	#424×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095511	C₂²	1	1	yes	characteristic
13	#15690818	—	2	4	no	—
14	#15690816	—	2	4	no	—
15	C₂×#13095495	C₂²	1	1	yes	characteristic
16	C₂×#13095494	C₂²	1	2	yes	—
17	C₂×#13095494	C₂²	1	2	yes	—
18	C₂²×#11715895	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₉₇	C₂²	1	2	yes	—
20	C₂×C₈×D₉₇	C₂²	1	2	yes	—
21	#15690775	C₄	1	1	yes	characteristic
22	#15690775	C₄	1	1	yes	characteristic
23	#15690772	C₄	1	1	yes	characteristic
24	#15690772	C₄	1	1	yes	characteristic
25	#15690770	C₄	1	2	yes	—
26	#15690770	C₄	1	2	yes	—
27	#15690767	C₂²	1	2	yes	—
28	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
29	C₄²×D₉₇	C₂²	1	2	yes	—
30	C₄⋊#11715895	C₂²	1	2	yes	—
31	C₄⋊#11715895	C₂²	1	2	yes	—
32	#15690739	C₂²	1	1	yes	characteristic
33	#15690739	C₂²	1	2	yes	—
34	#15690739	C₂²	1	1	yes	characteristic
35	#15690739	C₂²	1	1	yes	characteristic
36	#15690739	C₂²	1	1	yes	characteristic
37	#15690739	C₂²	1	2	yes	—
38	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
40	#117×C₉₇	C₂²	1	1	yes	characteristic
41	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
42	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
43	#13095511	D₄	1	4	yes	—
44	#13095511	Q₈	1	4	yes	—
45	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
46	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
48	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₉₇	C₂³	1	2	yes	—
53	#13095495	—	2	2	no	—
54	#13095494	—	2	4	no	—
55	#13095494	—	2	4	no	—
56	C₂×#11715895	—	2	2	no	—
57	C₂×#11715895	C₂³	1	1	yes	characteristic
58	C₂×#11715895	C₂³	1	2	yes	—
59	C₂×#11715895	C₂³	1	1	yes	characteristic
60	C₈×D₉₇	—	2	4	no	—
61	C₈×D₉₇	—	2	4	no	—
62	#13095490	C₂×C₄	1	2	yes	—
63	#13095490	C₂×C₄	1	2	yes	—
64	#13095488	C₂×C₄	1	2	yes	—
65	#13095488	C₂×C₄	1	2	yes	—
66	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
67	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
68	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
71	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
73	M₁₆×C₉₇	—	2	2	no	—
74	C₂×C₈×C₉₇	—	2	2	no	—
75	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₉₇	C₂³	1	2	yes	—
77	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
81	C₄²×C₉₇	C₂³	1	2	yes	—
82	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
89	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₉₇	C₄⋊C₄	1	8	yes	—
91	C₂×Dic₉₇	—	2	2	no	—
92	C₂×Dic₉₇	—	4	1	no	—
93	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
97	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
98	D₃₈₈	C₄⋊C₄	1	4	yes	—
99	C₄×D₉₇	—	2	4	no	—
100	C₄×D₉₇	—	4	2	no	—
101	C₄×D₉₇	C₂×D₄	1	4	yes	—
102	C₄×D₉₇	C₂×Q₈	1	4	yes	—
103	Dic₁₉₄	C₄⋊C₄	1	4	yes	—
104	C₈×C₉₇	—	2	1	no	—
105	C₈×C₉₇	—	2	1	no	—
106	C₈×C₉₇	—	2	2	no	—
107	#11715895	—	2	1	no	—
108	#11715895	—	2	2	no	—
109	#11715895	—	2	1	no	—
110	C₂²×C₉₇	—	2	1	no	—
111	C₂²×C₉₇	—	2	1	no	—
112	C₂²×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×D₉₇	—	2	2	no	—
115	C₂×D₉₇	—	2	2	no	—
116	C₂×D₉₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₄×C₉₇	—	2	2	no	—
118	C₄×C₉₇	—	2	2	no	—
119	C₄×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
122	Dic₉₇	—	2	1	no	—
123	Dic₉₇	—	2	1	no	—
124	Dic₉₇	—	2	1	no	—
125	Dic₉₇	C₂×C₄⋊C₄	1	4	yes	—
126	Dic₉₇	#131	1	2	yes	complemented
127	C₂×C₉₇	—	2	2	no	—
128	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
130	C₂×C₉₇	#568	1	1	yes	characteristic
131	D₉₇	—	2	4	no	—
132	#2906	—	97	1	no	maximal, Sylow
133	C₉₇	#2906	1	1	yes	characteristic, complemented, Sylow
134	C₂×#131	—	97	1	no	—
135	C₂×#131	—	97	1	no	—
136	C₄×#41	—	97	1	no	—
137	#424	—	97	1	no	—
138	#424	—	97	2	no	—
139	#424	—	97	2	no	—
140	#424	D₉₇	1	1	yes	characteristic, complemented
141	#423	—	97	1	no	—
142	#423	—	97	2	no	—
143	C₂×C₄⋊C₈	—	97	1	no	—
144	C₂×C₄⋊C₈	—	97	2	no	—
145	C₂×#41	—	97	1	no	—
146	#131	—	194	4	no	—
147	#131	—	194	4	no	—
