Subgroups — #27141445

Order 12416.

404 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13095495	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690769	C₂	1	1	yes	maximal, characteristic
3	#20260494	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	—	2	2	no	—
6	C₂×#13095495	—	2	1	no	—
7	C₂×#13095495	C₄	1	4	yes	complemented
8	C₂×#13095495	C₂²	1	2	yes	—
9	C₂²×#11715895	—	2	1	no	—
10	#15690769	—	2	2	no	—
11	#15690769	C₄	1	4	yes	complemented
12	C₂×C₄⋊Dic₉₇	—	2	1	no	—
13	C₂×C₂²⋊Dic₉₇	—	2	1	no	—
14	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
15	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
16	#117×C₉₇	—	2	1	no	—
17	#117×C₉₇	—	2	1	no	—
18	#117×C₉₇	C₄	1	2	yes	complemented
19	#117×C₉₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₉₇	C₄	1	2	yes	complemented
21	C₂×C₂²⋊C₄×C₉₇	—	2	1	no	—
22	C₂×C₄²×C₉₇	—	2	1	no	—
23	C₂²×Dic₉₇	—	2	2	no	—
24	#13095495	—	2	4	no	—
25	#13095495	—	2	2	no	—
26	#13095495	C₂×C₄	1	4	yes	complemented
27	C₂×#11715895	—	2	4	no	—
28	C₂×#11715895	—	2	2	no	—
29	C₄⋊Dic₉₇	—	2	4	no	—
30	C₂²⋊Dic₉₇	—	2	4	no	—
31	C₄×Dic₉₇	—	2	2	no	—
32	C₄×Dic₉₇	C₂×C₄	1	4	yes	complemented
33	C₂⁴×C₉₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉₇	—	2	2	no	—
35	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₉₇	D₄	1	2	yes	—
40	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₉₇	Q₈	1	2	yes	—
44	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₉₇	—	2	2	no	—
46	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
47	C₂²⋊C₄×C₉₇	—	2	2	no	—
48	C₂²⋊C₄×C₉₇	—	2	2	no	—
49	C₄²×C₉₇	—	2	2	no	—
50	C₄²×C₉₇	—	2	2	no	—
51	C₂³×C₉₇	—	4	2	no	—
52	C₂³×C₉₇	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
56	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₉₇	—	2	4	no	—
58	C₂×C₄×C₉₇	—	4	4	no	—
59	C₂×C₄×C₉₇	C₄²	1	2	yes	—
60	C₂×C₄×C₉₇	C₄²	1	2	yes	—
61	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
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	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
77	C₂×Dic₉₇	—	2	4	no	—
78	C₂×Dic₉₇	—	4	4	no	—
79	#11715895	—	2	4	no	—
80	C₂²×C₉₇	—	2	2	no	—
81	C₂²×C₉₇	—	2	2	no	—
82	C₂²×C₉₇	—	2	2	no	—
83	C₂²×C₉₇	—	4	2	no	—
84	C₂²×C₉₇	—	4	2	no	—
85	C₂²×C₉₇	#95	1	2	yes	—
86	C₂²×C₉₇	#95	1	2	yes	—
87	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₉₇	—	4	4	no	—
91	C₄×C₉₇	#95	1	2	yes	—
92	C₄×C₉₇	#95	1	2	yes	—
93	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
94	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	Dic₉₇	—	4	4	no	—
98	C₂×C₉₇	—	2	1	no	—
99	C₂×C₉₇	—	2	2	no	—
100	C₂×C₉₇	—	2	2	no	—
101	C₂×C₉₇	—	2	1	no	—
102	C₂×C₉₇	C₂×#95	1	1	yes	characteristic
103	C₂×C₉₇	#421	1	2	yes	—
104	#1733	—	97	1	no	maximal, Sylow
105	C₉₇	#1733	1	1	yes	characteristic, complemented, Sylow
106	C₂²×M₁₆	—	97	1	no	—
107	C₂×#117	—	97	1	no	—
108	C₂×#117	D₉₇	1	1	yes	characteristic
109	#339	—	97	4	no	—
110	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
111	C₂×M₁₆	—	97	4	no	—
112	C₂×M₁₆	—	97	2	no	—
113	C₂×M₁₆	—	194	2	no	—
114	C₂×M₁₆	—	194	1	no	—
115	C₂²×C₈	—	194	1	no	—
116	#117	—	2	1	no	—
117	#117	—	2	1	no	—
118	#117	—	97	4	no	—
119	#117	—	194	2	no	—
120	#117	Dic₉₇	1	2	yes	complemented
121	#117	C₂×D₉₇	1	2	yes	—
122	C₂×C₄⋊C₄	—	194	1	no	—
123	C₂×C₄⋊C₄	Dic₉₇	1	2	yes	complemented
