Subgroups — #28365991

Order 13952.

480 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	#20294931	C₂	1	2	yes	maximal
3	#20294907	C₂	1	4	yes	maximal
4	#20294891	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₁₀₉	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₁₀₉	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#384×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₁₈	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×#13182008	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₀₉	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
18	#16812606	C₂²	1	4	yes	—
19	#16812606	C₂²	1	4	yes	—
20	#16812606	C₂²	1	2	yes	—
21	#16812606	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×Dic₂₁₈	—	2	2	no	—
27	C₂×Dic₂₁₈	—	2	2	no	—
28	C₂×Dic₂₁₈	C₂×C₄	1	4	yes	—
29	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
30	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
31	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
32	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
33	C₄⋊Dic₁₀₉	—	2	2	no	—
34	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
35	C₄⋊Dic₁₀₉	Q₈	1	4	yes	complemented
36	#13182008	C₂×C₄	1	8	yes	—
37	C₄×Dic₁₀₉	—	2	4	no	—
38	C₄×Dic₁₀₉	—	4	4	no	—
39	C₄×Dic₁₀₉	C₂×C₄	1	8	yes	—
40	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₁₀₉	—	2	4	no	—
45	C₄⋊C₄×C₁₀₉	—	2	4	no	—
46	C₄⋊C₄×C₁₀₉	—	4	2	no	—
47	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
48	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	—	2	1	no	—
50	C₂×C₄×C₁₀₉	—	2	1	no	—
51	C₂×C₄×C₁₀₉	—	2	4	no	—
52	C₂×C₄×C₁₀₉	—	2	4	no	—
53	C₂×C₄×C₁₀₉	—	2	4	no	—
54	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₀₉	#41	1	2	yes	—
59	C₂×C₄×C₁₀₉	#41	1	2	yes	—
60	C₂×C₄×C₁₀₉	#41	1	2	yes	—
61	C₂×Dic₁₀₉	—	2	2	no	—
62	C₂×Dic₁₀₉	—	2	2	no	—
63	C₂×Dic₁₀₉	—	2	2	no	—
64	C₂×Dic₁₀₉	—	2	2	no	—
65	C₂×Dic₁₀₉	—	2	2	no	—
66	C₂×Dic₁₀₉	—	2	2	no	—
67	C₂×Dic₁₀₉	—	2	2	no	—
68	C₂×Dic₁₀₉	—	2	2	no	—
69	C₂×Dic₁₀₉	—	2	2	no	—
70	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
71	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
72	C₂×Dic₁₀₉	C₂×Q₈	1	4	yes	—
73	Dic₂₁₈	—	2	8	no	—
74	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
77	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
78	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
79	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂²×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
81	C₄×C₁₀₉	—	2	2	no	—
82	C₄×C₁₀₉	—	4	4	no	—
83	C₄×C₁₀₉	#117	1	4	yes	—
84	C₄×C₁₀₉	C₄×Q₈	1	4	yes	—
85	Dic₁₀₉	—	2	4	no	—
86	Dic₁₀₉	—	2	4	no	—
87	Dic₁₀₉	—	2	4	no	—
88	Dic₁₀₉	—	4	2	no	—
89	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
90	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂×C₁₀₉	#520	1	1	yes	characteristic
92	C₂×C₁₀₉	#554	1	1	yes	characteristic
93	C₂×C₁₀₉	#556	1	1	yes	characteristic
94	C₂×C₁₀₉	#557	1	1	yes	characteristic
95	C₂×C₁₀₉	#559	1	1	yes	characteristic
96	#2355	—	109	1	no	maximal, Sylow
97	C₁₀₉	#2355	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄×Q₈	—	109	1	no	—
