Subgroups — #27150599

Order 12416.

456 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₉₇	C₂	1	1	yes	maximal, characteristic
2	#20261211	C₂	1	2	yes	maximal
3	#20260840	C₂	1	2	yes	maximal, complemented
4	#20260839	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260829	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260772	C₂	1	1	yes	maximal, characteristic
9	#20260742	C₂	1	2	yes	maximal
10	#20260689	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₉₇	C₂²	1	2	yes	—
13	C₂×Dic₃₈₈	C₂²	1	1	yes	characteristic
14	C₂×#13095501	C₂²	1	2	yes	—
15	Q₁₆×D₉₇	C₄	1	8	yes	complemented
16	C₂×#13095494	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
18	Q₈×Dic₉₇	C₂²	1	2	yes	—
19	C₄×Dic₁₉₄	—	2	2	no	—
20	C₄×Dic₁₉₄	C₂²	1	2	yes	—
21	#15690775	—	2	2	no	—
22	#15690775	C₂²	1	2	yes	—
23	#15690768	—	2	2	no	—
24	#15690768	C₂²	1	2	yes	—
25	#15690767	—	2	2	no	—
26	#15690767	C₂²	1	1	yes	characteristic
27	#15690744	C₂²	1	1	yes	characteristic
28	#15690741	C₂²	1	2	yes	—
29	#15690735	C₂²	1	2	yes	—
30	#15690734	C₂²	1	2	yes	—
31	#15690733	C₂²	1	2	yes	—
32	#15690721	C₂²	1	1	yes	characteristic
33	#15690720	C₂²	1	1	yes	characteristic
34	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₉₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₉₇	C₂²	1	2	yes	—
37	#107×C₉₇	C₂²	1	1	yes	characteristic
38	#103×C₉₇	C₂²	1	2	yes	—
39	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
40	C₂×Dic₁₉₄	—	2	2	no	—
41	C₂×Dic₁₉₄	C₂³	1	2	yes	—
42	Q₈×D₉₇	—	2	4	no	—
43	Q₈×D₉₇	C₂×C₄	1	8	yes	—
44	C₂×C₄×D₉₇	—	2	2	no	—
45	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
46	Dic₃₈₈	C₂×C₄	1	4	yes	—
47	#13095501	C₂×C₄	1	8	yes	—
48	#13095494	—	2	2	no	—
49	C₂×#11715895	C₂³	1	1	yes	characteristic
50	C₂×#11715895	C₂³	1	1	yes	characteristic
51	C₈×D₉₇	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₉₇	—	2	2	no	—
53	C₄⋊Dic₉₇	C₂³	1	2	yes	—
54	#13095490	—	2	2	no	—
55	#13095490	—	2	2	no	—
56	#13095490	D₄	1	1	yes	characteristic
57	#13095490	D₄	1	1	yes	characteristic
58	#13095488	—	2	2	no	—
59	#13095488	D₄	1	1	yes	characteristic
60	#13095488	D₄	1	1	yes	characteristic
61	C₄×Dic₉₇	—	2	2	no	—
62	C₄×Dic₉₇	—	2	2	no	—
63	C₄×Dic₉₇	—	2	2	no	—
64	C₄×Dic₉₇	—	2	2	no	—
65	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
67	Q₁₆×C₉₇	C₂×C₄	1	4	yes	—
68	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₉₇	—	2	2	no	—
71	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
72	C₄²×C₉₇	—	2	2	no	—
73	C₄²×C₉₇	C₂³	1	1	yes	characteristic
74	C₂²×D₉₇	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₉₇	—	2	2	no	—
76	Q₈×C₉₇	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₉₇	—	2	2	no	—
78	C₂×C₄×C₉₇	—	2	2	no	—
79	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₉₇	—	2	2	no	—
83	C₂×Dic₉₇	—	2	2	no	—
84	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
87	C₄×D₉₇	—	2	8	no	—
88	C₄×D₉₇	C₂²×C₄	1	2	yes	—
89	C₄×D₉₇	#41	1	2	yes	—
90	Dic₁₉₄	—	2	2	no	—
91	Dic₁₉₄	—	2	8	no	—
92	Dic₁₉₄	C₂²×C₄	1	4	yes	—
93	C₈×C₉₇	—	2	1	no	—
94	C₈×C₉₇	C₂²×C₄	1	2	yes	—
95	#11715895	—	2	1	no	—
96	#11715895	C₂²×C₄	1	2	yes	—
97	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×D₉₇	—	2	1	no	—
99	C₂×D₉₇	—	2	1	no	—
100	C₂×D₉₇	C₄×D₄	1	2	yes	—
101	C₄×C₉₇	—	2	4	no	—
102	C₄×C₉₇	—	2	1	no	—
103	C₄×C₉₇	—	2	1	no	—
104	C₄×C₉₇	—	4	2	no	—
105	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
107	Dic₉₇	—	2	4	no	—
108	Dic₉₇	—	2	1	no	—
109	Dic₉₇	—	2	1	no	—
110	Dic₉₇	—	4	2	no	—
111	Dic₉₇	C₄×D₄	1	2	yes	—
112	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₉₇	#578	1	1	yes	characteristic
114	C₂×C₉₇	#578	1	1	yes	characteristic
115	D₉₇	—	2	2	no	—
