Subgroups — C₂×#20295761 #28376620

Order 13952.

1240 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295761	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16812696	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812688	C₂	1	2	yes	maximal, complemented
4	C₂×#16812687	C₂	1	2	yes	maximal, complemented
5	C₂×#16812684	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₄₃₆	C₂	1	2	yes	maximal, complemented
8	C₂×#16812661	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
11	C₂²×D₄₃₆	C₂²	1	2	yes	—
12	C₂²×D₄₃₆	C₂²	1	2	yes	—
13	C₂²×D₄₃₆	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₉	C₂²	1	4	yes	—
15	#16812696	C₂²	1	8	yes	—
16	#16812688	C₂²	1	16	yes	complemented
17	#16812687	C₂²	1	16	yes	complemented
18	#16812684	C₂²	1	32	yes	complemented
19	C₄×Dic₂₁₈	C₂²	1	8	yes	complemented
20	C₄×D₄₃₆	C₂²	1	16	yes	complemented
21	#16812661	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×#13182008	C₂²	1	2	yes	—
24	C₂×#13182006	C₂²	1	4	yes	—
25	C₂×#13182006	C₂²	1	4	yes	—
26	C₂×#13182006	C₂²	1	4	yes	—
27	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
28	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₄×Q₈×C₁₀₉	C₂²	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₂³×D₁₀₉	C₂³	1	4	yes	—
35	C₂×Dic₂₁₈	—	2	4	no	—
36	C₂×Dic₂₁₈	C₂³	1	4	yes	—
37	C₂×D₄₃₆	—	2	8	no	—
38	C₂×D₄₃₆	—	2	8	no	—
39	C₂×D₄₃₆	—	4	4	no	—
40	C₂×D₄₃₆	C₂³	1	8	yes	—
41	C₂×D₄₃₆	C₂³	1	8	yes	—
42	C₂×D₄₃₆	C₂³	1	8	yes	—
43	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
44	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
45	C₂×C₄×D₁₀₉	—	4	8	no	—
46	C₂×C₄×D₁₀₉	C₂³	1	16	yes	—
47	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
48	#13182008	C₂³	1	8	yes	—
49	#13182006	C₂³	1	16	yes	—
50	#13182006	C₂³	1	16	yes	—
51	#13182006	C₂³	1	16	yes	—
52	C₄×Dic₁₀₉	C₂³	1	8	yes	—
53	C₂×Q₈×C₁₀₉	—	2	4	no	—
54	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
55	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
61	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
62	C₄²×C₁₀₉	C₂³	1	8	yes	—
63	C₄²×C₁₀₉	C₂³	1	4	yes	—
64	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
65	C₂²×D₁₀₉	—	2	8	no	—
66	C₂²×D₁₀₉	—	2	8	no	—
67	C₂²×D₁₀₉	—	2	8	no	—
68	C₂²×D₁₀₉	C₂⁴	1	8	yes	—
69	Q₈×C₁₀₉	—	2	8	no	—
70	C₂×C₄×C₁₀₉	—	2	4	no	—
71	C₂×C₄×C₁₀₉	—	2	4	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	#41	1	4	yes	—
74	C₂×C₄×C₁₀₉	#41	1	4	yes	—
75	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
76	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
77	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
78	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
80	C₂×Dic₁₀₉	—	2	4	no	—
81	C₂×Dic₁₀₉	—	2	4	no	—
82	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
83	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
84	D₄₃₆	—	2	16	no	—
85	D₄₃₆	—	2	16	no	—
86	D₄₃₆	—	4	8	no	—
87	C₄×D₁₀₉	—	4	16	no	—
88	Dic₂₁₈	—	2	8	no	—
89	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
90	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
91	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
92	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
93	C₂×D₁₀₉	—	2	8	no	—
94	C₂×D₁₀₉	—	2	8	no	—
95	C₂×D₁₀₉	—	2	8	no	—
96	C₂×D₁₀₉	—	4	16	no	—
97	C₄×C₁₀₉	—	2	4	no	—
98	C₄×C₁₀₉	—	2	4	no	—
99	C₄×C₁₀₉	—	2	2	no	—
100	C₄×C₁₀₉	C₂×#41	1	4	yes	—
101	C₄×C₁₀₉	C₂×#41	1	4	yes	—
102	Dic₁₀₉	—	2	4	no	—
103	Dic₁₀₉	—	2	4	no	—
104	C₂×C₁₀₉	#559	1	4	yes	complemented
105	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
106	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
107	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
108	D₁₀₉	—	4	8	no	—
109	C₂×#559	—	109	1	no	maximal, Sylow
110	C₁₀₉	C₂×#559	1	1	yes	characteristic, complemented, Sylow
111	#559	—	109	16	no	—
112	C₂×#127	—	109	1	no	—
113	C₂×#124	—	109	2	no	—
114	C₂×#124	—	109	2	no	—
115	C₂×#123	—	109	4	no	—
116	C₂×C₄×Q₈	—	109	1	no	—
117	C₂×C₄×Q₈	D₁₀₉	1	1	yes	characteristic, complemented
118	C₂×C₄×D₄	—	109	2	no	—
119	C₂×C₄×D₄	—	109	2	no	—
