Subgroups — #28371778

Order 13952.

492 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×#16812707	C₂	1	2	yes	maximal, complemented
2	#20295576	C₂	1	1	yes	maximal, characteristic
3	#20295105	C₂	1	2	yes	maximal, complemented
4	#20295099	C₂	1	4	yes	maximal, complemented
5	C₈×D₄₃₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
7	#20294994	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13182029	C₂²	1	2	yes	—
10	#16812707	—	2	8	no	—
11	C₂×#13182012	C₂²	1	4	yes	—
12	C₂²×#11724784	C₂²	1	2	yes	—
13	C₂×C₈×D₁₀₉	C₂²	1	4	yes	—
14	#16812699	C₄	1	1	yes	characteristic
15	#16812696	C₄	1	1	yes	characteristic
16	#16812687	C₄	1	2	yes	—
17	#16812673	C₄	1	2	yes	—
18	#16812673	C₄	1	2	yes	—
19	C₄×D₄₃₆	C₂²	1	2	yes	—
20	C₄×D₄₃₆	C₂²	1	2	yes	—
21	C₄⋊#11724784	C₂²	1	1	yes	characteristic
22	C₄⋊#11724784	C₂²	1	2	yes	—
23	C₄⋊#11724784	C₂²	1	1	yes	characteristic
24	#16812611	C₂²	1	4	yes	—
25	#16812611	C₂²	1	4	yes	—
26	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
29	#97×C₁₀₉	C₂²	1	2	yes	—
30	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
31	#13182029	—	2	4	no	—
32	#13182029	—	2	4	no	—
33	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
34	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
35	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
36	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	4	yes	—
38	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
39	#13182012	—	2	8	no	—
40	C₂×#11724784	—	2	4	no	—
41	C₂×#11724784	C₂³	1	2	yes	—
42	C₂×#11724784	C₂³	1	2	yes	—
43	C₈×D₁₀₉	—	2	8	no	—
44	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
46	#13182006	C₂×C₄	1	4	yes	—
47	#13182006	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
51	M₁₆×C₁₀₉	—	2	2	no	—
52	M₁₆×C₁₀₉	D₄	1	4	yes	complemented
53	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
54	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
55	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₁₀₉	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₁₀₉	—	2	2	no	—
62	C₂×C₄×C₁₀₉	—	4	1	no	—
63	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
67	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₀₉	#41	1	2	yes	—
70	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₁₀₉	—	2	8	no	—
72	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
73	D₄₃₆	—	2	4	no	—
74	D₄₃₆	—	2	4	no	—
75	D₄₃₆	—	4	2	no	—
76	C₄×D₁₀₉	—	2	4	no	—
77	C₄×D₁₀₉	—	2	4	no	—
78	Dic₂₁₈	—	2	4	no	—
79	C₈×C₁₀₉	—	2	2	no	—
80	C₈×C₁₀₉	C₂×D₄	1	4	yes	—
81	#11724784	—	2	2	no	—
82	#11724784	—	2	2	no	—
83	C₂²×C₁₀₉	—	2	1	no	—
84	C₂²×C₁₀₉	—	2	1	no	—
85	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
86	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₁₀₉	—	2	4	no	—
88	C₂×D₁₀₉	—	2	4	no	—
89	C₂×D₁₀₉	—	2	4	no	—
90	C₄×C₁₀₉	—	2	1	no	—
91	C₄×C₁₀₉	—	2	1	no	—
92	C₄×C₁₀₉	—	2	1	no	—
93	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
94	C₄×C₁₀₉	#131	1	2	yes	complemented
95	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
97	Dic₁₀₉	—	2	4	no	—
98	C₂×C₁₀₉	—	2	2	no	—
99	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
101	C₂×C₁₀₉	#568	1	1	yes	characteristic
102	D₁₀₉	—	4	4	no	—
103	#2935	—	109	1	no	maximal, Sylow
104	C₁₀₉	#2935	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	—	109	2	no	—
106	#532	—	109	1	no	—
107	#436	—	109	2	no	—
108	#435	—	109	4	no	—
109	C₈×D₄	—	109	2	no	—
