Subgroups — C₄×#15690889 #27158168

Order 12416.

752 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690889	C₂	1	2	yes	maximal
2	C₄×C₄⋊#87604	C₂	1	4	yes	maximal, complemented
3	C₄×C₂²⋊#87604	C₂	1	4	yes	maximal
4	#20261671	C₂	1	2	yes	maximal, complemented
5	C₄²×#87604	C₂	1	2	yes	maximal, complemented
6	C₂×C₄²×D₉₇	C₂	1	1	yes	maximal, characteristic
7	#15690889	C₄	1	16	yes	direct product factor
8	C₂×C₄⋊#87604	C₂²	1	4	yes	—
9	C₂×C₂²⋊#87604	C₂²	1	4	yes	—
10	C₂×C₄×#87604	C₂²	1	8	yes	—
11	C₂×C₄×#87604	C₂²	1	4	yes	—
12	(C₂×#87604)⋊C₄	C₂²	1	8	yes	—
13	C₂²×C₄×D₉₇	C₂²	1	2	yes	—
14	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₉₇	C₄	1	2	yes	complemented
16	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic, complemented
17	C₄²×D₉₇	C₄	1	1	yes	characteristic, complemented
18	C₄²×D₉₇	C₄	1	2	yes	complemented
19	C₄²×D₉₇	C₄	1	1	yes	characteristic, complemented
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, complemented
24	C₂²×#87604	C₂³	1	8	yes	—
25	C₄⋊#87604	C₂×C₄	1	16	yes	complemented
26	C₂²⋊#87604	C₂×C₄	1	16	yes	—
27	C₄×#87604	—	2	16	no	—
28	C₄×#87604	C₂×C₄	1	16	yes	complemented
29	C₂³×D₉₇	C₂³	1	1	yes	characteristic
30	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
31	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₉₇	—	2	8	no	—
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	C₂³	1	2	yes	—
39	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×D₉₇	C₂³	1	2	yes	—
42	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
44	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
45	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
46	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
47	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
48	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
50	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₉₇	C₂×C₄	1	2	yes	—
53	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×D₉₇	—	2	4	no	—
56	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×D₉₇	C₂²×C₄	1	4	yes	—
58	C₂²×D₉₇	C₂⁴	1	1	yes	characteristic
59	C₂×#87604	—	2	16	no	—
60	C₂×#87604	C₂²×C₄	1	16	yes	—
61	C₂×C₄×C₉₇	—	2	4	no	—
62	C₂×C₄×C₉₇	C₄²	1	4	yes	complemented
63	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
67	C₂×Dic₉₇	—	2	4	no	—
68	C₂×Dic₉₇	C₄²	1	4	yes	complemented
69	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
70	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
73	C₄×D₉₇	—	2	4	no	—
74	C₄×D₉₇	—	2	4	no	—
75	C₄×D₉₇	—	2	4	no	—
76	C₄×D₉₇	C₄²	1	4	yes	complemented
77	C₄×D₉₇	C₄²	1	4	yes	complemented
78	C₄×D₉₇	C₂²×C₄	1	4	yes	—
79	C₄×D₉₇	C₂²×C₄	1	4	yes	—
80	C₄×D₉₇	#41	1	8	yes	—
81	C₂²×C₉₇	—	2	2	no	—
82	C₂²×C₉₇	C₂×C₄²	1	2	yes	—
83	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×D₉₇	—	2	2	no	—
85	C₂×D₉₇	—	2	8	no	—
86	C₂×D₉₇	C₂×C₄²	1	2	yes	—
87	C₂×D₉₇	C₂³×C₄	1	2	yes	—
88	C₂×D₉₇	C₂×#41	1	4	yes	—
89	#87604	—	2	16	no	—
90	C₄×C₉₇	—	2	2	no	—
91	C₄×C₉₇	C₂×C₄²	1	2	yes	—
92	C₄×C₉₇	C₂×C₄²	1	2	yes	—
93	C₄×C₉₇	#117	1	4	yes	complemented
94	Dic₉₇	—	2	2	no	—
95	Dic₉₇	C₂×C₄²	1	2	yes	—
96	Dic₉₇	C₂×C₄²	1	2	yes	—
97	Dic₉₇	#117	1	4	yes	complemented
98	C₂×C₉₇	—	2	2	no	—
99	C₂×C₉₇	C₂²×C₄²	1	1	yes	characteristic
100	C₂×C₉₇	C₂×#117	1	2	yes	—
101	D₉₇	—	2	2	no	—
102	D₉₇	C₄×#41	1	4	yes	—
103	C₄×#117	—	97	1	no	maximal, Sylow
104	C₉₇	C₄×#117	1	1	yes	characteristic, complemented, Sylow
105	C₂×#117	—	97	2	no	—
106	C₂²×C₄²	—	97	1	no	—
107	C₄×C₄⋊C₄	—	97	4	no	—
108	C₄×C₂²⋊C₄	—	97	4	no	—
109	#376	—	97	2	no	—
110	C₄³	—	97	2	no	—
111	C₂³×C₄	—	97	1	no	—
112	C₂³×C₄	—	97	2	no	—
