Subgroups — #28379948

Order 13952.

432 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296075	C₂	1	2	yes	maximal, complemented
2	#20296053	C₂	1	4	yes	maximal, complemented
3	D₄×#11724786	C₂	1	2	yes	maximal, complemented
4	#20296044	C₂	1	1	yes	maximal, characteristic, complemented
5	#20296043	C₂	1	2	yes	maximal, complemented
6	C₂×#16812766	C₂	1	2	yes	maximal, complemented
7	D₁₀₉⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13182028	C₂	1	1	yes	maximal, characteristic
9	C₂×D₁₀₉⋊C₈	C₂²	1	2	yes	—
10	C₂²×#11724786	C₂²	1	4	yes	—
11	#16812766	C₂²	1	2	yes	—
12	#16812766	C₂²	1	1	yes	characteristic
13	#16812766	C₂²	1	4	yes	—
14	#16812766	C₂²	1	1	yes	characteristic
15	C₄⋊#11724786	C₂²	1	2	yes	—
16	#16812756	C₂²	1	2	yes	—
17	C₂²⋊#11724786	C₂²	1	4	yes	—
18	C₄×#11724786	C₂²	1	4	yes	—
19	C₄×#11724786	C₂²	1	2	yes	—
20	C₂×#13182028	C₄	1	1	yes	characteristic
21	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
22	#16812664	C₂²	1	1	yes	characteristic
23	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₀₉	C₄	1	2	yes	—
25	#16812659	C₂²	1	2	yes	—
26	#16812657	C₄	1	1	yes	characteristic
27	#16812655	C₄	1	2	yes	—
28	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
29	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
31	D₁₀₉⋊C₈	—	4	2	no	—
32	C₂×#11724786	—	4	4	no	—
33	C₂×#11724786	C₂³	1	2	yes	—
34	C₂×#11724786	C₂³	1	4	yes	—
35	C₂×#11724786	C₂³	1	2	yes	—
36	#13182028	C₈	1	8	yes	complemented
37	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
39	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
40	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
41	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
44	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
45	#13182008	C₂×C₄	1	2	yes	—
46	#13182008	C₂×C₄	1	2	yes	—
47	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
48	#13182006	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
53	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
54	C₄×Dic₁₀₉	C₂³	1	2	yes	—
55	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
57	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
59	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
61	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	D₄×C₁₀₉	C₂×C₈	1	4	yes	complemented
63	C₂×C₄×C₁₀₉	—	4	2	no	—
64	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²⋊D₁₀₉	C₂×C₈	1	8	yes	complemented
69	C₂×Dic₁₀₉	—	2	2	no	—
70	C₂×Dic₁₀₉	—	2	2	no	—
71	C₂×Dic₁₀₉	C₂×C₈	1	8	yes	complemented
72	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
77	C₄×D₁₀₉	—	2	1	no	—
78	C₄×D₁₀₉	—	2	1	no	—
79	C₄×D₁₀₉	—	4	1	no	—
80	C₄×D₁₀₉	C₂×C₈	1	4	yes	complemented
81	Dic₂₁₈	C₂×C₈	1	4	yes	complemented
82	#11724786	—	2	2	no	—
83	#11724786	—	2	4	no	—
84	#11724786	—	2	2	no	—
85	C₂²×C₁₀₉	—	2	2	no	—
86	C₂²×C₁₀₉	—	2	2	no	—
87	C₂²×C₁₀₉	C₂²×C₈	1	4	yes	—
88	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₁₀₉	—	2	1	no	—
90	C₂×D₁₀₉	—	2	1	no	—
91	C₂×D₁₀₉	C₂²×C₈	1	2	yes	—
92	C₄×C₁₀₉	—	2	1	no	—
93	C₄×C₁₀₉	—	2	1	no	—
94	C₄×C₁₀₉	—	2	2	no	—
95	C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
96	Dic₁₀₉	—	2	2	no	—
97	Dic₁₀₉	—	2	1	no	—
98	Dic₁₀₉	C₂²×C₈	1	4	yes	—
99	Dic₁₀₉	C₂²×C₈	1	2	yes	—
100	Dic₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
101	Dic₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×C₁₀₉	—	2	4	no	—
103	C₂×C₁₀₉	#520	1	1	yes	characteristic
104	C₂×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
105	C₂×C₁₀₉	#568	1	1	yes	characteristic
106	D₁₀₉	—	2	2	no	—
107	#2988	—	109	1	no	maximal, Sylow
108	C₁₀₉	#2988	1	1	yes	characteristic, complemented, Sylow
109	C₄×#41	—	109	1	no	—
