Subgroups — #27150899

Order 12416.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261272	C₂	1	1	yes	maximal, characteristic
2	Q₈×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260812	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260808	C₂	1	2	yes	maximal, complemented
5	C₈×D₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260800	C₂	1	2	yes	maximal, complemented
7	#20260797	C₂	1	2	yes	maximal, complemented
8	#20260786	C₂	1	1	yes	maximal, characteristic, complemented
9	#20260760	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×#13095494	C₂²	1	2	yes	—
13	C₂×C₈×D₉₇	C₂²	1	2	yes	—
14	C₂×C₈×D₉₇	C₂²	1	2	yes	—
15	#15690805	C₄	1	2	yes	—
16	#15690801	C₄	1	1	yes	characteristic
17	#15690799	C₄	1	1	yes	characteristic
18	#15690795	C₄	1	2	yes	—
19	#15690786	C₄	1	1	yes	characteristic
20	#15690785	C₄	1	1	yes	characteristic
21	C₄×D₃₈₈	C₂²	1	2	yes	—
22	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
23	#15690767	C₂²	1	2	yes	—
24	C₄²×D₉₇	C₂²	1	1	yes	characteristic
25	C₄⋊#11715895	C₂²	1	1	yes	characteristic
26	C₄⋊#11715895	C₂²	1	2	yes	—
27	#15690739	C₂²	1	2	yes	—
28	#15690739	C₂²	1	1	yes	characteristic
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	C₄×#11715895	C₂²	1	1	yes	characteristic
35	C₄×#11715895	C₂²	1	2	yes	—
36	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
39	#97×C₉₇	C₂²	1	2	yes	—
40	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
41	C₂×D₃₈₈	C₂×C₄	1	2	yes	—
42	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₉₇	C₂³	1	2	yes	—
47	#13095494	—	4	2	no	—
48	C₂×#11715895	C₂³	1	2	yes	—
49	C₂×#11715895	C₂³	1	1	yes	characteristic
50	C₂×#11715895	C₂³	1	1	yes	characteristic
51	C₈×D₉₇	—	4	2	no	—
52	C₈×D₉₇	—	4	2	no	—
53	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
55	#13095490	C₂×C₄	1	2	yes	—
56	#13095490	C₂×C₄	1	2	yes	—
57	#13095488	C₂×C₄	1	2	yes	—
58	#13095488	C₂×C₄	1	2	yes	—
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	1	yes	characteristic
64	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₉₇	C₂³	1	2	yes	—
68	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
71	C₄²×C₉₇	C₂³	1	1	yes	characteristic
72	C₄²×C₉₇	C₂³	1	2	yes	—
73	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
74	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₉₇	—	2	2	no	—
76	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
85	D₃₈₈	—	2	4	no	—
86	D₃₈₈	—	2	2	no	—
87	C₄×D₉₇	—	2	2	no	—
88	C₄×D₉₇	—	2	2	no	—
89	C₄×D₉₇	—	2	1	no	—
90	C₄×D₉₇	—	2	2	no	—
91	C₄×D₉₇	—	2	1	no	—
92	C₄×D₉₇	—	2	2	no	—
93	C₈×C₉₇	—	2	1	no	—
94	C₈×C₉₇	—	2	2	no	—
95	C₈×C₉₇	—	2	1	no	—
96	#11715895	—	2	2	no	—
97	#11715895	—	2	1	no	—
98	#11715895	—	2	1	no	—
99	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₉₇	—	2	1	no	—
101	C₂×D₉₇	—	2	1	no	—
102	C₂×D₉₇	—	2	2	no	—
103	C₂×D₉₇	—	2	2	no	—
104	C₂×D₉₇	—	2	2	no	—
105	C₂×D₉₇	#131	1	2	yes	—
106	C₄×C₉₇	—	2	1	no	—
107	C₄×C₉₇	—	2	2	no	—
108	C₄×C₉₇	—	2	2	no	—
109	C₄×C₉₇	#131	1	2	yes	—
110	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
111	C₄×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
112	Dic₉₇	—	2	1	no	—
113	Dic₉₇	—	2	2	no	—
114	Dic₉₇	—	2	1	no	—
115	Dic₉₇	—	2	1	no	—
116	C₂×C₉₇	#520	1	1	yes	characteristic
117	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
118	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
119	D₉₇	—	2	2	no	—
120	D₉₇	—	4	2	no	—
121	#2994	—	97	1	no	maximal, Sylow
122	C₉₇	#2994	1	1	yes	characteristic, complemented, Sylow
123	#529	—	97	1	no	—
