Subgroups — #27150751

Order 12416.

406 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	C₄×#13095511	C₂	1	1	yes	maximal, characteristic
2	#20260865	C₂	1	2	yes	maximal, complemented
3	#20260804	C₂	1	4	yes	maximal, complemented
4	C₈×D₃₈₈	C₂	1	2	yes	maximal, complemented
5	#20260792	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260790	C₂	1	2	yes	maximal, complemented
7	#97×D₉₇	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13095511	C₄	1	1	yes	characteristic
10	C₂×C₈×D₉₇	C₂²	1	4	yes	—
11	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
12	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
13	C₄×D₃₈₈	C₄	1	1	yes	characteristic
14	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₉₇	C₄	1	2	yes	—
16	C₄×C₂²⋊D₉₇	C₂²	1	2	yes	—
17	#15690769	C₄	1	1	yes	characteristic
18	#15690767	C₄	1	2	yes	—
19	C₄²×D₉₇	C₂²	1	2	yes	—
20	C₄⋊#11715895	C₂²	1	2	yes	—
21	#15690739	C₂²	1	4	yes	—
22	C₂²⋊#11715895	C₂²	1	2	yes	—
23	#15690722	C₂²	1	4	yes	—
24	#15690721	C₂²	1	2	yes	—
25	#15690720	C₂²	1	4	yes	—
26	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
28	#97×C₉₇	C₂²	1	1	yes	characteristic
29	#97×C₉₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₉₇	C₂²	1	2	yes	—
31	#13095511	—	2	4	no	—
32	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
33	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₉₇	C₂³	1	2	yes	—
38	C₂×#11715895	C₂³	1	4	yes	—
39	C₈×D₉₇	—	2	8	no	—
40	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
42	#13095490	C₂×C₄	1	2	yes	—
43	#13095490	C₂×C₄	1	2	yes	—
44	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
45	#13095488	C₂×C₄	1	2	yes	—
46	#13095488	C₂×C₄	1	2	yes	—
47	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
48	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
49	C₄×Dic₉₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
53	M₁₆×C₉₇	—	2	4	no	—
54	C₂×C₈×C₉₇	C₂³	1	2	yes	—
55	C₂×C₈×C₉₇	C₂³	1	2	yes	—
56	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₉₇	C₂³	1	1	yes	characteristic
59	C₄²×C₉₇	C₂³	1	1	yes	characteristic
60	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₉₇	—	2	2	no	—
63	C₂×C₄×C₉₇	—	2	2	no	—
64	C₂×C₄×C₉₇	—	4	1	no	—
65	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₉₇	—	2	4	no	—
72	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
73	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
74	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
75	D₃₈₈	M₁₆	1	4	yes	complemented
76	C₄×D₉₇	—	2	4	no	—
77	C₄×D₉₇	—	2	4	no	—
78	C₄×D₉₇	—	4	2	no	—
79	Dic₁₉₄	—	2	2	no	—
80	C₈×C₉₇	—	2	2	no	—
81	C₈×C₉₇	#41	1	4	yes	complemented
82	#11715895	—	2	4	no	—
83	C₂²×C₉₇	—	2	1	no	—
84	C₂²×C₉₇	—	2	1	no	—
85	C₂²×C₉₇	—	2	1	no	—
86	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₉₇	—	2	2	no	—
88	C₂×D₉₇	—	2	2	no	—
89	C₂×D₉₇	C₂×M₁₆	1	4	yes	—
90	C₄×C₉₇	—	2	1	no	—
91	C₄×C₉₇	—	2	1	no	—
92	C₄×C₉₇	—	2	1	no	—
93	C₄×C₉₇	—	2	1	no	—
94	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
95	C₄×C₉₇	C₂×#41	1	2	yes	—
96	Dic₉₇	—	2	2	no	—
97	Dic₉₇	—	2	2	no	—
98	Dic₉₇	—	2	2	no	—
99	C₂×C₉₇	—	4	1	no	—
100	C₂×C₉₇	C₄×#41	1	1	yes	characteristic
101	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₉₇	#568	1	1	yes	characteristic