120	C₂²×Q₈	—	109	1	no	—
121	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
122	C₂²×D₄	—	109	2	no	—
123	C₂²×D₄	—	109	2	no	—
124	C₂²×D₄	—	109	2	no	—
125	C₂³×C₄	—	109	4	no	—
126	#127	—	109	8	no	—
127	#124	—	109	16	no	—
128	#124	—	109	16	no	—
129	#123	—	109	32	no	—
130	C₄×Q₈	—	109	8	no	—
131	C₄×Q₈	C₂×D₁₀₉	1	8	yes	complemented
132	C₄×D₄	—	109	16	no	—
133	C₄×D₄	—	109	16	no	—
134	C₂×C₄⋊C₄	—	109	1	no	—
135	C₂×C₄⋊C₄	—	109	2	no	—
136	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
138	C₂×C₂²⋊C₄	—	109	4	no	—
139	C₂×C₂²⋊C₄	—	109	4	no	—
140	C₂×C₂²⋊C₄	—	109	4	no	—
141	C₂×C₄²	—	109	2	no	—
142	C₂×C₄²	C₂×D₁₀₉	1	2	yes	—
143	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
144	C₂⁴	—	109	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	109	4	no	—
147	C₂×Q₈	—	218	4	no	—
148	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	—
149	C₂×D₄	—	109	8	no	—
150	C₂×D₄	—	109	8	no	—
151	C₂×D₄	—	109	8	no	—
152	C₂×D₄	—	436	4	no	—
153	C₂×D₄	—	218	8	no	—
154	C₂×D₄	—	218	8	no	—
155	C₂²×C₄	—	109	2	no	—
156	C₂²×C₄	—	109	2	no	—
157	C₂²×C₄	—	109	16	no	—
158	C₂²×C₄	—	436	8	no	—
159	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
161	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
164	C₄⋊C₄	—	109	8	no	—
165	C₄⋊C₄	—	109	4	no	—
166	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
167	C₄⋊C₄	C₂²×D₁₀₉	1	8	yes	—
168	C₂²⋊C₄	—	109	16	no	—
169	C₂²⋊C₄	—	109	16	no	—
170	C₂²⋊C₄	—	109	16	no	—
171	C₄²	—	109	8	no	—
172	C₄²	C₂²×D₁₀₉	1	8	yes	—
173	C₄²	C₂²×D₁₀₉	1	4	yes	—
174	C₂³	—	109	8	no	—
175	C₂³	—	218	8	no	—
176	C₂³	—	218	8	no	—
177	C₂³	—	218	8	no	—
178	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
179	Q₈	—	2	8	no	—
180	Q₈	—	218	8	no	—
181	D₄	—	436	8	no	—
182	D₄	—	218	16	no	—
183	D₄	—	218	16	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	109	4	no	—
188	C₂×C₄	—	109	4	no	—
189	C₂×C₄	—	436	16	no	—
190	C₂×C₄	—	218	4	no	—
191	C₂×C₄	—	218	4	no	—
192	C₂×C₄	#13182029	1	4	yes	—
193	C₂×C₄	#13182030	1	4	yes	—
194	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
195	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
196	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
197	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
198	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
199	C₂²	—	436	16	no	—
200	C₂²	—	218	8	no	—
201	C₂²	—	218	8	no	—
202	C₂²	—	218	8	no	—
203	C₂²	C₂×#13182029	1	2	yes	—
204	C₂²	C₂×#13182030	1	2	yes	—
205	C₂²	#16812745	1	2	yes	—
206	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	2	no	—
210	C₄	—	218	4	no	—
211	C₄	—	218	4	no	—
212	C₄	C₂×#13182029	1	4	yes	—
213	C₄	C₂×#13182030	1	4	yes	—
214	C₂	—	436	8	no	—
215	C₂	#20295761	1	4	yes	direct product factor
216	C₂	C₂²×#13182029	1	1	yes	characteristic
217	C₂	C₂²×#13182030	1	1	yes	characteristic
218	C₂	C₂×#16812745	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×C₄×Q₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂×#16812696	C₂
hypercenter	C₂×C₄×Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#559
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20295761	C₂	direct (N × H)
3. C₂×#16812688	C₂	semidirect
4. C₂×#16812687	C₂	semidirect
5. C₂×#16812684	C₂	semidirect
6. C₂×C₄×Dic₂₁₈	C₂	semidirect
7. C₂×C₄×D₄₃₆	C₂	semidirect
8. C₂×#16812661	C₂	semidirect
9. C₂×C₄×Q₈×C₁₀₉	C₂	semidirect
16. #16812688	C₂²	semidirect
17. #16812687	C₂²	semidirect
18. #16812684	C₂²	semidirect
19. C₄×Dic₂₁₈	C₂²	semidirect
20. C₄×D₄₃₆	C₂²	semidirect
21. #16812661	C₂²	semidirect
29. C₄×Q₈×C₁₀₉	C₂²	semidirect
104. C₂×C₁₀₉	#559	semidirect
110. C₁₀₉	C₂×#559	semidirect
117. C₂×C₄×Q₈	D₁₀₉	semidirect
131. C₄×Q₈	C₂×D₁₀₉	semidirect
215. C₂	#20295761	direct (N × H)

Maximal subgroups

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

subgroup	index
#20295761	2
C₂×#16812696	2
C₂×#16812688	2
C₂×#16812687	2
C₂×#16812684	2
C₂×C₄×Dic₂₁₈	2
C₂×C₄×D₄₃₆	2
C₂×#16812661	2
C₂×C₄×Q₈×C₁₀₉	2
C₂×#559	109

Subgroups — C2×#20295761 #28376620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295761 #28376620