Subgroups — C₂×C₄×#13182030 #28376213

Order 13952.

1728 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13182030	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182030	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₄₃₆	C₂	1	3	yes	maximal, complemented
5	C₂×#16812661	C₂	1	3	yes	maximal, complemented
6	C₂×#16812657	C₂	1	3	yes	maximal, complemented
7	C₂×C₄²×D₁₀₉	C₂	1	3	yes	maximal, complemented
8	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13182030	C₄	1	16	yes	direct product factor
10	C₂×#13182030	C₂²	1	8	yes	—
11	C₂²×D₄₃₆	C₂²	1	3	yes	—
12	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
13	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
14	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
15	Q₈×Dic₁₀₉	C₂²	1	8	yes	complemented
16	C₄×D₄₃₆	C₂²	1	24	yes	complemented
17	#16812661	C₂²	1	24	yes	complemented
18	#16812657	C₂²	1	24	yes	complemented
19	C₂×C₄⋊Dic₁₀₉	C₂²	1	3	yes	—
20	C₂×#13182006	C₂²	1	6	yes	—
21	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₀₉	C₂²	1	3	yes	—
23	C₂×C₄×Dic₁₀₉	C₂²	1	3	yes	—
24	C₄²×D₁₀₉	C₂²	1	24	yes	complemented
25	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₀₉	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
28	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
29	C₂³×D₁₀₉	C₂³	1	3	yes	—
30	#13182030	C₂×C₄	1	64	yes	direct product factor
31	C₂×D₄₃₆	C₂×C₄	1	24	yes	complemented
32	C₂×D₄₃₆	C₂³	1	12	yes	—
33	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
34	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₁₀₉	—	2	12	no	—
37	C₂×C₄×D₁₀₉	—	2	12	no	—
38	C₂×C₄×D₁₀₉	C₂×C₄	1	24	yes	complemented
39	C₂×C₄×D₁₀₉	C₂³	1	12	yes	—
40	C₂×C₄×D₁₀₉	C₂³	1	12	yes	—
41	C₂×C₄×D₁₀₉	C₂³	1	12	yes	—
42	C₄⋊Dic₁₀₉	C₂³	1	12	yes	—
43	#13182006	C₂³	1	24	yes	—
44	C₄×Dic₁₀₉	C₂³	1	4	yes	—
45	C₄×Dic₁₀₉	C₂³	1	12	yes	—
46	C₄×Dic₁₀₉	C₂³	1	12	yes	—
47	C₂×Q₈×C₁₀₉	C₂×C₄	1	8	yes	complemented
48	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
49	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
50	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
51	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₁₀₉	C₂³	1	12	yes	—
53	C₄²×C₁₀₉	C₂³	1	12	yes	—
54	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
55	C₂²×D₁₀₉	—	2	6	no	—
56	C₂²×D₁₀₉	—	2	6	no	—
57	C₂²×D₁₀₉	C₂²×C₄	1	12	yes	—
58	C₂²×D₁₀₉	C₂⁴	1	6	yes	—
59	Q₈×C₁₀₉	C₂²×C₄	1	16	yes	complemented
60	C₂×C₄×C₁₀₉	—	2	6	no	—
61	C₂×C₄×C₁₀₉	C₂²×C₄	1	12	yes	—
62	C₂×C₄×C₁₀₉	#41	1	4	yes	complemented
63	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
64	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
65	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
66	C₂×Dic₁₀₉	—	2	6	no	—
67	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
68	C₂×Dic₁₀₉	#41	1	4	yes	complemented
69	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