124	#446	D₉₇	1	1	yes	characteristic, complemented
125	C₈×Q₈	—	97	1	no	—
126	#436	—	97	1	no	—
127	#435	—	97	2	no	—
128	C₈×D₄	—	97	2	no	—
129	C₈×D₄	—	97	1	no	—
130	#433	—	97	2	no	—
131	#431	—	97	1	no	—
132	#424	—	97	2	no	—
133	C₂×C₄⋊C₈	—	97	1	no	—
134	C₂×M₁₆	—	97	2	no	—
135	C₂²×C₈	—	97	2	no	—
136	C₂²×C₈	—	97	2	no	—
137	#126	—	97	2	no	—
138	#125	—	97	1	no	—
139	#124	—	97	1	no	—
140	#123	—	97	2	no	—
141	C₂²⋊Q₈	—	97	1	no	—
142	#121	—	97	1	no	—
143	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic
144	C₄×D₄	—	97	2	no	—
145	C₄×D₄	—	97	1	no	—
146	#117	—	97	2	no	—
147	C₂×C₄²	—	97	1	no	—
148	C₄⋊C₈	—	97	2	no	—
149	C₄⋊C₈	—	97	1	no	—
150	C₄⋊C₈	—	97	1	no	—
151	C₄⋊C₈	—	97	1	no	—
152	C₄⋊C₈	—	97	2	no	—
153	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
155	C₂²⋊C₈	—	97	1	no	—
156	C₂²⋊C₈	—	97	1	no	—
157	C₂²⋊C₈	—	97	2	no	—
158	C₂²⋊C₈	—	97	2	no	—
159	#97	C₂×D₉₇	1	2	yes	—
160	C₄×C₈	—	97	1	no	—
161	C₄×C₈	—	97	2	no	—
162	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
163	C₂×Q₈	C₄×D₉₇	1	1	yes	characteristic
164	C₂×D₄	—	97	2	no	—
165	C₂×D₄	—	97	1	no	—
166	C₂²×C₄	—	97	1	no	—
167	C₂²×C₄	—	97	1	no	—
168	C₂²×C₄	—	97	1	no	—
169	C₂²×C₄	—	97	2	no	—
170	M₁₆	—	388	2	no	—
171	C₂×C₈	—	97	1	no	—
172	C₂×C₈	—	97	2	no	—
173	C₂×C₈	—	97	1	no	—
174	C₂×C₈	—	388	2	no	—
175	C₂×C₈	—	388	2	no	—
176	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₉₇	1	2	yes	—
179	C₄⋊C₄	—	97	2	no	—
180	C₄⋊C₄	—	97	2	no	—
181	C₄⋊C₄	—	97	1	no	—
182	C₄⋊C₄	—	97	2	no	—
183	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
184	C₄⋊C₄	C₄×D₉₇	1	2	yes	—
185	C₂²⋊C₄	—	97	2	no	—
186	C₂²⋊C₄	—	97	2	no	—
187	C₂²⋊C₄	—	97	2	no	—
188	C₂²⋊C₄	—	97	2	no	—
189	C₂²⋊C₄	—	97	2	no	—
190	C₄²	—	97	1	no	—
191	C₄²	—	97	1	no	—
192	C₄²	—	97	1	no	—
193	C₄²	C₂²×D₉₇	1	1	yes	characteristic
194	C₄²	C₂²×D₉₇	1	2	yes	—
195	C₂³	—	97	1	no	—
196	C₂³	—	97	2	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	194	4	no	—
199	D₄	—	194	2	no	—
200	C₂×C₄	—	97	1	no	—
201	C₂×C₄	—	97	1	no	—
202	C₂×C₄	—	97	1	no	—
203	C₂×C₄	—	97	2	no	—
204	C₂×C₄	—	194	2	no	—
205	C₂×C₄	—	194	2	no	—
206	C₂×C₄	—	194	1	no	—
207	C₂×C₄	—	194	2	no	—
208	C₂×C₄	—	194	2	no	—
209	C₂×C₄	—	194	1	no	—
210	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	—	194	2	no	—
219	C₈	—	194	1	no	—
220	C₈	—	194	1	no	—
221	C₂²	—	97	2	no	—
222	C₂²	—	194	2	no	—
223	C₂²	—	194	2	no	—
224	C₂²	—	194	2	no	—
225	C₂²	—	194	1	no	—
226	C₂²	—	194	1	no	—
227	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	194	1	no	—
232	C₄	—	194	2	no	—
233	C₄	—	194	1	no	—
234	C₄	—	194	1	no	—
235	C₄	#15690818	1	2	yes	—
236	C₄	#15690856	1	1	yes	characteristic
237	C₄	#15690858	1	1	yes	characteristic
238	C₂	—	388	2	no	—
239	C₂	—	194	2	no	—
240	C₂	#20261217	1	1	yes	characteristic
241	C₂	#131×D₉₇	1	1	yes	characteristic
242	C₂	C₂×#15690818	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	#15690785	C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#446	D₉₇
hypocenter	C₉₇	#2994
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. Q₈×#11715895	C₂	semidirect
3. #20260812	C₂	semidirect
4. #20260808	C₂	semidirect
5. C₈×D₃₈₈	C₂	semidirect
6. #20260800	C₂	semidirect
7. #20260797	C₂	semidirect
8. #20260786	C₂	semidirect
9. #20260760	C₂	semidirect
10. C₄⋊C₈×D₉₇	C₂	semidirect
11. #446×C₉₇	C₂	semidirect
122. C₉₇	#2994	semidirect
124. #446	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261272	2
Q₈×#11715895	2
#20260812	2
#20260808	2
C₈×D₃₈₈	2
#20260800	2
#20260797	2
#20260786	2
#20260760	2
C₄⋊C₈×D₉₇	2
#446×C₉₇	2
#2994	97