Subgroups — #117×Dic₁₉₇ #38745055

Order 25216.

752 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263285	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₉₇	C₂	1	4	yes	maximal, complemented
4	C₂²⋊C₄×Dic₁₉₇	C₂	1	4	yes	maximal
5	#27160925	C₂	1	2	yes	maximal, complemented
6	C₄²×Dic₁₉₇	C₂	1	2	yes	maximal, complemented
7	C₂×#117×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₁₉₇	C₂²	1	2	yes	—
9	#20263285	C₄	1	8	yes	complemented
10	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
12	C₂×C₄×Dic₁₉₇	C₄	1	8	yes	complemented
13	C₂×C₄×Dic₁₉₇	C₂²	1	8	yes	—
14	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
18	#20263233	C₂²	1	8	yes	—
19	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
20	#117×C₁₉₇	C₄	1	8	yes	complemented
21	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₉₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₉₇	C₂²	1	2	yes	—
24	C₂²×Dic₁₉₇	—	2	8	no	—
25	C₂²×Dic₁₉₇	C₂×C₄	1	8	yes	—
26	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
27	C₂²×Dic₁₉₇	C₂³	1	4	yes	—
28	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
29	C₄⋊Dic₁₉₇	C₂×C₄	1	8	yes	complemented
30	C₂²⋊Dic₁₉₇	C₂×C₄	1	8	yes	—
31	C₄×Dic₁₉₇	—	2	16	no	—
32	C₄×Dic₁₉₇	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₉₇	C₂×C₄	1	8	yes	—
34	C₄×Dic₁₉₇	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₉₇	C₂×C₄	1	8	yes	complemented
36	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₁₉₇	C₂×C₄	1	8	yes	complemented
44	C₂²⋊C₄×C₁₉₇	C₂×C₄	1	8	yes	—
45	C₄²×C₁₉₇	C₂×C₄	1	8	yes	complemented
46	C₂×Dic₁₉₇	—	2	8	no	—
47	C₂×Dic₁₉₇	—	2	4	no	—
48	C₂×Dic₁₉₇	—	2	16	no	—
49	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
50	C₂×Dic₁₉₇	C₂²×C₄	1	8	yes	—
51	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
52	C₂×Dic₁₉₇	#41	1	8	yes	—
53	C₂³×C₁₉₇	—	2	4	no	—
54	C₂³×C₁₉₇	C₂²×C₄	1	2	yes	—
55	C₂³×C₁₉₇	C₂²×C₄	1	2	yes	—
56	C₂³×C₁₉₇	C₂²×C₄	1	2	yes	—
57	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
58	C₂×C₄×C₁₉₇	—	2	8	no	—
59	C₂×C₄×C₁₉₇	C₄²	1	16	yes	complemented
60	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₁₉₇	C₂²×C₄	1	8	yes	—
65	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₁₉₇	—	2	4	no	—
68	C₂²×C₁₉₇	—	2	4	no	—
69	C₂²×C₁₉₇	—	2	4	no	—
70	C₂²×C₁₉₇	C₂×C₄²	1	4	yes	—
71	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
72	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
73	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₁₉₇	C₂×#41	1	4	yes	—
75	C₄×C₁₉₇	—	2	8	no	—
76	C₄×C₁₉₇	C₂×C₄²	1	4	yes	—
77	C₄×C₁₉₇	C₂×C₄²	1	4	yes	—
78	Dic₁₉₇	—	2	4	no	—
79	Dic₁₉₇	—	2	8	no	—
80	Dic₁₉₇	#117	1	8	yes	direct product factor
81	C₂×C₁₉₇	—	2	4	no	—
82	C₂×C₁₉₇	C₂²×C₄²	1	1	yes	characteristic
83	C₂×C₁₉₇	C₂×#117	1	2	yes	—
84	C₂×C₁₉₇	C₄×#41	1	2	yes	—
85	C₂×C₁₉₇	C₄×#41	1	2	yes	—
86	C₁₉₇	C₄×#117	1	1	yes	characteristic, complemented, Sylow
87	C₄×#117	—	197	1	no	maximal, Sylow
88	C₂×#117	—	197	1	no	—
89	C₂×#117	D₁₉₇	1	1	yes	characteristic
90	C₂²×C₄²	—	197	1	no	—
91	C₄×C₄⋊C₄	—	197	4	no	—
92	C₄×C₂²⋊C₄	—	197	4	no	—
93	#376	—	197	2	no	—
94	C₄³	—	197	2	no	—
95	C₂³×C₄	—	197	2	no	—
96	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
97	#117	—	197	8	no	—
98	#117	Dic₁₉₇	1	8	yes	direct product factor
99	C₂×C₄⋊C₄	—	197	2	no	—
100	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
