Subgroups — #27150305

Order 12416.

484 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690818	C₂	1	2	yes	maximal
2	C₄×#13095511	C₂	1	1	yes	maximal, characteristic
3	#20260763	C₂	1	2	yes	maximal, complemented
4	#20260760	C₂	1	4	yes	maximal, complemented
5	#20260757	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₈×D₉₇	C₂	1	2	yes	maximal, complemented
8	#423×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13095511	C₂²	1	1	yes	characteristic
10	#15690818	—	2	8	no	—
11	C₂×#13095494	C₂²	1	4	yes	—
12	C₂²×#11715895	C₂²	1	2	yes	—
13	C₂×C₈×D₉₇	C₂²	1	4	yes	—
14	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
15	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
16	C₄×D₃₈₈	C₄	1	1	yes	characteristic
17	C₄×D₃₈₈	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₉₇	C₄	1	2	yes	—
19	C₄×C₂²⋊D₉₇	C₄	1	2	yes	—
20	#15690769	C₂²	1	1	yes	characteristic
21	#15690767	C₂²	1	2	yes	—
22	C₄²×D₉₇	C₂²	1	2	yes	—
23	C₄⋊#11715895	C₂²	1	1	yes	characteristic
24	C₄⋊#11715895	C₂²	1	1	yes	characteristic
25	C₄⋊#11715895	C₂²	1	2	yes	—
26	#15690739	C₂²	1	4	yes	—
27	#15690739	C₂²	1	4	yes	—
28	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
29	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
31	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
32	#13095511	D₄	1	4	yes	—
33	#13095511	Q₈	1	4	yes	—
34	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
35	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₉₇	C₂³	1	2	yes	—
40	#13095494	—	2	8	no	—
41	C₂×#11715895	—	2	4	no	—
42	C₂×#11715895	C₂³	1	2	yes	—
43	C₂×#11715895	C₂³	1	2	yes	—
44	C₈×D₉₇	—	2	8	no	—
45	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
47	#13095490	C₂×C₄	1	2	yes	—
48	#13095490	C₂×C₄	1	2	yes	—
49	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
50	#13095488	C₂×C₄	1	2	yes	—
51	#13095488	C₂×C₄	1	2	yes	—
52	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
53	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
54	C₄×Dic₉₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
58	M₁₆×C₉₇	—	2	4	no	—
59	C₂×C₈×C₉₇	C₂³	1	4	yes	—
60	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
62	C₄²×C₉₇	C₂³	1	1	yes	characteristic
63	C₄²×C₉₇	C₂³	1	1	yes	characteristic
64	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₉₇	—	2	2	no	—
67	C₂×C₄×C₉₇	—	4	1	no	—
68	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
76	C₂²⋊D₉₇	C₄⋊C₄	1	8	yes	—
77	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
80	D₃₈₈	C₄⋊C₄	1	4	yes	—
81	C₄×D₉₇	—	2	4	no	—
82	C₄×D₉₇	—	4	2	no	—
83	C₄×D₉₇	C₂×D₄	1	4	yes	—
84	C₄×D₉₇	C₂×Q₈	1	4	yes	—
85	Dic₁₉₄	C₄⋊C₄	1	4	yes	—
86	C₈×C₉₇	—	2	4	no	—
87	#11715895	—	2	2	no	—
88	#11715895	—	2	2	no	—
89	C₂²×C₉₇	—	2	1	no	—
90	C₂²×C₉₇	—	2	1	no	—
91	C₂²×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₉₇	—	2	2	no	—
94	C₂×D₉₇	—	2	2	no	—
95	C₂×D₉₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₄×C₉₇	—	2	1	no	—
97	C₄×C₉₇	—	2	1	no	—
98	C₄×C₉₇	—	2	1	no	—
99	C₄×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₄×C₉₇	#131	1	2	yes	complemented
101	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
103	Dic₉₇	—	2	2	no	—
104	Dic₉₇	—	2	2	no	—
105	Dic₉₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₉₇	—	2	2	no	—
107	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
109	C₂×C₉₇	#568	1	1	yes	characteristic
110	D₉₇	—	2	4	no	—
111	#2906	—	97	1	no	maximal, Sylow
112	C₉₇	#2906	1	1	yes	characteristic, complemented, Sylow
113	C₂×#131	—	97	2	no	—