103	D₉₇	—	2	4	no	—
104	#2970	—	97	1	no	maximal, Sylow
105	C₉₇	#2970	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	—	97	1	no	—
107	#446	—	97	2	no	—
108	#435	—	97	4	no	—
109	C₈×D₄	—	97	2	no	—
110	#432	—	97	1	no	—
111	#432	—	97	2	no	—
112	C₄×M₁₆	D₉₇	1	1	yes	characteristic, complemented
113	C₂×#97	—	97	2	no	—
114	C₂×#41	—	97	1	no	—
115	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
116	C₂²×C₈	—	97	4	no	—
117	C₄×Q₈	—	97	1	no	—
118	C₄×Q₈	—	97	1	no	—
119	C₄×D₄	—	97	2	no	—
120	C₄×D₄	—	97	2	no	—
121	C₄×D₄	—	97	1	no	—
122	C₄×D₄	—	97	1	no	—
123	#117	—	97	1	no	—
124	#117	—	97	2	no	—
125	C₂×C₄²	—	97	2	no	—
126	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
127	C₄⋊C₈	—	97	4	no	—
128	C₄⋊C₈	—	97	2	no	—
129	C₂²⋊C₈	—	97	2	no	—
130	C₂²⋊C₈	—	97	4	no	—
131	#97	—	97	4	no	—
132	#97	—	97	2	no	—
133	#97	C₂×D₉₇	1	1	yes	characteristic
134	#97	C₂×D₉₇	1	1	yes	characteristic
135	C₄×C₈	C₂×D₉₇	1	2	yes	—
136	#41	—	194	4	no	—
137	C₂×Q₈	—	97	1	no	—
138	C₂×D₄	—	97	2	no	—
139	C₂×D₄	—	97	1	no	—
140	C₂²×C₄	—	97	2	no	—
141	C₂²×C₄	—	97	2	no	—
142	C₂²×C₄	—	97	2	no	—
143	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	97	4	no	—
148	C₂×C₈	—	194	8	no	—
149	C₂×C₈	C₂²×D₉₇	1	2	yes	—
150	C₂×C₈	C₂²×D₉₇	1	2	yes	—
151	C₄⋊C₄	—	97	2	no	—
152	C₄⋊C₄	—	97	2	no	—
153	C₄⋊C₄	—	97	1	no	—
154	C₄⋊C₄	—	97	1	no	—
155	C₂²⋊C₄	—	97	2	no	—
156	C₂²⋊C₄	—	97	2	no	—
157	C₂²⋊C₄	—	97	2	no	—
158	C₄²	—	97	2	no	—
159	C₄²	—	97	2	no	—
160	C₄²	—	97	2	no	—
161	C₄²	C₄×D₉₇	1	1	yes	characteristic
162	C₄²	C₄×D₉₇	1	1	yes	characteristic
163	C₄²	C₂²×D₉₇	1	1	yes	characteristic
164	C₄²	C₂²×D₉₇	1	1	yes	characteristic
165	C₂³	—	97	2	no	—
166	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
167	Q₈	—	194	2	no	—
168	D₄	—	97	4	no	—
169	D₄	—	194	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	97	2	no	—
174	C₂×C₄	—	97	2	no	—
175	C₂×C₄	—	97	2	no	—
176	C₂×C₄	—	388	2	no	—
177	C₂×C₄	—	194	4	no	—
178	C₂×C₄	—	194	4	no	—
179	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	194	4	no	—
187	C₈	#13095511	1	4	yes	complemented
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	97	4	no	—
192	C₂²	—	194	2	no	—
193	C₂²	—	194	2	no	—
194	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	194	2	no	—
200	C₄	—	194	2	no	—
201	C₄	—	194	2	no	—
202	C₄	M₁₆×D₉₇	1	2	yes	—
203	C₄	C₂×#13095511	1	2	yes	—
204	C₂	—	4	1	no	—
205	C₂	—	194	4	no	—
206	C₂	C₄×#13095511	1	1	yes	characteristic
207	C₂	C₂×M₁₆×D₉₇	1	1	yes	characteristic
208	C₂	#20261501	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	C₄×M₁₆×C₉₇	C₂
prime radical	C₂×#13095511	C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₄×M₁₆	D₉₇
hypocenter	C₉₇	#2970
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20260865	C₂	semidirect
3. #20260804	C₂	semidirect
4. C₈×D₃₈₈	C₂	semidirect
5. #20260792	C₂	semidirect
6. #20260790	C₂	semidirect
7. #97×D₉₇	C₂	semidirect
8. C₄×M₁₆×C₉₇	C₂	semidirect
75. D₃₈₈	M₁₆	semidirect
81. C₈×C₉₇	#41	semidirect
105. C₉₇	#2970	semidirect
112. C₄×M₁₆	D₉₇	semidirect
187. C₈	#13095511	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13095511	2
#20260865	2
#20260804	2
C₈×D₃₈₈	2
#20260792	2
#20260790	2
#97×D₉₇	2
C₄×M₁₆×C₉₇	2
#2970	97