Subgroups — C₄×Q₁₆×D₉₇ #27150497

Order 12416.

488 conjugacy classes of subgroups.

The subgroup lattice has 240 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	C₄×Q₈×D₉₇	C₂	1	2	yes	maximal
3	C₄×Dic₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260829	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13095501	C₂	1	2	yes	maximal, complemented
7	#20260827	C₂	1	2	yes	maximal, complemented
8	#107×D₉₇	C₂	1	1	yes	maximal, characteristic
9	#103×D₉₇	C₂	1	2	yes	maximal
10	C₄×C₈×D₉₇	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	direct product factor
16	C₂×C₈×D₉₇	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	C₄⋊C₄×D₉₇	—	2	2	no	—
24	C₄⋊C₄×D₉₇	C₂²	1	2	yes	—
25	C₄²×D₉₇	—	2	2	no	—
26	C₄²×D₉₇	C₂²	1	1	yes	characteristic
27	#15690744	C₂²	1	1	yes	characteristic
28	#15690743	C₂²	1	2	yes	—
29	#15690735	C₂²	1	2	yes	—
30	#15690734	C₂²	1	2	yes	—
31	#15690733	C₂²	1	2	yes	—
32	C₄×#11715895	C₂²	1	1	yes	characteristic
33	C₈×Dic₉₇	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	—	2	2	no	—
46	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
47	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
48	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
49	Dic₃₈₈	C₂×C₄	1	4	yes	complemented
50	#13095501	C₂×C₄	1	8	yes	complemented
51	C₂×#11715895	C₂³	1	1	yes	characteristic
52	C₂×#11715895	C₂³	1	1	yes	characteristic
53	C₈×D₉₇	—	2	2	no	—
54	C₈×D₉₇	C₂×C₄	1	4	yes	—
55	C₄⋊Dic₉₇	—	2	2	no	—
56	C₄⋊Dic₉₇	C₂³	1	2	yes	—
57	#13095490	—	2	2	no	—
58	#13095490	—	2	2	no	—
59	#13095490	—	2	2	no	—
60	C₄×Dic₉₇	—	2	2	no	—
61	C₄×Dic₉₇	—	2	2	no	—
62	C₄×Dic₉₇	—	2	2	no	—
63	C₄×Dic₉₇	D₄	1	1	yes	characteristic
64	C₄×Dic₉₇	D₄	1	1	yes	characteristic
65	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
67	Q₁₆×C₉₇	C₂×C₄	1	4	yes	complemented
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	—	2	2	no	—
89	C₄×D₉₇	—	4	4	no	—
90	C₄×D₉₇	Q₁₆	1	4	yes	direct product factor
91	C₄×D₉₇	C₂²×C₄	1	2	yes	—
92	C₄×D₉₇	#41	1	2	yes	—
93	Dic₁₉₄	—	2	2	no	—
94	Dic₁₉₄	—	2	8	no	—
95	Dic₁₉₄	C₂²×C₄	1	4	yes	—
96	C₈×C₉₇	—	2	1	no	—
97	C₈×C₉₇	C₂²×C₄	1	2	yes	—
98	#11715895	—	2	1	no	—
99	#11715895	C₂²×C₄	1	2	yes	—
100	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×D₉₇	C₄×D₄	1	2	yes	—
102	C₂×D₉₇	C₂×Q₁₆	1	2	yes	—
103	C₂×D₉₇	#135	1	2	yes	—
104	C₄×C₉₇	—	2	4	no	—
105	C₄×C₉₇	—	2	1	no	—
106	C₄×C₉₇	—	4	2	no	—
107	C₄×C₉₇	C₂×Q₁₆	1	2	yes	complemented
108	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
109	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
110	Dic₉₇	—	2	4	no	—
111	Dic₉₇	—	2	1	no	—
112	Dic₉₇	—	4	2	no	—
113	Dic₉₇	C₄×D₄	1	2	yes	—
114	Dic₉₇	C₂×Q₁₆	1	2	yes	complemented
115	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
116	C₂×C₉₇	C₂²×Q₁₆	1	1	yes	characteristic
117	C₂×C₉₇	C₂×#135	1	1	yes	characteristic
118	D₉₇	C₄×Q₁₆	1	4	yes	direct product factor
119	C₂×C₄×Q₁₆	—	97	1	no	maximal, Sylow
120	C₉₇	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₁₆	—	97	1	no	—
122	C₂×C₄×Q₈	—	97	2	no	—
123	C₄×Q₁₆	—	97	1	no	—
124	C₄×Q₁₆	—	97	2	no	—
125	C₄×Q₁₆	—	97	1	no	—
126	C₄×Q₁₆	—	97	2	no	—
127	C₄×Q₁₆	—	97	1	no	—
128	C₄×Q₁₆	D₉₇	1	1	yes	characteristic, direct product factor
129	C₂×#107	—	97	1	no	—
130	C₂×#103	—	97	2	no	—
131	C₂×C₄×C₈	—	97	1	no	—
132	C₂²×Q₈	—	97	2	no	—
133	C₂×Q₁₆	—	97	2	no	—
134	C₂×Q₁₆	—	97	8	no	—
135	C₂×Q₁₆	—	97	1	no	—
136	C₂×Q₁₆	C₂×D₉₇	1	1	yes	characteristic
137	C₂²×C₈	—	97	1	no	—
138	C₂²×C₈	—	97	1	no	—
139	C₄×Q₈	—	97	2	no	—
140	C₄×Q₈	—	97	2	no	—
141	C₄×Q₈	—	97	2	no	—
