Subgroups — #27150398

Order 12416.

392 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₉₇	C₂	1	1	yes	maximal, characteristic
2	#20260793	C₂	1	2	yes	maximal, complemented
3	#20260793	C₂	1	2	yes	maximal, complemented
4	#20260789	C₂	1	2	yes	maximal, complemented
5	#20260785	C₂	1	2	yes	maximal, complemented
6	#20260758	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15690739	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260702	C₂	1	1	yes	maximal, characteristic
9	#20260693	C₂	1	1	yes	maximal, characteristic, complemented
10	#20260690	C₂	1	1	yes	maximal, characteristic, complemented
11	#408×C₉₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#13095495	C₂²	1	1	yes	characteristic
13	C₂²×#11715895	C₂²	1	1	yes	characteristic
14	Q₈×Dic₉₇	C₄	1	1	yes	characteristic
15	Q₈×Dic₉₇	C₄	1	1	yes	characteristic
16	D₄×Dic₉₇	C₄	1	2	yes	—
17	D₄×Dic₉₇	C₄	1	1	yes	characteristic
18	D₄×Dic₉₇	C₄	1	2	yes	—
19	D₄×Dic₉₇	C₄	1	1	yes	characteristic
20	#15690769	C₂²	1	1	yes	characteristic
21	#15690769	C₂²	1	2	yes	—
22	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
24	#15690739	C₂²	1	1	yes	characteristic
25	#15690739	C₂²	1	1	yes	characteristic
26	#15690739	C₂²	1	1	yes	characteristic
27	#15690739	C₂²	1	1	yes	characteristic
28	#15690739	C₂²	1	2	yes	—
29	#15690739	C₂²	1	2	yes	—
30	C₂²⋊#11715895	C₂²	1	2	yes	—
31	C₂²⋊#11715895	C₂²	1	2	yes	—
32	#15690720	C₂²	1	2	yes	—
33	#15690720	C₂²	1	2	yes	—
34	C₈×Dic₉₇	C₂²	1	2	yes	—
35	C₈×Dic₉₇	C₂²	1	2	yes	—
36	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
41	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
42	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
44	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
45	#13095495	—	2	2	no	—
46	C₂×#11715895	—	2	2	no	—
47	C₂×#11715895	C₂³	1	1	yes	characteristic
48	C₂×#11715895	C₂³	1	1	yes	characteristic
49	C₂×#11715895	C₂³	1	2	yes	—
50	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
54	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
57	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
58	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
60	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
62	C₄×Dic₉₇	C₂³	1	2	yes	—
63	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
64	#41×C₉₇	—	2	2	no	—
65	#41×C₉₇	—	2	2	no	—
66	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
67	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
68	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
70	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
71	M₁₆×C₉₇	—	2	2	no	—
72	C₂×C₈×C₉₇	—	2	2	no	—
73	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₉₇	C₂³	1	2	yes	—
76	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
77	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
78	Q₈×C₉₇	—	2	2	no	—
79	D₄×C₉₇	—	2	4	no	—
80	D₄×C₉₇	—	2	2	no	—
81	C₂×C₄×C₉₇	—	2	2	no	—
82	C₂×C₄×C₉₇	—	2	2	no	—
83	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉₇	#41	1	2	yes	—
86	C₂×C₄×C₉₇	#41	1	2	yes	—
87	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₉₇	—	2	2	no	—
89	C₂×Dic₉₇	—	2	4	no	—
90	C₂×Dic₉₇	—	4	1	no	—
91	C₂×Dic₉₇	—	4	2	no	—
92	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
94	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
96	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
98	C₈×C₉₇	—	2	1	no	—
99	C₈×C₉₇	—	2	1	no	—
100	C₈×C₉₇	—	2	2	no	—
101	#11715895	—	2	1	no	—
102	#11715895	—	2	1	no	—
103	#11715895	—	2	2	no	—
104	C₂²×C₉₇	—	2	1	no	—
105	C₂²×C₉₇	—	2	1	no	—
106	C₂²×C₉₇	—	2	2	no	—
107	C₂²×C₉₇	—	2	2	no	—
108	C₂²×C₉₇	—	2	2	no	—
109	C₂²×C₉₇	#117	1	2	yes	—
110	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
111	C₄×C₉₇	—	2	2	no	—
112	C₄×C₉₇	#117	1	2	yes	—
113	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
114	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
115	Dic₉₇	—	2	1	no	—
116	Dic₉₇	—	2	2	no	—
117	Dic₉₇	—	2	1	no	—
118	Dic₉₇	—	2	1	no	—
119	Dic₉₇	—	2	2	no	—
120	Dic₉₇	#131	1	2	yes	—
121	C₂×C₉₇	—	2	2	no	—
122	C₂×C₉₇	—	4	2	no	—
123	C₂×C₉₇	C₂×#117	1	1	yes	characteristic