70	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
71	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
72	D₄₃₆	C₂²×C₄	1	48	yes	complemented
73	C₄×D₁₀₉	—	2	24	no	—
74	C₄×D₁₀₉	—	2	24	no	—
75	C₄×D₁₀₉	C₂²×C₄	1	48	yes	complemented
76	C₂²×C₁₀₉	C₂³×C₄	1	2	yes	—
77	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
78	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
79	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
80	C₂×D₁₀₉	—	2	24	no	—
81	C₂×D₁₀₉	—	2	6	no	—
82	C₂×D₁₀₉	—	2	6	no	—
83	C₂×D₁₀₉	C₂³×C₄	1	12	yes	—
84	C₄×C₁₀₉	—	2	6	no	—
85	C₄×C₁₀₉	C₂³×C₄	1	12	yes	—
86	C₄×C₁₀₉	C₂×#41	1	4	yes	complemented
87	Dic₁₀₉	—	2	6	no	—
88	Dic₁₀₉	C₂³×C₄	1	4	yes	—
89	Dic₁₀₉	C₂×#41	1	4	yes	complemented
90	C₂×C₁₀₉	C₄×#41	1	4	yes	complemented
91	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
92	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
93	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
94	D₁₀₉	—	2	12	no	—
95	C₂×C₄×#41	—	109	1	no	maximal, Sylow
96	C₁₀₉	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
97	C₂²×#41	—	109	1	no	—
98	C₄×#41	—	109	16	no	—
99	C₂×C₄×Q₈	—	109	1	no	—
100	C₂×C₄×Q₈	D₁₀₉	1	1	yes	characteristic, complemented
101	C₂×C₄×D₄	—	109	3	no	—
102	C₂×C₄×D₄	—	109	3	no	—
103	C₂×#117	—	109	3	no	—
104	C₂²×C₄²	—	109	3	no	—
105	C₂×#41	—	109	16	no	—
106	C₂×#41	—	109	8	no	—
107	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
108	C₂²×D₄	—	109	3	no	—
109	C₂³×C₄	—	109	3	no	—
110	C₂³×C₄	—	109	3	no	—
111	C₂³×C₄	—	109	3	no	—
112	C₄×Q₈	—	109	8	no	—
113	C₄×Q₈	C₂×D₁₀₉	1	8	yes	complemented
114	C₄×D₄	—	109	24	no	—
115	C₄×D₄	—	109	24	no	—
116	#117	—	109	24	no	—
117	C₂×C₄⋊C₄	—	109	3	no	—
118	C₂×C₄⋊C₄	C₂×D₁₀₉	1	3	yes	—
119	C₂×C₂²⋊C₄	—	109	6	no	—
120	C₂×C₄²	—	109	3	no	—
121	C₂×C₄²	—	109	3	no	—
122	C₂×C₄²	—	109	1	no	—
123	C₂×C₄²	—	109	24	no	—
124	C₂×C₄²	C₂×D₁₀₉	1	3	yes	—
125	C₂⁴	—	109	3	no	—
126	#41	—	109	64	no	—
127	C₂×Q₈	C₄×D₁₀₉	1	8	yes	complemented
128	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	—
129	C₂×D₄	—	109	24	no	—
130	C₂×D₄	—	109	12	no	—
131	C₂²×C₄	—	109	1	no	—
132	C₂²×C₄	—	109	1	no	—
133	C₂²×C₄	—	109	24	no	—
134	C₂²×C₄	—	109	12	no	—
135	C₂²×C₄	—	109	3	no	—
136	C₂²×C₄	—	109	12	no	—
137	C₂²×C₄	—	109	12	no	—
138	C₂²×C₄	—	218	12	no	—
139	C₂²×C₄	—	218	12	no	—
140	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
141	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
142	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
143	C₄⋊C₄	—	109	12	no	—
144	C₄⋊C₄	C₂²×D₁₀₉	1	12	yes	—
145	C₂²⋊C₄	—	109	24	no	—
146	C₄²	—	109	4	no	—
147	C₄²	—	109	12	no	—
148	C₄²	—	109	12	no	—
149	C₄²	C₂²×D₁₀₉	1	12	yes	—
150	C₂³	—	109	12	no	—
151	C₂³	—	109	6	no	—
152	C₂³	—	218	6	no	—
153	C₂³	—	218	6	no	—
154	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
155	Q₈	C₂×C₄×D₁₀₉	1	16	yes	complemented