116	#2950	—	97	1	no	maximal, Sylow
117	C₉₇	#2950	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	—	97	1	no	—
119	#519	—	97	2	no	—
120	#441	—	97	2	no	—
121	#441	—	97	2	no	—
122	C₄×Q₁₆	—	97	1	no	—
123	C₄×Q₁₆	—	97	1	no	—
124	C₄×Q₁₆	—	97	1	no	—
125	C₄×Q₁₆	D₉₇	1	1	yes	characteristic, complemented
126	#427	—	97	1	no	—
127	#419	—	97	2	no	—
128	#405	—	97	1	no	—
129	C₂²×Q₈	—	97	2	no	—
130	C₂×Q₁₆	—	97	2	no	—
131	C₂×Q₁₆	—	97	8	no	—
132	C₂×Q₁₆	—	97	1	no	—
133	C₂×Q₁₆	C₂×D₉₇	1	1	yes	characteristic
134	C₂×M₁₆	—	97	1	no	—
135	C₂²×C₈	—	97	1	no	—
136	C₄×Q₈	—	97	2	no	—
137	C₄×Q₈	—	97	2	no	—
138	C₄×Q₈	—	97	2	no	—
139	C₄×Q₈	—	194	2	no	—
140	C₄×Q₈	—	194	2	no	—
141	C₄×Q₈	C₂×D₉₇	1	2	yes	—
142	#117	—	97	1	no	—
143	#117	—	97	2	no	—
144	#117	—	194	2	no	—
145	#117	—	194	2	no	—
146	#107	—	97	1	no	—
147	#107	C₂×D₉₇	1	1	yes	characteristic
148	#106	—	97	2	no	—
149	#103	—	97	2	no	—
150	#103	—	97	2	no	—
151	#103	—	97	2	no	—
152	#103	C₂×D₉₇	1	2	yes	—
153	#97	—	97	1	no	—
154	#97	—	97	1	no	—
155	C₄×C₈	—	97	1	no	—
156	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
157	C₂×Q₈	—	97	8	no	—
158	C₂×Q₈	—	97	2	no	—
159	C₂×Q₈	—	194	2	no	—
160	C₂×Q₈	—	194	4	no	—
161	C₂×Q₈	C₂²×D₉₇	1	2	yes	—
162	C₂²×C₄	—	97	1	no	—
163	C₂²×C₄	—	194	2	no	—
164	Q₁₆	—	97	8	no	—
165	Q₁₆	—	97	4	no	—
166	Q₁₆	C₄×D₉₇	1	4	yes	—
167	M₁₆	—	194	2	no	—
168	C₂×C₈	—	97	1	no	—
169	C₂×C₈	—	97	4	no	—
170	C₂×C₈	—	97	1	no	—
171	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	97	1	no	—
175	C₄⋊C₄	—	97	1	no	—
176	C₄⋊C₄	—	97	2	no	—
177	C₄⋊C₄	—	194	2	no	—
178	C₄⋊C₄	—	194	2	no	—
179	C₄⋊C₄	—	194	2	no	—
180	C₄⋊C₄	C₂²×D₉₇	1	2	yes	—
181	C₂²⋊C₄	—	97	1	no	—
182	C₂²⋊C₄	—	97	1	no	—
183	C₂²⋊C₄	—	194	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	97	1	no	—
186	C₄²	—	194	2	no	—
187	C₄²	—	194	2	no	—
188	C₄²	—	194	2	no	—
189	C₄²	—	194	2	no	—
190	C₄²	C₂²×D₉₇	1	1	yes	characteristic
191	C₂³	—	97	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	97	4	no	—
194	Q₈	—	194	8	no	—
195	Q₈	—	194	2	no	—
196	Q₈	C₂×C₄×D₉₇	1	4	yes	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	97	1	no	—
200	C₂×C₄	—	97	2	no	—
201	C₂×C₄	—	97	2	no	—
202	C₂×C₄	—	97	1	no	—
203	C₂×C₄	—	97	1	no	—
204	C₂×C₄	—	194	8	no	—
205	C₂×C₄	—	194	2	no	—
206	C₂×C₄	—	194	2	no	—
207	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
208	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	97	2	no	—
212	C₈	—	194	1	no	—
213	C₈	C₂×C₄×D₉₇	1	2	yes	—
214	C₂²	—	97	2	no	—
215	C₂²	—	194	1	no	—
216	C₂²	—	194	1	no	—
217	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	97	2	no	—
223	C₄	—	388	2	no	—
224	C₄	—	194	4	no	—
225	C₄	—	194	1	no	—
226	C₄	—	194	1	no	—
227	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
228	C₄	#41×D₉₇	1	1	yes	characteristic
229	C₂	—	194	2	no	—
230	C₂	C₄×D₄×D₉₇	1	1	yes	characteristic
231	C₂	#20261548	1	1	yes	characteristic
232	C₂	#20261549	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₉₇
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₄×Q₁₆×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂²×D₉₇	C₂×D₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₄×Q₁₆	D₉₇
hypocenter	C₉₇	#2950
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #20260840	C₂	semidirect
4. #20260839	C₂	semidirect
5. C₄×Dic₃₈₈	C₂	semidirect
6. Q₁₆×Dic₉₇	C₂	semidirect
7. #20260829	C₂	semidirect
11. C₄×Q₁₆×C₉₇	C₂	semidirect
15. Q₁₆×D₉₇	C₄	semidirect
117. C₉₇	#2950	semidirect
125. C₄×Q₁₆	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×D₉₇	2
#20261211	2
#20260840	2
#20260839	2
C₄×Dic₃₈₈	2
Q₁₆×Dic₉₇	2
#20260829	2
#20260772	2
#20260742	2
#20260689	2
C₄×Q₁₆×C₉₇	2
#2950	97