99	#391	—	109	2	no	—
100	#387	—	109	4	no	—
101	#384	D₁₀₉	1	1	yes	characteristic
102	#382	—	109	2	no	—
103	C₄×C₄⋊C₄	—	109	2	no	—
104	C₄×C₄⋊C₄	—	109	2	no	—
105	C₄×C₄⋊C₄	—	109	1	no	—
106	C₂²×Q₈	—	109	1	no	—
107	C₄×Q₈	—	109	8	no	—
108	C₂×C₄⋊C₄	—	109	2	no	—
109	C₂×C₄⋊C₄	—	109	1	no	—
110	C₂×C₄⋊C₄	—	109	1	no	—
111	C₂×C₄⋊C₄	—	109	1	no	—
112	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
115	C₂×C₄²	—	109	2	no	—
116	C₂×C₄²	—	109	4	no	—
117	C₂×C₄²	—	109	2	no	—
118	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
119	#95	—	109	4	no	—
120	#95	—	109	2	no	—
121	#95	—	109	4	no	—
122	#95	—	109	4	no	—
123	C₂×Q₈	—	109	4	no	—
124	C₂×Q₈	—	218	2	no	—
125	C₂×Q₈	—	218	2	no	—
126	C₂²×C₄	—	109	2	no	—
127	C₂²×C₄	—	109	2	no	—
128	C₂²×C₄	—	109	2	no	—
129	C₂²×C₄	—	109	2	no	—
130	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
131	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	109	8	no	—
138	C₄⋊C₄	—	109	4	no	—
139	C₄⋊C₄	—	109	4	no	—
140	C₄⋊C₄	—	218	2	no	—
141	C₄²	—	109	8	no	—
142	C₄²	—	436	4	no	—
143	C₄²	—	218	4	no	—
144	C₄²	C₄×D₁₀₉	1	4	yes	—
145	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
146	Q₈	—	218	8	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	109	4	no	—
153	C₂×C₄	—	109	4	no	—
154	C₂×C₄	—	109	4	no	—
155	C₂×C₄	—	218	2	no	—
156	C₂×C₄	—	218	2	no	—
157	C₂×C₄	—	218	2	no	—
158	C₂×C₄	—	218	2	no	—
159	C₂×C₄	—	218	2	no	—
160	C₂×C₄	—	218	2	no	—
161	C₂×C₄	—	218	2	no	—
162	C₂×C₄	—	218	2	no	—
163	C₂×C₄	—	218	2	no	—
164	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
166	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
167	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
168	C₂×C₄	#13182028	1	2	yes	—
169	C₂×C₄	#13182028	1	2	yes	—
170	C₂×C₄	#13182030	1	2	yes	—
171	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
172	C₂²	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
173	C₂²	C₂×#13182028	1	1	yes	characteristic
174	C₂²	C₂×#13182028	1	1	yes	characteristic
175	C₂²	C₂×#13182030	1	1	yes	characteristic
176	C₂²	#16812744	1	1	yes	characteristic
177	C₂²	#16812747	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	4	4	no	—
180	C₄	—	436	2	no	—
181	C₄	—	218	4	no	—
182	C₄	—	218	4	no	—
183	C₄	—	218	4	no	—
184	C₄	#16812657	1	4	yes	—
185	C₄	#16812664	1	4	yes	—
186	C₂	C₂×#16812657	1	1	yes	characteristic
187	C₂	C₂×#16812664	1	1	yes	characteristic
188	C₂	#20295507	1	1	yes	characteristic
189	C₂	#20295737	1	1	yes	characteristic
190	C₂	#20295750	1	1	yes	characteristic
191	C₂	#20295751	1	1	yes	characteristic
192	C₂	#20295762	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator	C₂²×C₁₀₉	C₂³×C₄
Fitting	#384×C₁₀₉	C₂
socle, prime radical, cosocle	C₂³×C₁₀₉	C₂⁴
hypercenter	#384	D₁₀₉
hypocenter	C₁₀₉	#2355
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₂₁₈	C₄	semidirect
35. C₄⋊Dic₁₀₉	Q₈	semidirect
97. C₁₀₉	#2355	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₈	2
#20294931	2
#20294907	2
#20294891	2
C₄⋊C₄×Dic₁₀₉	2
C₄×C₄⋊Dic₁₀₉	2
#384×C₁₀₉	2
#2355	109