Subgroups — C₂×#20261745 #27158387

Order 12416.

512 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690889	C₂	1	2	yes	maximal
2	#20261745	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15690879	C₂	1	2	yes	maximal, complemented
4	C₂×#15690878	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic
6	#15690889	—	2	4	no	—
7	#15690889	C₂²	1	8	yes	—
8	C₂×C₄⋊#87604	C₂²	1	4	yes	—
9	C₂×C₂²⋊#87604	—	2	2	no	—
10	C₂×C₄×#87604	—	2	2	no	—
11	#15690879	C₂²	1	8	yes	complemented
12	#15690878	C₂²	1	8	yes	complemented
13	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
14	C₂²×#11715895	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×D₉₇	C₄	1	1	yes	characteristic, complemented
16	C₂×C₈×D₉₇	C₄	1	1	yes	characteristic, complemented
17	C₂×C₈×D₉₇	C₄	1	4	yes	complemented
18	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₉₇	C₂²	1	4	yes	—
21	C₂²×C₈×C₉₇	C₄	1	1	yes	characteristic, complemented
22	C₂²×#87604	—	2	4	no	—
23	C₄⋊#87604	—	2	4	no	—
24	C₄⋊#87604	C₂³	1	8	yes	—
25	C₂²⋊#87604	—	2	8	no	—
26	C₄×#87604	—	2	8	no	—
27	C₂³×D₉₇	D₄	1	1	yes	characteristic
28	C₂²×Dic₉₇	Q₈	1	1	yes	characteristic
29	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
32	C₂×C₄×D₉₇	D₄	1	2	yes	—
33	C₂×C₄×D₉₇	Q₈	1	2	yes	—
34	C₂×C₄×D₉₇	Q₈	1	1	yes	characteristic
35	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₉₇	C₂³	1	2	yes	—
37	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
38	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
39	C₂×#11715895	C₂×C₄	1	4	yes	complemented
40	C₈×D₉₇	C₂×C₄	1	2	yes	—
41	C₈×D₉₇	C₂×C₄	1	2	yes	—
42	C₈×D₉₇	C₂×C₄	1	4	yes	complemented
43	C₈×D₉₇	C₂×C₄	1	4	yes	complemented
44	C₈×D₉₇	C₂³	1	2	yes	—
45	C₈×D₉₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	complemented
50	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×D₉₇	—	2	4	no	—
52	C₂²×D₉₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×D₉₇	C₄⋊C₄	1	2	yes	—
54	C₂²×D₉₇	C₂×D₄	1	2	yes	—
55	C₂²×D₉₇	C₂×D₄	1	1	yes	characteristic
56	C₂×#87604	—	2	8	no	—
57	C₂×#87604	—	4	8	no	—
58	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
62	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
63	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂×Dic₉₇	C₂×Q₈	1	2	yes	—
65	C₂×Dic₉₇	C₂×Q₈	1	1	yes	characteristic
66	C₄×D₉₇	C₄⋊C₄	1	2	yes	—
67	C₄×D₉₇	C₄⋊C₄	1	2	yes	—
68	C₄×D₉₇	C₄⋊C₄	1	2	yes	—
69	C₄×D₉₇	C₂²×C₄	1	1	yes	characteristic
70	C₄×D₉₇	C₂²×C₄	1	2	yes	—
71	C₄×D₉₇	C₂×D₄	1	2	yes	—
72	C₄×D₉₇	C₂×D₄	1	1	yes	characteristic
73	C₄×D₉₇	C₂×Q₈	1	2	yes	—
74	C₄×D₉₇	C₂×Q₈	1	1	yes	characteristic
75	C₄×D₉₇	C₂⁴	1	1	yes	characteristic
76	C₈×C₉₇	C₂²×C₄	1	2	yes	—
77	C₈×C₉₇	C₂²×C₄	1	2	yes	—
78	#11715895	C₂²×C₄	1	2	yes	—
79	#11715895	C₂²×C₄	1	2	yes	—
80	C₂²×C₉₇	—	2	2	no	—
81	C₂²×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×D₉₇	—	2	2	no	—
84	C₂×D₉₇	—	2	4	no	—
85	C₂×D₉₇	—	2	4	no	—
86	C₂×D₉₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×D₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₉₇	#135	1	4	yes	—
89	C₂×D₉₇	C₂²×D₄	1	1	yes	characteristic
90	#87604	—	4	8	no	—
91	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
94	Dic₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	Dic₉₇	C₂×C₄⋊C₄	1	2	yes	—
96	Dic₉₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₉₇	—	2	2	no	—
98	C₂×C₉₇	#427	1	2	yes	complemented
99	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	D₉₇	—	2	2	no	—
101	D₉₇	#427	1	2	yes	complemented
102	D₉₇	C₂×#135	1	2	yes	—
103	C₂×#427	—	97	1	no	maximal, Sylow
104	C₉₇	C₂×#427	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₈	—	97	1	no	—
106	C₂×#117	—	97	2	no	—
107	#427	—	97	8	no	—
108	C₂×#107	—	97	2	no	—
109	C₂×#106	—	97	2	no	—
110	C₂³×C₄	—	97	1	no	—
