Subgroups — #27150711

Order 12416.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261269	C₂	1	1	yes	maximal, characteristic
2	#20260865	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260814	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260806	C₂	1	2	yes	maximal, complemented
6	C₈×D₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₉₇	C₂	1	2	yes	maximal, complemented
8	#20260794	C₂	1	2	yes	maximal, complemented
9	#20260792	C₂	1	1	yes	maximal, characteristic, complemented
10	#20260689	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095494	C₂²	1	2	yes	—
13	C₂×C₈×D₉₇	C₂²	1	2	yes	—
14	#15690804	C₄	1	2	yes	—
15	#15690802	C₄	1	1	yes	characteristic
16	#15690798	C₄	1	1	yes	characteristic
17	#15690793	C₄	1	2	yes	—
18	#15690789	C₄	1	1	yes	characteristic
19	#15690784	C₄	1	1	yes	characteristic
20	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
21	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₉₇	C₂²	1	2	yes	—
23	#15690767	C₂²	1	2	yes	—
24	C₄⋊#11715895	C₂²	1	1	yes	characteristic
25	C₄⋊#11715895	C₂²	1	1	yes	characteristic
26	#15690739	C₂²	1	2	yes	—
27	#15690739	C₂²	1	2	yes	—
28	C₂²⋊#11715895	C₂²	1	2	yes	—
29	#15690722	C₂²	1	2	yes	—
30	#15690722	C₂²	1	2	yes	—
31	#15690721	C₂²	1	2	yes	—
32	#15690720	C₂²	1	2	yes	—
33	C₈×Dic₉₇	C₂²	1	2	yes	—
34	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
42	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₉₇	C₂³	1	2	yes	—
45	#13095494	—	2	4	no	—
46	C₂×#11715895	C₂³	1	2	yes	—
47	C₂×#11715895	C₂³	1	2	yes	—
48	C₈×D₉₇	—	2	4	no	—
49	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
51	#13095490	C₂×C₄	1	2	yes	—
52	#13095490	C₂×C₄	1	2	yes	—
53	#13095490	C₂×C₄	1	2	yes	—
54	#13095490	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
56	#13095488	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	C₄×Dic₉₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₉₇	—	2	2	no	—
65	C₂×C₈×C₉₇	—	2	2	no	—
66	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
70	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₉₇	C₂³	1	1	yes	characteristic
73	C₄²×C₉₇	C₂³	1	1	yes	characteristic
74	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₉₇	—	2	2	no	—
77	C₂×C₄×C₉₇	—	2	2	no	—
78	C₂×C₄×C₉₇	—	2	1	no	—
79	C₂×C₄×C₉₇	—	2	1	no	—
80	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₉₇	—	2	4	no	—
87	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
88	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
89	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
90	D₃₈₈	—	2	2	no	—
91	C₄×D₉₇	—	2	2	no	—
92	C₄×D₉₇	—	2	2	no	—
93	Dic₁₉₄	—	2	2	no	—
94	C₈×C₉₇	—	2	1	no	—
95	C₈×C₉₇	—	2	1	no	—
96	C₈×C₉₇	#41	1	2	yes	—
97	C₈×C₉₇	#41	1	2	yes	—
98	#11715895	—	2	2	no	—
99	#11715895	—	2	2	no	—
100	C₂²×C₉₇	—	2	1	no	—
101	C₂²×C₉₇	—	2	1	no	—
102	C₂²×C₉₇	#131	1	2	yes	—
103	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₉₇	—	2	2	no	—
105	C₂×D₉₇	—	2	2	no	—
106	C₂×D₉₇	—	2	2	no	—
107	C₄×C₉₇	—	2	1	no	—
108	C₄×C₉₇	—	2	1	no	—
109	C₄×C₉₇	—	2	1	no	—
110	C₄×C₉₇	—	2	1	no	—
111	C₄×C₉₇	#131	1	2	yes	—
112	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
113	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
114	Dic₉₇	—	2	2	no	—
115	Dic₉₇	—	2	2	no	—
116	Dic₉₇	—	2	2	no	—
117	C₂×C₉₇	—	2	2	no	—
118	C₂×C₉₇	C₄×#41	1	1	yes	characteristic
119	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
120	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
121	D₉₇	—	4	2	no	—
122	#2966	—	97	1	no	maximal, Sylow
123	C₉₇	#2966	1	1	yes	characteristic, complemented, Sylow
124	#529	—	97	1	no	—
125	#446	—	97	1	no	—