124	C₂×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₂²⋊C₄	—	194	1	no	—
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	—	97	2	no	—
128	C₂⁴	C₂²⋊D₉₇	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	194	2	no	—
131	C₂²×C₄	Dic₁₉₄	1	2	yes	—
132	C₂²×C₄	C₄×D₉₇	1	2	yes	—
133	C₂²×C₄	C₄×D₉₇	1	2	yes	—
134	C₂²×C₄	D₃₈₈	1	2	yes	—
135	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
137	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
138	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
141	M₁₆	—	97	4	no	—
142	M₁₆	—	194	4	no	—
143	M₁₆	—	194	2	no	—
144	C₂×C₈	—	194	4	no	—
145	C₂×C₈	—	194	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	194	4	no	—
148	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
149	C₂²⋊C₄	—	2	2	no	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₂²⋊C₄	—	194	4	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	97	4	no	—
155	C₄²	—	194	2	no	—
156	C₂³	—	4	2	no	—
157	C₂³	#13095488	1	2	yes	—
158	C₂³	C₂²⋊Dic₉₇	1	1	yes	characteristic
159	C₂³	C₂²⋊Dic₉₇	1	1	yes	characteristic
160	C₂³	#13095490	1	2	yes	—
161	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	388	4	no	—
165	C₂×C₄	—	194	4	no	—
166	C₂×C₄	C₄×Dic₉₇	1	2	yes	—
167	C₂×C₄	C₄×Dic₉₇	1	2	yes	—
168	C₂×C₄	#13095488	1	2	yes	—
169	C₂×C₄	#13095488	1	2	yes	—
170	C₂×C₄	#13095488	1	2	yes	—
171	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
172	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
173	C₂×C₄	#13095490	1	2	yes	—
174	C₂×C₄	#13095490	1	2	yes	—
175	C₂×C₄	#13095490	1	2	yes	—
176	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
177	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
178	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
180	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
181	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
182	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
183	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
184	C₈	—	194	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	#15690717	1	2	yes	—
191	C₂²	#15690717	1	2	yes	—
192	C₂²	C₂×#13095488	1	1	yes	characteristic
193	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
194	C₂²	C₂×#13095490	1	1	yes	characteristic
195	C₄	—	4	4	no	—
196	C₄	—	388	4	no	—
197	C₄	#15690717	1	2	yes	—
198	C₄	#15690717	1	2	yes	—
199	C₄	C₂×C₄×Dic₉₇	1	1	yes	characteristic
200	C₄	C₂×#13095488	1	1	yes	characteristic
201	C₄	C₂×#13095490	1	1	yes	characteristic
202	C₄	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	—	2	2	no	—
206	C₂	—	2	2	no	—
207	C₂	C₂×#15690717	1	1	yes	characteristic
208	C₂	#20260752	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₉₇
commutator	C₄×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#117×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₉₇	D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#117	D₉₇
hypocenter	C₉₇	#1733
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #20260494	C₂	semidirect
7. C₂×#13095495	C₄	semidirect
11. #15690769	C₄	semidirect
18. #117×C₉₇	C₄	semidirect
20. C₂×C₄⋊C₄×C₉₇	C₄	semidirect
26. #13095495	C₂×C₄	semidirect
32. C₄×Dic₉₇	C₂×C₄	semidirect
105. C₉₇	#1733	semidirect
120. #117	Dic₉₇	semidirect
123. C₂×C₄⋊C₄	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13095495	2
C₂×#15690769	2
#20260494	2
C₂×#117×C₉₇	2
#1733	97