101	C₂×C₂²⋊C₄	—	197	2	no	—
102	C₂×C₂²⋊C₄	C₂×D₁₉₇	1	2	yes	—
103	C₂×C₄²	—	197	8	no	—
104	C₂×C₄²	—	197	8	no	—
105	C₂×C₄²	—	197	2	no	—
106	C₂×C₄²	—	197	2	no	—
107	C₂×C₄²	—	197	1	no	—
108	C₂×C₄²	—	197	1	no	—
109	C₂×C₄²	C₂×D₁₉₇	1	2	yes	—
110	#95	—	197	8	no	—
111	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
112	C₂²×C₄	—	394	8	no	—
113	C₂²×C₄	—	197	8	no	—
114	C₂²×C₄	—	197	2	no	—
115	C₂²×C₄	—	197	4	no	—
116	C₂²×C₄	—	197	2	no	—
117	C₂²×C₄	C₄×D₁₉₇	1	4	yes	—
118	C₂²×C₄	C₄×D₁₉₇	1	4	yes	—
119	C₂²×C₄	C₂×Dic₁₉₇	1	4	yes	—
120	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
121	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
123	C₄⋊C₄	—	197	8	no	—
124	C₄⋊C₄	C₂×Dic₁₉₇	1	8	yes	complemented
125	C₂²⋊C₄	—	197	8	no	—
126	C₂²⋊C₄	C₂×Dic₁₉₇	1	8	yes	—
127	C₄²	—	394	16	no	—
128	C₄²	—	197	4	no	—
129	C₄²	—	197	8	no	—
130	C₄²	—	197	4	no	—
131	C₄²	—	197	8	no	—
132	C₄²	C₂×Dic₁₉₇	1	8	yes	complemented
133	C₂³	—	2	4	no	—
134	C₂³	C₂×C₄×D₁₉₇	1	2	yes	—
135	C₂³	C₂×C₄×D₁₉₇	1	2	yes	—
136	C₂³	C₂²×Dic₁₉₇	1	2	yes	—
137	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	394	8	no	—
140	C₂×C₄	—	394	4	no	—
141	C₂×C₄	—	394	16	no	—
142	C₂×C₄	—	197	4	no	—
143	C₂×C₄	—	197	8	no	—
144	C₂×C₄	—	197	4	no	—
145	C₂×C₄	—	197	8	no	—
146	C₂×C₄	C₄×Dic₁₉₇	1	16	yes	complemented
147	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
148	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
149	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
150	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
151	C₂×C₄	C₂²×Dic₁₉₇	1	8	yes	—
152	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
153	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	4	no	—
157	C₂²	C₂×C₄×Dic₁₉₇	1	4	yes	—
158	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
159	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
160	C₂²	C₂³×Dic₁₉₇	1	1	yes	characteristic
161	C₂²	#41×D₁₉₇	1	4	yes	—
162	C₄	—	2	8	no	—
163	C₄	—	394	4	no	—
164	C₄	—	394	8	no	—
165	C₄	—	197	8	no	—
166	C₄	C₂×C₄×Dic₁₉₇	1	4	yes	—
167	C₄	C₂×C₄×Dic₁₉₇	1	4	yes	—
168	C₂	—	2	4	no	—
169	C₂	C₂²×C₄×Dic₁₉₇	1	1	yes	characteristic
170	C₂	#117×D₁₉₇	1	2	yes	—
171	C₂	#27161547	1	2	yes	—
172	C₂	#41×Dic₁₉₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉₇
commutator	C₂×C₁₉₇	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₉₇
Fitting	C₂×#117×C₁₉₇	C₂
socle, cosocle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂⁴×C₁₉₇	C₂³
hypercenter	C₂×#117	D₁₉₇
hypocenter	C₁₉₇	C₄×#117
LCS	G ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×Dic₁₉₇	C₂	semidirect
5. #27160925	C₂	semidirect
6. C₄²×Dic₁₉₇	C₂	semidirect
9. #20263285	C₄	semidirect
12. C₂×C₄×Dic₁₉₇	C₄	semidirect
20. #117×C₁₉₇	C₄	semidirect
29. C₄⋊Dic₁₉₇	C₂×C₄	semidirect
35. C₄×Dic₁₉₇	C₂×C₄	semidirect
43. C₄⋊C₄×C₁₉₇	C₂×C₄	semidirect
45. C₄²×C₁₉₇	C₂×C₄	semidirect
59. C₂×C₄×C₁₉₇	C₄²	semidirect
80. Dic₁₉₇	#117	direct (N × H)
86. C₁₉₇	C₄×#117	semidirect
98. #117	Dic₁₉₇	direct (N × H)
124. C₄⋊C₄	C₂×Dic₁₉₇	semidirect
132. C₄²	C₂×Dic₁₉₇	semidirect
146. C₂×C₄	C₄×Dic₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263285	2
C₂²×C₄×Dic₁₉₇	2
C₄⋊C₄×Dic₁₉₇	2
C₂²⋊C₄×Dic₁₉₇	2
#27160925	2
C₄²×Dic₁₉₇	2
C₂×#117×C₁₉₇	2
C₄×#117	197

Subgroups — #117×Dic197 #38745055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #117×Dic₁₉₇ #38745055