Subgroups — C₄×D₉₇⋊M₁₆ #27158292

Order 12416.

656 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₉₇⋊M₁₆	C₂	1	2	yes	maximal
2	C₄×#13095521	C₂	1	4	yes	maximal, complemented
3	C₄×#13095520	C₂	1	4	yes	maximal, complemented
4	#20261694	C₂	1	2	yes	maximal, complemented
5	C₄×#13095519	C₂	1	2	yes	maximal, complemented
6	C₂×C₄²×D₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13095521	C₂²	1	4	yes	—
8	D₉₇⋊M₁₆	C₄	1	16	yes	direct product factor
9	C₂×#13095520	C₂²	1	4	yes	—
10	C₂×#13095519	C₂²	1	4	yes	—
11	#15690865	C₂²	1	8	yes	complemented
12	C₄×#11715897	C₂²	1	8	yes	complemented
13	C₂²×C₄×D₉₇	C₄	1	2	yes	—
14	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
17	C₄²×D₉₇	C₄	1	1	yes	characteristic
18	C₄²×D₉₇	C₄	1	2	yes	—
19	C₄²×D₉₇	C₄	1	1	yes	characteristic
20	C₄²×D₉₇	C₂²	1	1	yes	characteristic
21	C₄²×D₉₇	C₂²	1	1	yes	characteristic
22	C₄²×D₉₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
24	#13095521	C₂×C₄	1	16	yes	complemented
25	#13095520	C₂×C₄	1	16	yes	complemented
26	#13095519	C₂×C₄	1	16	yes	complemented
27	C₂×#11715897	C₂³	1	8	yes	—
28	C₂³×D₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
30	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×D₉₇	—	2	8	no	—
32	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₉₇	C₂×C₄	1	8	yes	—
38	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
41	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
42	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
43	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
45	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
46	C₄×Dic₉₇	C₂³	1	2	yes	—
47	C₄×Dic₉₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₉₇	C₂×C₄	1	2	yes	—
52	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₉₇	—	2	4	no	—
55	C₂²×D₉₇	C₄²	1	4	yes	—
56	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₉₇	—	2	4	no	—
59	C₂×C₄×C₉₇	C₄²	1	4	yes	—
60	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
64	C₂×Dic₉₇	—	2	4	no	—
65	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
66	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
68	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
69	C₂×Dic₉₇	C₂⁴	1	1	yes	characteristic
70	C₄×D₉₇	—	2	4	no	—
71	C₄×D₉₇	—	2	4	no	—
72	C₄×D₉₇	—	2	4	no	—
73	C₄×D₉₇	C₄²	1	4	yes	—
74	C₄×D₉₇	C₄²	1	4	yes	—
75	C₄×D₉₇	M₁₆	1	8	yes	complemented
76	C₄×D₉₇	C₂²×C₄	1	4	yes	—
77	C₄×D₉₇	C₂²×C₄	1	4	yes	—
78	#11715897	C₂²×C₄	1	16	yes	complemented
79	C₂²×C₉₇	—	2	2	no	—
80	C₂²×C₉₇	C₂×C₄²	1	2	yes	—
81	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×D₉₇	—	2	2	no	—
83	C₂×D₉₇	—	2	8	no	—
84	C₂×D₉₇	C₂×C₄²	1	2	yes	—
85	C₂×D₉₇	C₂×C₄²	1	2	yes	—
86	C₂×D₉₇	C₂×M₁₆	1	4	yes	—
87	C₄×C₉₇	—	2	2	no	—
88	C₄×C₉₇	C₂×C₄²	1	2	yes	—
89	C₄×C₉₇	C₂×C₄²	1	2	yes	—
90	C₄×C₉₇	C₂×M₁₆	1	4	yes	complemented
91	Dic₉₇	—	2	2	no	—
92	Dic₉₇	C₂×C₄²	1	2	yes	—
93	Dic₉₇	C₂×M₁₆	1	4	yes	complemented
94	Dic₉₇	C₂³×C₄	1	2	yes	—
95	C₂×C₉₇	—	2	2	no	—
96	C₂×C₉₇	C₂²×C₄²	1	1	yes	characteristic
97	C₂×C₉₇	C₂²×M₁₆	1	2	yes	—
98	D₉₇	—	2	2	no	—
99	D₉₇	C₄×M₁₆	1	4	yes	complemented
100	C₂×C₄×M₁₆	—	97	1	no	maximal, Sylow
101	C₉₇	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
102	C₂²×M₁₆	—	97	2	no	—
103	C₂²×C₄²	—	97	1	no	—
104	C₄×M₁₆	—	97	4	no	—
105	C₄×M₁₆	—	97	4	no	—
106	C₂×#97	—	97	2	no	—
107	C₂×C₄×C₈	—	97	2	no	—
108	C₂³×C₄	—	97	1	no	—
109	C₂³×C₄	—	97	2	no	—
110	C₂×M₁₆	—	97	4	no	—
