Subgroups — C₂×#20260689 #27150147

Order 12416.

592 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13095494	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15690767	C₂	1	1	yes	maximal, characteristic
4	#20260689	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15690721	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15690720	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
10	C₂×#13095494	C₄	1	8	yes	complemented
11	C₂×#13095494	C₂²	1	4	yes	—
12	C₂²×#11715895	C₂²	1	1	yes	characteristic
13	C₂²×#11715895	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₉₇	C₄	1	8	yes	complemented
15	C₂×C₈×D₉₇	C₂²	1	4	yes	—
16	#15690767	C₄	1	4	yes	—
17	#15690767	C₂²	1	4	yes	—
18	C₂×#13095490	C₄	1	2	yes	—
19	C₂×#13095488	C₄	1	2	yes	—
20	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
21	#15690721	C₂²	1	4	yes	complemented
22	#15690720	C₂²	1	4	yes	complemented
23	C₈×Dic₉₇	C₂²	1	4	yes	complemented
24	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₉₇	C₂²	1	4	yes	complemented
28	C₂³×D₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
30	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₉₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₉₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₉₇	C₂³	1	2	yes	—
35	#13095494	C₂×C₄	1	16	yes	complemented
36	C₂×#11715895	C₂×C₄	1	4	yes	—
37	C₂×#11715895	C₂×C₄	1	4	yes	—
38	C₂×#11715895	C₂³	1	2	yes	—
39	C₂×#11715895	C₂³	1	2	yes	—
40	C₈×D₉₇	C₂×C₄	1	16	yes	complemented
41	#13095490	C₂×C₄	1	8	yes	—
42	#13095488	C₂×C₄	1	8	yes	—
43	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
44	C₄×Dic₉₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
48	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
49	C₂×C₈×C₉₇	C₂³	1	2	yes	—
50	C₂×C₈×C₉₇	C₂³	1	2	yes	—
51	C₄²×C₉₇	C₂×C₄	1	2	yes	—
52	C₄²×C₉₇	C₂³	1	2	yes	—
53	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₉₇	—	2	4	no	—
55	C₂²×D₉₇	C₄²	1	4	yes	—
56	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₉₇	—	2	4	no	—
58	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₉₇	—	2	4	no	—
64	C₂×Dic₉₇	C₄²	1	4	yes	—
65	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
66	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
67	C₄×D₉₇	C₄²	1	8	yes	—
68	C₄×D₉₇	C₂²×C₄	1	4	yes	—
69	C₄×D₉₇	C₂²×C₄	1	4	yes	—
70	C₈×C₉₇	C₂²×C₄	1	4	yes	—
71	C₈×C₉₇	C₂²×C₄	1	4	yes	—
72	#11715895	C₂²×C₄	1	4	yes	—
73	#11715895	C₂²×C₄	1	4	yes	—
74	C₂²×C₉₇	C₂×C₄²	1	2	yes	—
75	C₂²×C₉₇	#131	1	4	yes	—
76	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
77	C₂×D₉₇	—	2	4	no	—
78	C₂×D₉₇	—	2	8	no	—
79	C₂×D₉₇	C₂×C₄²	1	4	yes	—
80	C₄×C₉₇	—	2	4	no	—
81	C₄×C₉₇	C₂×C₄²	1	2	yes	—
82	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
84	Dic₉₇	—	2	4	no	—
85	Dic₉₇	C₂×C₄²	1	4	yes	—
86	C₂×C₉₇	#405	1	4	yes	complemented
87	C₂×C₉₇	C₂²×C₄²	1	1	yes	characteristic
88	C₂×C₉₇	C₂×#131	1	2	yes	—
89	D₉₇	—	2	4	no	—
90	C₂×#405	—	97	1	no	maximal, Sylow
91	C₉₇	C₂×#405	1	1	yes	characteristic, complemented, Sylow
92	C₂²×M₁₆	—	97	1	no	—
93	C₂³×C₈	—	97	1	no	—
94	C₂×#117	—	97	1	no	—
95	#405	—	97	8	no	—
96	C₂×#97	—	97	1	no	—
97	C₂×#97	—	97	1	no	—
98	C₂×C₄×C₈	—	97	1	no	—
99	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic, complemented
100	C₂³×C₄	—	97	1	no	—