113	#117	—	97	16	no	—
114	C₂×C₄⋊C₄	—	97	4	no	—
115	C₂×C₂²⋊C₄	—	97	4	no	—
116	C₂×C₄²	—	97	4	no	—
117	C₂×C₄²	—	97	2	no	—
118	C₂×C₄²	—	97	1	no	—
119	C₂×C₄²	—	97	1	no	—
120	C₂×C₄²	—	97	2	no	—
121	C₂×C₄²	—	97	1	no	—
122	C₂×C₄²	—	97	2	no	—
123	C₂×C₄²	—	97	1	no	—
124	C₂×C₄²	—	97	1	no	—
125	C₂×C₄²	—	97	8	no	—
126	C₂×C₄²	#87604	1	1	yes	characteristic, complemented
127	#95	—	97	8	no	—
128	C₂⁴	—	97	1	no	—
129	C₂²×C₄	—	97	1	no	—
130	C₂²×C₄	—	97	8	no	—
131	C₂²×C₄	—	97	1	no	—
132	C₂²×C₄	—	97	1	no	—
133	C₂²×C₄	—	97	1	no	—
134	C₂²×C₄	—	97	2	no	—
135	C₂²×C₄	—	97	1	no	—
136	C₂²×C₄	—	97	8	no	—
137	C₂²×C₄	—	97	2	no	—
138	C₂²×C₄	—	97	2	no	—
139	C₂²×C₄	—	97	2	no	—
140	C₂²×C₄	—	97	2	no	—
141	C₂²×C₄	—	194	8	no	—
142	C₂²×C₄	C₂×#87604	1	2	yes	—
143	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
144	C₄⋊C₄	—	97	16	no	—
145	C₂²⋊C₄	—	97	16	no	—
146	C₄²	—	97	16	no	—
147	C₄²	—	97	2	no	—
148	C₄²	—	97	1	no	—
149	C₄²	—	97	2	no	—
150	C₄²	—	97	2	no	—
151	C₄²	—	97	2	no	—
152	C₄²	—	97	2	no	—
153	C₄²	—	97	1	no	—
154	C₄²	—	194	16	no	—
155	C₄²	C₂×#87604	1	1	yes	characteristic
156	C₄²	C₂×#87604	1	2	yes	—
157	C₄²	C₂×#87604	1	1	yes	characteristic
158	C₂³	—	97	1	no	—
159	C₂³	—	97	4	no	—
160	C₂³	—	97	1	no	—
161	C₂³	—	194	4	no	—
162	C₂³	C₂²×#87604	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	97	4	no	—
165	C₂×C₄	—	97	1	no	—
166	C₂×C₄	—	97	1	no	—
167	C₂×C₄	—	97	2	no	—
168	C₂×C₄	—	97	4	no	—
169	C₂×C₄	—	97	4	no	—
170	C₂×C₄	—	97	4	no	—
171	C₂×C₄	—	97	4	no	—
172	C₂×C₄	—	97	8	no	—
173	C₂×C₄	—	97	16	no	—
174	C₂×C₄	—	97	2	no	—
175	C₂×C₄	—	194	4	no	—
176	C₂×C₄	—	194	4	no	—
177	C₂×C₄	—	194	4	no	—
178	C₂×C₄	—	194	16	no	—
179	C₂×C₄	—	194	4	no	—
180	C₂×C₄	C₄×#87604	1	4	yes	complemented
181	C₂×C₄	C₂²×#87604	1	2	yes	—
182	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
183	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
184	C₂×C₄	C₂²×#87604	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	97	2	no	—
187	C₂²	—	97	4	no	—
188	C₂²	—	97	2	no	—
189	C₂²	—	194	8	no	—
190	C₂²	—	194	2	no	—
191	C₂²	C₂×C₄×#87604	1	2	yes	—
192	C₂²	C₂³×#87604	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	97	2	no	—
195	C₄	—	97	2	no	—
196	C₄	—	97	4	no	—
197	C₄	—	194	16	no	—
198	C₄	—	194	2	no	—
199	C₄	C₂×C₄×#87604	1	2	yes	—
200	C₄	C₂×C₄×#87604	1	2	yes	—
201	C₄	#15690889	1	4	yes	direct product factor
202	C₂	—	2	2	no	—
203	C₂	—	97	4	no	—
204	C₂	—	194	2	no	—
205	C₂	C₂²×C₄×#87604	1	1	yes	characteristic
206	C₂	C₂×#15690889	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	C₂³
cosocle	C₂²×D₉₇	C₂⁴
hypercenter	C₂×C₄²	#87604
hypocenter	C₉₇	C₄×#117
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×C₄⋊#87604	C₂	semidirect
4. #20261671	C₂	semidirect
5. C₄²×#87604	C₂	semidirect
7. #15690889	C₄	direct (N × H)
15. C₂×C₄×Dic₉₇	C₄	semidirect
16. C₂×C₄×Dic₉₇	C₄	semidirect
17. C₄²×D₉₇	C₄	semidirect
18. C₄²×D₉₇	C₄	semidirect
19. C₄²×D₉₇	C₄	semidirect
23. C₂×C₄²×C₉₇	C₄	semidirect
25. C₄⋊#87604	C₂×C₄	semidirect
28. C₄×#87604	C₂×C₄	semidirect
62. C₂×C₄×C₉₇	C₄²	semidirect
68. C₂×Dic₉₇	C₄²	semidirect
76. C₄×D₉₇	C₄²	semidirect
77. C₄×D₉₇	C₄²	semidirect
93. C₄×C₉₇	#117	semidirect
97. Dic₉₇	#117	semidirect
104. C₉₇	C₄×#117	semidirect
126. C₂×C₄²	#87604	semidirect
180. C₂×C₄	C₄×#87604	semidirect
201. C₄	#15690889	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×#15690889	2
C₄×C₄⋊#87604	2
C₄×C₂²⋊#87604	2
#20261671	2
C₄²×#87604	2
C₂×C₄²×D₉₇	2
C₄×#117	97

Subgroups — C4×#15690889 #27158168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15690889 #27158168