111	C₂×M₁₆	—	97	16	no	—
112	C₂×M₁₆	—	97	4	no	—
113	C₂²×C₈	—	97	4	no	—
114	C₂×C₄²	—	97	1	no	—
115	C₂×C₄²	—	97	2	no	—
116	C₂×C₄²	—	97	2	no	—
117	C₂×C₄²	—	97	1	no	—
118	C₂×C₄²	—	97	1	no	—
119	C₂×C₄²	—	97	2	no	—
120	C₂×C₄²	—	97	1	no	—
121	C₂×C₄²	—	97	1	no	—
122	C₂×C₄²	#87604	1	1	yes	characteristic
123	#97	—	97	8	no	—
124	C₄×C₈	—	97	8	no	—
125	C₂⁴	—	97	1	no	—
126	C₂²×C₄	—	97	1	no	—
127	C₂²×C₄	—	97	8	no	—
128	C₂²×C₄	—	97	1	no	—
129	C₂²×C₄	—	97	1	no	—
130	C₂²×C₄	—	97	2	no	—
131	C₂²×C₄	—	97	1	no	—
132	C₂²×C₄	—	97	1	no	—
133	C₂²×C₄	—	97	2	no	—
134	C₂²×C₄	—	97	2	no	—
135	C₂²×C₄	—	97	2	no	—
136	C₂²×C₄	—	97	2	no	—
137	C₂²×C₄	—	194	8	no	—
138	C₂²×C₄	C₂×#87604	1	2	yes	—
139	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
140	M₁₆	—	97	16	no	—
141	M₁₆	—	97	16	no	—
142	C₂×C₈	—	97	8	no	—
143	C₂×C₈	—	97	16	no	—
144	C₄²	—	97	2	no	—
145	C₄²	—	97	1	no	—
146	C₄²	—	97	2	no	—
147	C₄²	—	97	2	no	—
148	C₄²	—	97	1	no	—
149	C₄²	—	97	2	no	—
150	C₄²	—	97	2	no	—
151	C₄²	C₂×#87604	1	1	yes	characteristic
152	C₄²	C₂×#87604	1	2	yes	—
153	C₄²	C₂×#87604	1	1	yes	characteristic
154	C₂³	—	97	1	no	—
155	C₂³	—	97	4	no	—
156	C₂³	—	97	1	no	—
157	C₂³	—	194	4	no	—
158	C₂³	C₂²×#87604	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	97	4	no	—
161	C₂×C₄	—	97	1	no	—
162	C₂×C₄	—	97	1	no	—
163	C₂×C₄	—	97	2	no	—
164	C₂×C₄	—	97	4	no	—
165	C₂×C₄	—	97	4	no	—
166	C₂×C₄	—	97	4	no	—
167	C₂×C₄	—	97	4	no	—
168	C₂×C₄	—	97	8	no	—
169	C₂×C₄	—	97	2	no	—
170	C₂×C₄	—	194	4	no	—
171	C₂×C₄	—	194	4	no	—
172	C₂×C₄	—	194	4	no	—
173	C₂×C₄	—	194	4	no	—
174	C₂×C₄	C₄×#87604	1	4	yes	—
175	C₂×C₄	C₂²×#87604	1	2	yes	—
176	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
177	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
178	C₂×C₄	C₂²×#87604	1	2	yes	—
179	C₈	—	97	16	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	97	2	no	—
182	C₂²	—	97	4	no	—
183	C₂²	—	97	2	no	—
184	C₂²	—	194	8	no	—
185	C₂²	—	194	2	no	—
186	C₂²	C₂×C₄×#87604	1	2	yes	—
187	C₂²	C₂³×#87604	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	97	2	no	—
190	C₄	—	97	2	no	—
191	C₄	—	97	4	no	—
192	C₄	—	194	2	no	—
193	C₄	C₂×C₄×#87604	1	2	yes	—
194	C₄	C₂×C₄×#87604	1	2	yes	—
195	C₄	D₉₇⋊M₁₆	1	4	yes	direct product factor
196	C₂	—	2	2	no	—
197	C₂	—	97	4	no	—
198	C₂	—	194	2	no	—
199	C₂	C₂²×C₄×#87604	1	1	yes	characteristic
200	C₂	C₂×D₉₇⋊M₁₆	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×#87604
commutator	C₂×C₉₇	C₂²×C₄²
Frattini	C₂²	C₂³×#87604
Fitting	C₂×C₄²×C₉₇	C₄
socle	C₂²×C₉₇	C₂³×C₄
prime radical	C₂³×D₉₇	C₂×C₄
cosocle	C₂×Dic₉₇	C₂⁴
hypercenter	C₂×C₄²	#87604
hypocenter	C₉₇	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#13095521	C₂	semidirect
3. C₄×#13095520	C₂	semidirect
4. #20261694	C₂	semidirect
5. C₄×#13095519	C₂	semidirect
8. D₉₇⋊M₁₆	C₄	direct (N × H)
11. #15690865	C₂²	semidirect
12. C₄×#11715897	C₂²	semidirect
24. #13095521	C₂×C₄	semidirect
25. #13095520	C₂×C₄	semidirect
26. #13095519	C₂×C₄	semidirect
75. C₄×D₉₇	M₁₆	semidirect
78. #11715897	C₂²×C₄	semidirect
90. C₄×C₉₇	C₂×M₁₆	semidirect
93. Dic₉₇	C₂×M₁₆	semidirect
99. D₉₇	C₄×M₁₆	semidirect
101. C₉₇	C₂×C₄×M₁₆	semidirect
195. C₄	D₉₇⋊M₁₆	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×D₉₇⋊M₁₆	2
C₄×#13095521	2
C₄×#13095520	2
#20261694	2
C₄×#13095519	2
C₂×C₄²×D₉₇	2
C₂×C₄×M₁₆	97

Subgroups — C4×D97⋊M16 #27158292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₉₇⋊M₁₆ #27158292