101	C₂×M₁₆	—	97	8	no	—
102	C₂×M₁₆	—	97	4	no	—
103	C₂²×C₈	—	97	1	no	—
104	C₂²×C₈	—	97	1	no	—
105	C₂²×C₈	—	97	8	no	—
106	C₂²×C₈	—	97	4	no	—
107	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
109	#117	—	97	4	no	—
110	#117	—	97	4	no	—
111	C₂×C₄⋊C₄	—	97	2	no	—
112	C₂×C₂²⋊C₄	—	97	2	no	—
113	C₂×C₄²	—	97	1	no	—
114	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
115	#97	—	97	4	no	—
116	#97	—	97	4	no	—
117	C₄×C₈	—	97	4	no	—
118	C₄×C₈	C₂×D₉₇	1	4	yes	complemented
119	C₂⁴	—	97	1	no	—
120	C₂²×C₄	—	97	1	no	—
121	C₂²×C₄	—	97	2	no	—
122	C₂²×C₄	—	97	4	no	—
123	C₂²×C₄	—	97	4	no	—
124	C₂²×C₄	—	97	2	no	—
125	C₂²×C₄	—	97	2	no	—
126	C₂²×C₄	C₄×D₉₇	1	2	yes	—
127	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
128	M₁₆	—	97	16	no	—
129	C₂×C₈	—	97	4	no	—
130	C₂×C₈	—	97	4	no	—
131	C₂×C₈	—	97	2	no	—
132	C₂×C₈	—	97	2	no	—
133	C₂×C₈	—	97	16	no	—
134	C₂×C₈	C₄×D₉₇	1	4	yes	—
135	C₂×C₈	C₄×D₉₇	1	4	yes	—
136	C₂×C₈	C₂²×D₉₇	1	2	yes	—
137	C₂×C₈	C₂²×D₉₇	1	2	yes	—
138	C₄⋊C₄	—	97	8	no	—
139	C₂²⋊C₄	—	97	8	no	—
140	C₄²	—	97	2	no	—
141	C₄²	—	97	2	no	—
142	C₄²	C₄×D₉₇	1	2	yes	—
143	C₄²	C₂²×D₉₇	1	2	yes	—
144	C₂³	—	97	4	no	—
145	C₂³	—	97	2	no	—
146	C₂³	—	194	4	no	—
147	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	97	4	no	—
150	C₂×C₄	—	97	2	no	—
151	C₂×C₄	—	97	8	no	—
152	C₂×C₄	—	97	4	no	—
153	C₂×C₄	—	97	4	no	—
154	C₂×C₄	—	97	4	no	—
155	C₂×C₄	—	194	4	no	—
156	C₂×C₄	C₂×C₄×D₉₇	1	4	yes	—
157	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
160	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
161	C₈	—	97	4	no	—
162	C₈	—	97	4	no	—
163	C₈	C₂×C₄×D₉₇	1	4	yes	—
164	C₈	C₂×C₄×D₉₇	1	4	yes	—
165	C₂²	—	97	4	no	—
166	C₂²	—	194	8	no	—
167	C₂²	—	194	4	no	—
168	C₂²	C₄²×D₉₇	1	2	yes	—
169	C₂²	#15690816	1	4	yes	—
170	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
171	C₄	—	2	4	no	—
172	C₄	—	97	4	no	—
173	C₄	—	194	4	no	—
174	C₄	C₄²×D₉₇	1	2	yes	—
175	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
176	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
177	C₂	—	194	4	no	—
178	C₂	#20260689	1	4	yes	direct product factor
179	C₂	C₂×C₄²×D₉₇	1	1	yes	characteristic
180	C₂	C₂×#15690816	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₉₇
commutator	C₂×C₉₇	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle	C₂³×C₉₇	C₂²×C₄
prime radical	C₂³×D₉₇	C₂×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	C₂×#405
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #20260689	C₂	direct (N × H)
5. C₂×#15690721	C₂	semidirect
6. C₂×#15690720	C₂	semidirect
7. C₂×C₈×Dic₉₇	C₂	semidirect
8. C₂×C₄×C₈×C₉₇	C₂	semidirect
10. C₂×#13095494	C₄	semidirect
14. C₂×C₈×D₉₇	C₄	semidirect
21. #15690721	C₂²	semidirect
22. #15690720	C₂²	semidirect
23. C₈×Dic₉₇	C₂²	semidirect
27. C₄×C₈×C₉₇	C₂²	semidirect
35. #13095494	C₂×C₄	semidirect
40. C₈×D₉₇	C₂×C₄	semidirect
86. C₂×C₉₇	#405	semidirect
91. C₉₇	C₂×#405	semidirect
99. C₂×C₄×C₈	D₉₇	semidirect
118. C₄×C₈	C₂×D₉₇	semidirect
178. C₂	#20260689	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13095494	2
C₂²×C₈×D₉₇	2
C₂×#15690767	2
#20260689	2
C₂×#15690721	2
C₂×#15690720	2
C₂×C₈×Dic₉₇	2
C₂×C₄×C₈×C₉₇	2
C₂×#405	97

Subgroups — C2×#20260689 #27150147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20260689 #27150147