142	C₄×Q₈	—	194	2	no	—
143	C₄×Q₈	—	194	2	no	—
144	C₄×Q₈	C₂×D₉₇	1	2	yes	—
145	C₂×C₄⋊C₄	—	97	2	no	—
146	C₂×C₄⋊C₄	—	194	2	no	—
147	C₂×C₄²	—	97	1	no	—
148	C₂×C₄²	—	194	2	no	—
149	#107	—	97	1	no	—
150	#107	—	97	2	no	—
151	#107	C₂×D₉₇	1	1	yes	characteristic
152	#103	—	97	2	no	—
153	#103	—	97	2	no	—
154	#103	—	97	2	no	—
155	#103	C₂×D₉₇	1	2	yes	—
156	C₄×C₈	—	97	1	no	—
157	C₄×C₈	—	97	1	no	—
158	C₄×C₈	—	97	1	no	—
159	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
160	C₂×Q₈	—	97	8	no	—
161	C₂×Q₈	—	97	2	no	—
162	C₂×Q₈	—	194	2	no	—
163	C₂×Q₈	—	194	4	no	—
164	C₂×Q₈	C₂²×D₉₇	1	2	yes	—
165	C₂²×C₄	—	97	1	no	—
166	C₂²×C₄	—	97	1	no	—
167	C₂²×C₄	—	97	1	no	—
168	C₂²×C₄	—	194	2	no	—
169	C₂²×C₄	—	194	2	no	—
170	Q₁₆	—	97	8	no	—
171	Q₁₆	—	97	4	no	—
172	Q₁₆	C₄×D₉₇	1	4	yes	direct product factor
173	C₂×C₈	—	97	1	no	—
174	C₂×C₈	—	97	4	no	—
175	C₂×C₈	—	97	1	no	—
176	C₂×C₈	—	194	2	no	—
177	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	97	2	no	—
181	C₄⋊C₄	—	194	2	no	—
182	C₄⋊C₄	—	194	2	no	—
183	C₄⋊C₄	—	194	2	no	—
184	C₄⋊C₄	—	194	2	no	—
185	C₄⋊C₄	C₂²×D₉₇	1	2	yes	—
186	C₄²	—	2	2	no	—
187	C₄²	—	97	1	no	—
188	C₄²	—	97	1	no	—
189	C₄²	—	97	1	no	—
190	C₄²	—	194	2	no	—
191	C₄²	—	194	2	no	—
192	C₄²	—	194	2	no	—
193	C₄²	C₂²×D₉₇	1	1	yes	characteristic
194	C₂³	—	97	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	97	4	no	—
197	Q₈	—	194	8	no	—
198	Q₈	—	194	2	no	—
199	Q₈	C₂×C₄×D₉₇	1	4	yes	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	97	1	no	—
203	C₂×C₄	—	97	1	no	—
204	C₂×C₄	—	97	4	no	—
205	C₂×C₄	—	97	2	no	—
206	C₂×C₄	—	97	2	no	—
207	C₂×C₄	—	97	1	no	—
208	C₂×C₄	—	388	4	no	—
209	C₂×C₄	—	194	2	no	—
210	C₂×C₄	—	194	8	no	—
211	C₂×C₄	—	194	2	no	—
212	C₂×C₄	—	194	2	no	—
213	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
214	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
215	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	97	2	no	—
218	C₈	—	194	1	no	—
219	C₈	C₂×C₄×D₉₇	1	2	yes	—
220	C₂²	—	97	2	no	—
221	C₂²	—	97	2	no	—
222	C₂²	—	97	2	no	—
223	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
224	C₄	—	2	4	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	2	no	—
227	C₄	—	97	2	no	—
228	C₄	—	97	2	no	—
229	C₄	—	388	2	no	—
230	C₄	—	194	4	no	—
231	C₄	—	194	1	no	—
232	C₄	Q₁₆×D₉₇	1	2	yes	direct product factor
233	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
234	C₄	#41×D₉₇	1	1	yes	characteristic
235	C₂	—	97	4	no	—
236	C₂	C₄×D₄×D₉₇	1	1	yes	characteristic
237	C₂	C₂×Q₁₆×D₉₇	1	1	yes	characteristic
238	C₂	#135×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	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	C₂×C₄×Q₁₆
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₃₈₈	C₂	semidirect
4. Q₁₆×Dic₉₇	C₂	semidirect
5. #20260829	C₂	semidirect
6. C₄×#13095501	C₂	semidirect
7. #20260827	C₂	semidirect
11. C₄×Q₁₆×C₉₇	C₂	semidirect
15. Q₁₆×D₉₇	C₄	direct (N × H)
49. Dic₃₈₈	C₂×C₄	semidirect
50. #13095501	C₂×C₄	semidirect
67. Q₁₆×C₉₇	C₂×C₄	semidirect
90. C₄×D₉₇	Q₁₆	direct (N × H)
107. C₄×C₉₇	C₂×Q₁₆	semidirect
114. Dic₉₇	C₂×Q₁₆	semidirect
118. D₉₇	C₄×Q₁₆	direct (N × H)
120. C₉₇	C₂×C₄×Q₁₆	semidirect
128. C₄×Q₁₆	D₉₇	direct (N × H)
172. Q₁₆	C₄×D₉₇	direct (N × H)
232. C₄	Q₁₆×D₉₇	direct (N × H)

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
C₄×Q₈×D₉₇	2
C₄×Dic₃₈₈	2
Q₁₆×Dic₉₇	2
#20260829	2
C₄×#13095501	2
#20260827	2
#107×D₉₇	2
#103×D₉₇	2
C₄×C₈×D₉₇	2
C₄×Q₁₆×C₉₇	2
C₂×C₄×Q₁₆	97

Subgroups — C4×Q16×D97 #27150497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₁₆×D₉₇ #27150497