114	C₄×#41	—	97	1	no	—
115	#424	—	97	2	no	—
116	#424	—	97	4	no	—
117	#423	—	97	2	no	—
118	#423	D₉₇	1	1	yes	characteristic, complemented
119	C₂×C₄⋊C₈	—	97	1	no	—
120	C₂×C₄⋊C₈	—	97	2	no	—
121	C₂×#41	—	97	1	no	—
122	#131	—	194	8	no	—
123	C₂×M₁₆	—	97	4	no	—
124	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
125	C₂²×C₈	—	97	2	no	—
126	C₂²×C₈	—	97	4	no	—
127	C₄×Q₈	—	97	1	no	—
128	C₄×Q₈	—	97	1	no	—
129	C₄×D₄	—	97	2	no	—
130	C₄×D₄	—	97	2	no	—
131	C₄×D₄	—	97	1	no	—
132	C₄×D₄	—	97	1	no	—
133	#117	—	97	1	no	—
134	#117	—	97	2	no	—
135	C₂×C₄²	—	97	2	no	—
136	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
137	C₄⋊C₈	—	97	4	no	—
138	C₄⋊C₈	—	97	4	no	—
139	C₄⋊C₈	—	97	1	no	—
140	C₄⋊C₈	—	97	1	no	—
141	C₄⋊C₈	—	97	2	no	—
142	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
143	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
144	#41	—	97	4	no	—
145	#41	—	97	4	no	—
146	C₂×Q₈	—	97	1	no	—
147	C₂×D₄	—	97	2	no	—
148	C₂×D₄	—	97	1	no	—
149	C₂²×C₄	—	97	2	no	—
150	C₂²×C₄	—	97	2	no	—
151	C₂²×C₄	—	97	2	no	—
152	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
153	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
155	M₁₆	—	2	4	no	—
156	M₁₆	—	194	8	no	—
157	C₂×C₈	—	97	2	no	—
158	C₂×C₈	—	97	2	no	—
159	C₂×C₈	—	194	8	no	—
160	C₂×C₈	—	194	4	no	—
161	C₂×C₈	C₂²×D₉₇	1	4	yes	—
162	C₄⋊C₄	—	97	2	no	—
163	C₄⋊C₄	—	97	2	no	—
164	C₄⋊C₄	—	97	1	no	—
165	C₄⋊C₄	—	97	1	no	—
166	C₂²⋊C₄	—	97	2	no	—
167	C₂²⋊C₄	—	97	2	no	—
168	C₂²⋊C₄	—	97	2	no	—
169	C₄²	—	97	2	no	—
170	C₄²	—	97	2	no	—
171	C₄²	—	97	2	no	—
172	C₄²	C₄×D₉₇	1	1	yes	characteristic
173	C₄²	C₄×D₉₇	1	1	yes	characteristic
174	C₄²	C₂²×D₉₇	1	1	yes	characteristic
175	C₄²	C₂²×D₉₇	1	1	yes	characteristic
176	C₂³	—	97	2	no	—
177	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
178	Q₈	—	97	4	no	—
179	D₄	—	97	8	no	—
180	D₄	—	97	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	97	2	no	—
184	C₂×C₄	—	97	4	no	—
185	C₂×C₄	—	97	4	no	—
186	C₂×C₄	—	97	2	no	—
187	C₂×C₄	—	97	2	no	—
188	C₂×C₄	—	388	2	no	—
189	C₂×C₄	—	194	4	no	—
190	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
195	C₂×C₄	D₄×D₉₇	1	2	yes	—
196	C₂×C₄	Q₈×D₉₇	1	2	yes	—
197	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	—	194	2	no	—
200	C₈	—	194	2	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	97	4	no	—
204	C₂²	—	194	2	no	—
205	C₂²	—	194	2	no	—
206	C₂²	C₄⋊C₄×D₉₇	1	2	yes	—
207	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	97	4	no	—
212	C₄	—	194	2	no	—
213	C₄	—	194	2	no	—
214	C₄	C₄⋊C₄×D₉₇	1	2	yes	—
215	C₄	#15690818	1	2	yes	complemented
216	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
217	C₄	C₂×Q₈×D₉₇	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	194	4	no	—
220	C₂	C₂×C₄⋊C₄×D₉₇	1	1	yes	characteristic
221	C₂	C₂×#15690818	1	1	yes	characteristic
222	C₂	#20261501	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#423×C₉₇	C₂
prime radical	C₂×#13095511	C₂²
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#423	D₉₇
hypocenter	C₉₇	#2906
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
3. #20260763	C₂	semidirect
4. #20260760	C₂	semidirect
5. #20260757	C₂	semidirect
6. C₂×C₄⋊#11715895	C₂	semidirect
7. C₄⋊C₈×D₉₇	C₂	semidirect
8. #423×C₉₇	C₂	semidirect
100. C₄×C₉₇	#131	semidirect
112. C₉₇	#2906	semidirect
118. #423	D₉₇	semidirect
215. C₄	#15690818	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690818	2
C₄×#13095511	2
#20260763	2
#20260760	2
#20260757	2
C₂×C₄⋊#11715895	2
C₄⋊C₈×D₉₇	2
#423×C₉₇	2
#2906	97