111	C₂²×C₈	—	97	1	no	—
112	C₂²×C₈	—	97	1	no	—
113	C₂²×C₈	—	97	1	no	—
114	C₂²×C₈	—	97	1	no	—
115	C₂²×C₈	—	97	1	no	—
116	C₂²×C₈	—	97	4	no	—
117	C₂²×C₈	—	97	4	no	—
118	C₂²×C₈	#87604	1	1	yes	characteristic, complemented
119	#117	—	97	8	no	—
120	#117	—	194	4	no	—
121	C₂×C₄⋊C₄	—	97	4	no	—
122	C₂×C₂²⋊C₄	—	194	2	no	—
123	C₂×C₄²	—	194	2	no	—
124	#107	—	97	8	no	—
125	#106	—	97	8	no	—
126	C₂⁴	—	97	1	no	—
127	C₂²×C₄	—	97	1	no	—
128	C₂²×C₄	—	97	2	no	—
129	C₂²×C₄	—	97	2	no	—
130	C₂²×C₄	—	97	1	no	—
131	C₂²×C₄	—	97	1	no	—
132	C₂²×C₄	—	97	1	no	—
133	C₂²×C₄	—	97	2	no	—
134	C₂²×C₄	—	97	1	no	—
135	C₂²×C₄	—	97	2	no	—
136	C₂²×C₄	—	194	4	no	—
137	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
138	C₂×C₈	—	97	1	no	—
139	C₂×C₈	—	97	1	no	—
140	C₂×C₈	—	97	4	no	—
141	C₂×C₈	—	97	2	no	—
142	C₂×C₈	—	97	2	no	—
143	C₂×C₈	—	97	4	no	—
144	C₂×C₈	—	97	4	no	—
145	C₂×C₈	—	97	2	no	—
146	C₂×C₈	—	97	2	no	—
147	C₂×C₈	C₂×#87604	1	1	yes	characteristic
148	C₂×C₈	C₂×#87604	1	1	yes	characteristic
149	C₂×C₈	C₂×#87604	1	4	yes	complemented
150	C₄⋊C₄	—	97	8	no	—
151	C₄⋊C₄	—	194	4	no	—
152	C₂²⋊C₄	—	194	8	no	—
153	C₄²	—	194	8	no	—
154	C₂³	—	97	1	no	—
155	C₂³	—	97	2	no	—
156	C₂³	—	97	2	no	—
157	C₂³	—	97	1	no	—
158	C₂³	—	194	4	no	—
159	C₂³	C₄⋊#87604	1	1	yes	characteristic
160	C₂×C₄	—	97	2	no	—
161	C₂×C₄	—	97	1	no	—
162	C₂×C₄	—	97	2	no	—
163	C₂×C₄	—	97	1	no	—
164	C₂×C₄	—	97	2	no	—
165	C₂×C₄	—	97	1	no	—
166	C₂×C₄	—	97	2	no	—
167	C₂×C₄	—	97	2	no	—
168	C₂×C₄	—	97	2	no	—
169	C₂×C₄	—	97	1	no	—
170	C₂×C₄	—	97	2	no	—
171	C₂×C₄	—	97	1	no	—
172	C₂×C₄	—	97	1	no	—
173	C₂×C₄	—	97	2	no	—
174	C₂×C₄	—	388	8	no	—
175	C₂×C₄	—	194	8	no	—
176	C₂×C₄	C₄⋊#87604	1	2	yes	—
177	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
178	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
179	C₂×C₄	C₂²×#87604	1	2	yes	—
180	C₈	—	97	2	no	—
181	C₈	—	97	2	no	—
182	C₈	C₂²×#87604	1	2	yes	—
183	C₈	C₂²×#87604	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	97	4	no	—
186	C₂²	—	97	1	no	—
187	C₂²	—	97	1	no	—
188	C₂²	—	97	2	no	—
189	C₂²	—	194	4	no	—
190	C₂²	—	194	4	no	—
191	C₂²	—	194	2	no	—
192	C₂²	C₂×C₄⋊#87604	1	2	yes	—
193	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
194	C₄	—	97	2	no	—
195	C₄	—	97	1	no	—
196	C₄	—	97	1	no	—
197	C₄	—	388	8	no	—
198	C₄	C₂×C₄⋊#87604	1	1	yes	characteristic
199	C₄	C₂×C₄⋊#87604	1	2	yes	—
200	C₄	C₂³×#87604	1	1	yes	characteristic
201	C₂	—	2	2	no	—
202	C₂	—	97	2	no	—
203	C₂	—	97	2	no	—
204	C₂	—	194	2	no	—
205	C₂	#20261745	1	2	yes	direct product factor
206	C₂	C₂²×C₄⋊#87604	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#87604
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₄	C₂³×#87604
Fitting	C₂²×C₈×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂³×D₉₇	D₄
cosocle	C₄×D₉₇	C₂⁴
hypercenter	C₂²×C₈	#87604
hypocenter	C₉₇	C₂×#427
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #20261745	C₂	direct (N × H)
3. C₂×#15690879	C₂	semidirect
4. C₂×#15690878	C₂	semidirect
11. #15690879	C₂²	semidirect
12. #15690878	C₂²	semidirect
14. C₂²×#11715895	C₄	semidirect
15. C₂×C₈×D₉₇	C₄	semidirect
16. C₂×C₈×D₉₇	C₄	semidirect
17. C₂×C₈×D₉₇	C₄	semidirect
21. C₂²×C₈×C₉₇	C₄	semidirect
39. C₂×#11715895	C₂×C₄	semidirect
42. C₈×D₉₇	C₂×C₄	semidirect
43. C₈×D₉₇	C₂×C₄	semidirect
49. C₂×C₈×C₉₇	C₂×C₄	semidirect
98. C₂×C₉₇	#427	semidirect
101. D₉₇	#427	semidirect
104. C₉₇	C₂×#427	semidirect
118. C₂²×C₈	#87604	semidirect
149. C₂×C₈	C₂×#87604	semidirect
205. C₂	#20261745	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15690889	2
#20261745	2
C₂×#15690879	2
C₂×#15690878	2
C₂²×C₈×D₉₇	2
C₂×#427	97

Subgroups — C2×#20261745 #27158387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261745 #27158387