124	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
125	C₂×C₉₇	#568	1	1	yes	characteristic
126	#2925	—	97	1	no	maximal, Sylow
127	C₉₇	#2925	1	1	yes	characteristic, complemented, Sylow
128	C₄×#41	—	97	1	no	—
129	#433	—	97	2	no	—
130	#433	—	97	2	no	—
131	#432	—	97	2	no	—
132	#431	—	97	2	no	—
133	#423	—	97	1	no	—
134	C₂×C₄⋊C₈	—	97	1	no	—
135	#408	—	97	1	no	—
136	#408	D₉₇	1	1	yes	characteristic
137	#405	—	97	1	no	—
138	#405	—	97	1	no	—
139	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
140	C₂×M₁₆	—	97	1	no	—
141	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
142	C₂²×C₈	—	97	1	no	—
143	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
144	C₄×Q₈	—	97	1	no	—
145	C₄×Q₈	—	97	1	no	—
146	C₄×D₄	—	97	2	no	—
147	C₄×D₄	—	97	1	no	—
148	C₄×D₄	—	97	2	no	—
149	C₄×D₄	—	97	1	no	—
150	#117	—	97	1	no	—
151	#117	—	97	2	no	—
152	C₂×C₄²	—	97	1	no	—
153	C₂×C₄²	—	97	2	no	—
154	C₄⋊C₈	—	97	1	no	—
155	C₄⋊C₈	—	97	1	no	—
156	C₄⋊C₈	—	97	1	no	—
157	C₄⋊C₈	—	97	1	no	—
158	C₄⋊C₈	—	97	2	no	—
159	C₄⋊C₈	—	97	2	no	—
160	C₂²⋊C₈	—	97	2	no	—
161	C₂²⋊C₈	—	97	2	no	—
162	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
163	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
164	#97	—	97	2	no	—
165	#97	—	97	2	no	—
166	C₄×C₈	—	97	2	no	—
167	C₄×C₈	—	97	2	no	—
168	#41	—	2	2	no	—
169	#41	—	2	2	no	—
170	C₂×Q₈	C₄×D₉₇	1	1	yes	characteristic
171	C₂×D₄	C₄×D₉₇	1	2	yes	—
172	C₂×D₄	C₄×D₉₇	1	1	yes	characteristic
173	C₂²×C₄	—	97	2	no	—
174	C₂²×C₄	—	97	1	no	—
175	C₂²×C₄	—	97	2	no	—
176	C₂²×C₄	—	97	1	no	—
177	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
178	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	194	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	97	1	no	—
183	C₂×C₈	—	97	2	no	—
184	C₂×C₈	—	97	1	no	—
185	C₂×C₈	—	194	2	no	—
186	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
187	C₂×C₈	C₂²×D₉₇	1	2	yes	—
188	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
189	C₄⋊C₄	—	97	2	no	—
190	C₄⋊C₄	—	97	2	no	—
191	C₄⋊C₄	—	97	1	no	—
192	C₄⋊C₄	—	97	1	no	—
193	C₂²⋊C₄	—	97	2	no	—
194	C₂²⋊C₄	—	97	2	no	—
195	C₂²⋊C₄	—	97	2	no	—
196	C₄²	—	97	2	no	—
197	C₄²	—	97	1	no	—
198	C₄²	—	97	2	no	—
199	C₄²	—	97	1	no	—
200	C₄²	—	97	2	no	—
201	C₄²	—	97	1	no	—
202	C₄²	—	97	1	no	—
203	C₂³	C₂×C₄×D₉₇	1	2	yes	—
204	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
205	Q₈	—	2	2	no	—
206	D₄	—	2	4	no	—
207	D₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	97	1	no	—
211	C₂×C₄	—	97	1	no	—
212	C₂×C₄	—	97	2	no	—
213	C₂×C₄	—	97	1	no	—
214	C₂×C₄	—	97	2	no	—
215	C₂×C₄	—	97	1	no	—
216	C₂×C₄	—	388	1	no	—
217	C₂×C₄	—	388	2	no	—
218	C₂×C₄	—	194	2	no	—
219	C₂×C₄	—	194	4	no	—
220	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
222	C₂×C₄	#13095510	1	2	yes	—
223	C₂×C₄	#13095510	1	2	yes	—
224	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	2	no	—
228	C₈	—	194	1	no	—
229	C₈	—	194	1	no	—
230	C₈	—	194	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	#15690766	1	2	yes	—
237	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	97	2	no	—
240	C₄	—	194	2	no	—
241	C₄	—	194	1	no	—
242	C₄	—	194	1	no	—
243	C₄	—	194	2	no	—
244	C₄	—	194	1	no	—
245	C₄	#15690766	1	2	yes	—
246	C₄	C₂×#13095510	1	1	yes	characteristic
247	C₄	C₂×#13095510	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×#15690766	1	1	yes	characteristic
251	C₂	#131×D₉₇	1	1	yes	characteristic
252	C₂	#20261500	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#408×C₉₇	C₂
prime radical	C₂×#41×C₉₇	C₂²
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#408	D₉₇
hypocenter	C₉₇	#2925
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. #20260793	C₂	semidirect
3. #20260793	C₂	semidirect
4. #20260789	C₂	semidirect
5. #20260785	C₂	semidirect
6. #20260758	C₂	semidirect
7. C₂×#15690739	C₂	semidirect
9. #20260693	C₂	semidirect
10. #20260690	C₂	semidirect
127. C₉₇	#2925	semidirect

Maximal subgroups

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

subgroup	index
#41×Dic₉₇	2
#20260793	2
#20260789	2
#20260785	2
#20260758	2
C₂×#15690739	2
#20260702	2
#20260693	2
#20260690	2
#408×C₉₇	2
#2925	97