156	D₄	—	109	48	no	—
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	109	4	no	—
159	C₂×C₄	—	109	4	no	—
160	C₂×C₄	—	109	2	no	—
161	C₂×C₄	—	109	2	no	—
162	C₂×C₄	—	109	48	no	—
163	C₂×C₄	—	109	6	no	—
164	C₂×C₄	—	218	24	no	—
165	C₂×C₄	—	218	6	no	—
166	C₂×C₄	—	218	24	no	—
167	C₂×C₄	C₂×C₄×D₁₀₉	1	12	yes	—
168	C₂×C₄	#13182030	1	4	yes	direct product factor
169	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
170	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
171	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
172	C₂²	—	109	12	no	—
173	C₂²	—	218	24	no	—
174	C₂²	—	218	6	no	—
175	C₂²	—	218	6	no	—
176	C₂²	C₂²×C₄×D₁₀₉	1	2	yes	—
177	C₂²	#41×D₁₀₉	1	2	yes	—
178	C₂²	C₂×#13182030	1	2	yes	—
179	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
180	C₄	—	2	6	no	—
181	C₄	—	109	4	no	—
182	C₄	—	109	4	no	—
183	C₄	—	218	6	no	—
184	C₄	C₂²×C₄×D₁₀₉	1	12	yes	—
185	C₄	C₂×#13182030	1	4	yes	direct product factor
186	C₂	—	218	12	no	—
187	C₂	C₄×#13182030	1	4	yes	direct product factor
188	C₂	C₂³×C₄×D₁₀₉	1	1	yes	characteristic
189	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
190	C₂	C₂²×#13182030	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×C₄×Q₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂²×#13182030	C₂
cosocle	C₂²×C₁₀₉	C₂⁵
hypercenter	C₂×C₄×Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×C₄×#41
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×#13182030	C₂	direct (N × H)
3. C₂×Q₈×Dic₁₀₉	C₂	semidirect
4. C₂×C₄×D₄₃₆	C₂	semidirect
5. C₂×#16812661	C₂	semidirect
6. C₂×#16812657	C₂	semidirect
7. C₂×C₄²×D₁₀₉	C₂	semidirect
8. C₂×C₄×Q₈×C₁₀₉	C₂	semidirect
9. C₂×#13182030	C₄	direct (N × H)
15. Q₈×Dic₁₀₉	C₂²	semidirect
16. C₄×D₄₃₆	C₂²	semidirect
17. #16812661	C₂²	semidirect
18. #16812657	C₂²	semidirect
24. C₄²×D₁₀₉	C₂²	semidirect
26. C₄×Q₈×C₁₀₉	C₂²	semidirect
30. #13182030	C₂×C₄	direct (N × H)
31. C₂×D₄₃₆	C₂×C₄	semidirect
38. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
47. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
59. Q₈×C₁₀₉	C₂²×C₄	semidirect
62. C₂×C₄×C₁₀₉	#41	semidirect
68. C₂×Dic₁₀₉	#41	semidirect
72. D₄₃₆	C₂²×C₄	semidirect
75. C₄×D₁₀₉	C₂²×C₄	semidirect
86. C₄×C₁₀₉	C₂×#41	semidirect
89. Dic₁₀₉	C₂×#41	semidirect
90. C₂×C₁₀₉	C₄×#41	semidirect
96. C₁₀₉	C₂×C₄×#41	semidirect
100. C₂×C₄×Q₈	D₁₀₉	semidirect
113. C₄×Q₈	C₂×D₁₀₉	semidirect
127. C₂×Q₈	C₄×D₁₀₉	semidirect
155. Q₈	C₂×C₄×D₁₀₉	semidirect
168. C₂×C₄	#13182030	direct (N × H)
185. C₄	C₂×#13182030	direct (N × H)
187. C₂	C₄×#13182030	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×#13182030	2
C₄×#13182030	2
C₂×Q₈×Dic₁₀₉	2
C₂×C₄×D₄₃₆	2
C₂×#16812661	2
C₂×#16812657	2
C₂×C₄²×D₁₀₉	2
C₂×C₄×Q₈×C₁₀₉	2
C₂×C₄×#41	109

Subgroups — C2×C4×#13182030 #28376213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13182030 #28376213