110	C₈×Q₈	—	109	2	no	—
111	C₈×D₄	—	109	2	no	—
112	C₈×D₄	—	109	4	no	—
113	#431	—	109	2	no	—
114	#431	—	109	1	no	—
115	C₂×C₄⋊C₈	—	109	1	no	—
116	C₂×C₄⋊C₈	—	109	2	no	—
117	C₂×#41	—	109	1	no	—
118	C₂²×C₈	—	109	4	no	—
119	C₂²×C₈	—	109	2	no	—
120	C₄×Q₈	—	109	1	no	—
121	C₄×Q₈	—	109	1	no	—
122	C₄×D₄	—	109	1	no	—
123	C₄×D₄	—	109	2	no	—
124	C₄×D₄	—	109	2	no	—
125	C₄×D₄	#89137	1	1	yes	characteristic
126	#117	—	109	1	no	—
127	#117	—	109	2	no	—
128	C₂×C₄²	—	109	2	no	—
129	C₂×C₄²	—	109	1	no	—
130	C₄⋊C₈	—	109	2	no	—
131	C₄⋊C₈	—	109	1	no	—
132	C₄⋊C₈	—	109	2	no	—
133	C₄⋊C₈	—	109	4	no	—
134	C₄⋊C₈	—	109	1	no	—
135	C₂²⋊C₈	—	109	4	no	—
136	C₂²⋊C₈	—	109	2	no	—
137	C₄×C₈	—	109	4	no	—
138	C₄×C₈	—	109	2	no	—
139	#41	—	109	8	no	—
140	C₂×Q₈	—	109	1	no	—
141	C₂×D₄	—	109	2	no	—
142	C₂×D₄	C₂×#89137	1	1	yes	characteristic
143	C₂²×C₄	—	109	2	no	—
144	C₂²×C₄	—	109	1	no	—
145	C₂²×C₄	—	109	2	no	—
146	C₂²×C₄	—	109	1	no	—
147	C₂²×C₄	—	109	1	no	—
148	C₂²×C₄	C₂×#89137	1	2	yes	—
149	C₂×C₈	—	109	2	no	—
150	C₂×C₈	—	109	4	no	—
151	C₂×C₈	—	109	2	no	—
152	C₂×C₈	—	436	4	no	—
153	C₂×C₈	—	436	2	no	—
154	C₄⋊C₄	—	109	2	no	—
155	C₄⋊C₄	—	109	2	no	—
156	C₄⋊C₄	—	109	1	no	—
157	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic
158	C₂²⋊C₄	—	109	2	no	—
159	C₂²⋊C₄	—	109	2	no	—
160	C₂²⋊C₄	C₂×#89137	1	2	yes	—
161	C₄²	—	109	2	no	—
162	C₄²	—	109	1	no	—
163	C₄²	—	109	2	no	—
164	C₄²	—	109	2	no	—
165	C₄²	—	109	1	no	—
166	C₄²	—	109	1	no	—
167	C₄²	C₂×#89137	1	1	yes	characteristic
168	C₂³	—	109	1	no	—
169	C₂³	C₂²×#89137	1	2	yes	—
170	Q₈	—	109	4	no	—
171	D₄	—	109	8	no	—
172	D₄	D₁₀₉⋊C₈	1	4	yes	complemented
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	109	8	no	—
175	C₂×C₄	—	109	1	no	—
176	C₂×C₄	—	109	1	no	—
177	C₂×C₄	—	109	2	no	—
178	C₂×C₄	—	109	4	no	—
179	C₂×C₄	—	109	2	no	—
180	C₂×C₄	—	109	1	no	—
181	C₂×C₄	—	436	1	no	—
182	C₂×C₄	—	218	1	no	—
183	C₂×C₄	—	218	1	no	—
184	C₂×C₄	—	218	2	no	—
185	C₂×C₄	—	218	2	no	—
186	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
187	C₂×C₄	C₂²×#89137	1	2	yes	—
188	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
189	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
190	C₈	—	218	2	no	—
191	C₈	—	218	4	no	—
192	C₈	—	218	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	109	2	no	—
196	C₂²	—	218	1	no	—
197	C₂²	—	218	1	no	—
198	C₂²	C₂×D₁₀₉⋊C₈	1	4	yes	—
199	C₂²	C₂³×#89137	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	2	no	—
203	C₄	—	109	1	no	—
204	C₄	—	109	1	no	—
205	C₄	—	109	2	no	—
206	C₄	—	109	4	no	—
207	C₄	—	218	2	no	—
208	C₄	—	218	1	no	—
209	C₄	C₂×D₁₀₉⋊C₈	1	2	yes	—
210	C₂	—	2	4	no	—
211	C₂	—	218	2	no	—
212	C₂	#20296095	1	1	yes	characteristic
213	C₂	C₂²×D₁₀₉⋊C₈	1	1	yes	characteristic
214	C₂	#20296104	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator	C₂×C₁₀₉	C₂³×C₈
Fitting	C₄×D₄×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂×#13182028	C₄
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	C₄×D₄	#89137
hypocenter	C₁₀₉	#2988
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20296075	C₂	semidirect
2. #20296053	C₂	semidirect
3. D₄×#11724786	C₂	semidirect
4. #20296044	C₂	semidirect
5. #20296043	C₂	semidirect
6. C₂×#16812766	C₂	semidirect
7. D₁₀₉⋊(C₄⋊C₈)	C₂	semidirect
36. #13182028	C₈	semidirect
62. D₄×C₁₀₉	C₂×C₈	semidirect
68. C₂²⋊D₁₀₉	C₂×C₈	semidirect
71. C₂×Dic₁₀₉	C₂×C₈	semidirect
80. C₄×D₁₀₉	C₂×C₈	semidirect
81. Dic₂₁₈	C₂×C₈	semidirect
108. C₁₀₉	#2988	semidirect
172. D₄	D₁₀₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20296075	2
#20296053	2
D₄×#11724786	2
#20296044	2
#20296043	2
C₂×#16812766	2
D₁₀₉⋊(C₄⋊C₈)	2
C₄×#13182028	2
#2988	109