148	C₂×M₁₆	—	97	1	no	—
149	C₂×M₁₆	—	97	2	no	—
150	C₂×M₁₆	—	97	2	no	—
151	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
152	C₂²×C₈	—	97	1	no	—
153	C₂²×C₈	—	97	2	no	—
154	C₂²×C₈	—	97	2	no	—
155	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
156	C₄×Q₈	—	97	1	no	—
157	C₄×Q₈	—	97	1	no	—
158	C₄×D₄	—	97	1	no	—
159	C₄×D₄	—	97	1	no	—
160	C₄×D₄	—	97	2	no	—
161	C₄×D₄	—	97	2	no	—
162	#117	—	97	2	no	—
163	#117	C₂×D₉₇	1	1	yes	characteristic
164	C₂×C₄²	—	97	1	no	—
165	C₂×C₄²	—	97	2	no	—
166	C₄⋊C₈	—	97	1	no	—
167	C₄⋊C₈	—	97	2	no	—
168	C₄⋊C₈	—	97	1	no	—
169	C₄⋊C₈	—	97	1	no	—
170	C₄⋊C₈	—	97	1	no	—
171	C₄⋊C₈	—	97	2	no	—
172	C₄⋊C₈	—	97	2	no	—
173	C₄⋊C₈	—	97	2	no	—
174	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
175	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
176	#41	—	97	4	no	—
177	#41	—	97	4	no	—
178	C₂×Q₈	—	97	1	no	—
179	C₂×D₄	—	97	2	no	—
180	C₂×D₄	—	97	1	no	—
181	C₂²×C₄	—	97	2	no	—
182	C₂²×C₄	—	97	1	no	—
183	C₂²×C₄	—	97	1	no	—
184	C₂²×C₄	—	97	2	no	—
185	C₂²×C₄	—	97	2	no	—
186	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
187	M₁₆	—	2	2	no	—
188	M₁₆	—	194	4	no	—
189	M₁₆	—	194	4	no	—
190	M₁₆	—	194	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	97	1	no	—
193	C₂×C₈	—	97	2	no	—
194	C₂×C₈	—	97	1	no	—
195	C₂×C₈	—	194	4	no	—
196	C₂×C₈	—	194	4	no	—
197	C₂×C₈	—	194	2	no	—
198	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
199	C₂×C₈	C₂²×D₉₇	1	2	yes	—
200	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
201	C₄⋊C₄	—	97	2	no	—
202	C₄⋊C₄	—	97	2	no	—
203	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
204	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
205	C₂²⋊C₄	—	97	2	no	—
206	C₂²⋊C₄	—	97	2	no	—
207	C₂²⋊C₄	C₄×D₉₇	1	2	yes	—
208	C₄²	—	97	2	no	—
209	C₄²	—	97	1	no	—
210	C₄²	—	97	2	no	—
211	C₄²	—	97	1	no	—
212	C₄²	—	97	1	no	—
213	C₄²	—	97	1	no	—
214	C₄²	C₂²×D₉₇	1	2	yes	—
215	C₂³	—	97	2	no	—
216	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
217	Q₈	—	97	4	no	—
218	D₄	—	97	8	no	—
219	D₄	—	97	4	no	—
220	C₂×C₄	—	97	1	no	—
221	C₂×C₄	—	97	2	no	—
222	C₂×C₄	—	97	4	no	—
223	C₂×C₄	—	97	4	no	—
224	C₂×C₄	—	97	1	no	—
225	C₂×C₄	—	97	1	no	—
226	C₂×C₄	—	97	1	no	—
227	C₂×C₄	—	388	2	no	—
228	C₂×C₄	—	388	1	no	—
229	C₂×C₄	—	194	4	no	—
230	C₂×C₄	—	194	2	no	—
231	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
232	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
233	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
234	C₂×C₄	D₄×D₉₇	1	2	yes	—
235	C₂×C₄	Q₈×D₉₇	1	2	yes	—
236	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	2	no	—
240	C₈	—	194	1	no	—
241	C₈	—	194	2	no	—
242	C₈	—	194	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	97	4	no	—
246	C₂²	—	194	2	no	—
247	C₂²	—	194	2	no	—
248	C₂²	C₄⋊C₄×D₉₇	1	2	yes	—
249	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	97	4	no	—
253	C₄	—	97	2	no	—
254	C₄	—	194	1	no	—
255	C₄	—	194	1	no	—
256	C₄	—	194	1	no	—
257	C₄	C₄⋊C₄×D₉₇	1	2	yes	—
258	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
259	C₄	C₂×Q₈×D₉₇	1	1	yes	characteristic
260	C₂	—	2	2	no	—
261	C₂	—	194	4	no	—
262	C₂	C₂×C₄⋊C₄×D₉₇	1	1	yes	characteristic
263	C₂	#131×D₉₇	1	1	yes	characteristic
264	C₂	#20261500	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#424×C₉₇	C₂
prime radical	C₂×#13095511	C₂²
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#424	D₉₇
hypocenter	C₉₇	#2906
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. #20260762	C₂	semidirect
5. #20260760	C₂	semidirect
6. #20260760	C₂	semidirect
7. #20260758	C₂	semidirect
8. #20260757	C₂	semidirect
9. C₂×#15690739	C₂	semidirect
10. C₄⋊C₈×D₉₇	C₂	semidirect
11. #424×C₉₇	C₂	semidirect
126. Dic₉₇	#131	semidirect
133. C₉₇	#2906	semidirect
140. #424	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690818	2
C₂×#15690816	2
#20261203	2
#20260762	2
#20260760	2
#20260758	2
#20260757	2
C₂×#15690739	2
C₄⋊C₈×D₉₇	2
#424×C₉₇	2
#2906	97