110	C₂×C₄⋊C₈	—	109	1	no	—
111	#408	—	109	2	no	—
112	C₄×M₁₆	D₁₀₉	1	1	yes	characteristic, complemented
113	C₂×#41	—	109	2	no	—
114	#131	—	218	8	no	—
115	C₂×M₁₆	—	109	4	no	—
116	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
117	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
118	C₂²×C₈	—	109	2	no	—
119	C₂²×C₈	—	109	4	no	—
120	C₄⋊Q₈	—	109	1	no	—
121	#127	—	109	1	no	—
122	#124	—	109	2	no	—
123	#121	—	109	2	no	—
124	#121	—	109	2	no	—
125	C₄×D₄	—	109	2	no	—
126	C₄×D₄	—	109	2	no	—
127	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
128	C₄⋊C₈	—	109	1	no	—
129	C₄⋊C₈	—	109	2	no	—
130	C₄⋊C₈	—	109	1	no	—
131	C₂²⋊C₈	—	109	4	no	—
132	C₂²⋊C₈	—	109	4	no	—
133	#97	C₂×D₁₀₉	1	2	yes	—
134	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
135	#41	—	218	4	no	—
136	#41	—	218	4	no	—
137	C₂×Q₈	—	109	2	no	—
138	C₂×D₄	—	109	4	no	—
139	C₂×D₄	—	109	2	no	—
140	C₂×D₄	—	109	2	no	—
141	C₂×D₄	—	109	2	no	—
142	C₂²×C₄	—	109	4	no	—
143	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
146	M₁₆	—	2	2	no	—
147	M₁₆	—	218	8	no	—
148	M₁₆	D₄₃₆	1	4	yes	complemented
149	C₂×C₈	—	109	2	no	—
150	C₂×C₈	—	109	2	no	—
151	C₂×C₈	—	218	4	no	—
152	C₂×C₈	—	218	8	no	—
153	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
154	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
155	C₄⋊C₄	—	109	2	no	—
156	C₄⋊C₄	—	109	2	no	—
157	C₂²⋊C₄	—	109	4	no	—
158	C₂²⋊C₄	—	109	4	no	—
159	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
160	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
161	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
162	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
163	C₂³	—	109	4	no	—
164	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
165	Q₈	—	218	4	no	—
166	D₄	—	436	2	no	—
167	D₄	—	218	8	no	—
168	D₄	—	218	4	no	—
169	D₄	—	218	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	109	4	no	—
173	C₂×C₄	—	218	4	no	—
174	C₂×C₄	—	218	4	no	—
175	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
181	C₂×C₄	#13182029	1	2	yes	—
182	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	218	2	no	—
185	C₈	—	218	2	no	—
186	C₈	C₂×D₄₃₆	1	4	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	218	4	no	—
190	C₂²	—	218	4	no	—
191	C₂²	—	218	4	no	—
192	C₂²	C₄×D₄₃₆	1	2	yes	—
193	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	218	4	no	—
198	C₄	C₄×D₄₃₆	1	2	yes	—
199	C₄	#16812707	1	2	yes	complemented
200	C₄	C₂²×D₄₃₆	1	1	yes	characteristic
201	C₄	C₂×#13182029	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	436	4	no	—
204	C₂	C₂×C₄×D₄₃₆	1	1	yes	characteristic
205	C₂	C₂×#16812707	1	1	yes	characteristic
206	C₂	#20295792	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₄×M₁₆×C₁₀₉	C₂
prime radical	#20295576	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#2935
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16812707	C₂	semidirect
3. #20295105	C₂	semidirect
4. #20295099	C₂	semidirect
5. C₈×D₄₃₆	C₂	semidirect
6. C₂×C₄⋊#11724784	C₂	semidirect
7. #20294994	C₂	semidirect
8. C₄×M₁₆×C₁₀₉	C₂	semidirect
52. M₁₆×C₁₀₉	D₄	semidirect
94. C₄×C₁₀₉	#131	semidirect
104. C₁₀₉	#2935	semidirect
112. C₄×M₁₆	D₁₀₉	semidirect
148. M₁₆	D₄₃₆	semidirect
199. C₄	#16812707	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812707	2
#20295576	2
#20295105	2
#20295099	2
C₈×D₄₃₆	2
C₂×C₄⋊#11724784	2
#20294994	2
C₄×M₁₆×C₁₀₉	2
#2935	109