126	C₈×Q₈	—	97	1	no	—
127	#436	—	97	1	no	—
128	#435	—	97	2	no	—
129	C₈×D₄	—	97	2	no	—
130	C₈×D₄	—	97	1	no	—
131	#433	—	97	2	no	—
132	#432	—	97	1	no	—
133	#405	—	97	2	no	—
134	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic, complemented
135	C₂×M₁₆	—	97	2	no	—
136	C₂²×C₈	—	97	2	no	—
137	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
138	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
139	#126	—	97	2	no	—
140	#125	—	97	1	no	—
141	#124	—	97	1	no	—
142	#123	—	97	2	no	—
143	C₂²⋊Q₈	—	97	1	no	—
144	#121	—	97	1	no	—
145	C₄×Q₈	—	97	1	no	—
146	C₄×D₄	—	97	2	no	—
147	C₄×D₄	—	97	1	no	—
148	#117	—	97	2	no	—
149	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
150	C₄⋊C₈	—	97	2	no	—
151	C₄⋊C₈	—	97	2	no	—
152	C₄⋊C₈	—	97	1	no	—
153	C₄⋊C₈	—	97	1	no	—
154	C₂²⋊C₈	—	97	2	no	—
155	C₂²⋊C₈	—	97	2	no	—
156	C₂²⋊C₈	—	97	2	no	—
157	#97	—	97	2	no	—
158	#97	—	97	2	no	—
159	C₄×C₈	—	97	2	no	—
160	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
161	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
162	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
163	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
164	C₂×Q₈	—	97	1	no	—
165	C₂×D₄	—	97	2	no	—
166	C₂×D₄	—	97	1	no	—
167	C₂²×C₄	—	97	2	no	—
168	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
171	M₁₆	—	194	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	97	2	no	—
175	C₂×C₈	—	97	2	no	—
176	C₂×C₈	—	194	4	no	—
177	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
181	C₄⋊C₄	—	97	2	no	—
182	C₄⋊C₄	—	97	2	no	—
183	C₄⋊C₄	—	97	2	no	—
184	C₄⋊C₄	—	97	2	no	—
185	C₄⋊C₄	—	97	1	no	—
186	C₄⋊C₄	—	97	1	no	—
187	C₂²⋊C₄	—	97	2	no	—
188	C₂²⋊C₄	—	97	2	no	—
189	C₂²⋊C₄	—	97	2	no	—
190	C₂²⋊C₄	—	97	2	no	—
191	C₂²⋊C₄	—	97	2	no	—
192	C₄²	—	97	2	no	—
193	C₄²	C₄×D₉₇	1	1	yes	characteristic
194	C₄²	C₄×D₉₇	1	1	yes	characteristic
195	C₄²	C₂²×D₉₇	1	1	yes	characteristic
196	C₄²	C₂²×D₉₇	1	1	yes	characteristic
197	C₂³	—	97	2	no	—
198	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
199	Q₈	—	194	2	no	—
200	D₄	—	194	4	no	—
201	D₄	—	194	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	97	2	no	—
207	C₂×C₄	—	97	2	no	—
208	C₂×C₄	—	97	2	no	—
209	C₂×C₄	—	194	2	no	—
210	C₂×C₄	—	194	2	no	—
211	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	194	2	no	—
220	C₈	—	194	2	no	—
221	C₈	#13095511	1	2	yes	—
222	C₈	#13095511	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	194	2	no	—
226	C₂²	—	194	2	no	—
227	C₂²	—	194	2	no	—
228	C₂²	#15690816	1	2	yes	—
229	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	194	2	no	—
235	C₄	—	194	2	no	—
236	C₄	—	194	2	no	—
237	C₄	#15690816	1	2	yes	—
238	C₄	C₂×#13095511	1	1	yes	characteristic
239	C₄	C₂×#13095511	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	—	388	2	no	—
242	C₂	C₄×#13095511	1	1	yes	characteristic
243	C₂	C₂×#15690816	1	1	yes	characteristic
244	C₂	C₂×#15690816	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
prime radical	#15690784	C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#2966
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20260865	C₂	semidirect
3. C₈×Dic₁₉₄	C₂	semidirect
4. #20260814	C₂	semidirect
5. #20260806	C₂	semidirect
6. C₈×D₃₈₈	C₂	semidirect
7. C₈×C₂²⋊D₉₇	C₂	semidirect
8. #20260794	C₂	semidirect
9. #20260792	C₂	semidirect
10. #20260689	C₂	semidirect
11. C₂×C₄×C₈×C₉₇	C₂	semidirect
123. C₉₇	#2966	semidirect
134. C₂×C₄×C₈	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261269	2
#20260865	2
C₈×Dic₁₉₄	2
#20260814	2
#20260806	2
C₈×D₃₈₈	2
C₈×C₂²⋊D₉₇	2
#20260794	2
#20260792	2
#20260689	2
C₂×C₄×C₈×C₉₇	2
#2966	97