Subgroups — #557×D₁₀₉ #28376600

Order 13952.

1048 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295755	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295754	C₂	1	3	yes	maximal, complemented
3	#20295753	C₂	1	6	yes	maximal, complemented
4	#20295752	C₂	1	2	yes	maximal, complemented
5	#20295751	C₂	1	3	yes	maximal, complemented
6	C₄⋊Q₈×D₁₀₉	C₂	1	3	yes	maximal
7	#125×D₁₀₉	C₂	1	6	yes	maximal
8	C₄×Q₈×D₁₀₉	C₂	1	2	yes	maximal
9	C₄×Q₈×D₁₀₉	C₂	1	3	yes	maximal
10	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
11	#557×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₀₉	C₂²	1	3	yes	—
13	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	#16812699	C₂²	1	3	yes	—
15	#16812698	C₂²	1	6	yes	—
16	#16812698	C₂²	1	6	yes	—
17	#16812697	C₂²	1	6	yes	—
18	#16812691	C₂²	1	6	yes	—
19	#16812690	C₂²	1	6	yes	—
20	#16812690	C₂²	1	6	yes	—
21	#16812689	C₂²	1	12	yes	—
22	#16812689	C₂²	1	12	yes	—
23	Q₈×Dic₁₀₉	C₂²	1	3	yes	—
24	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
26	C₄×Dic₂₁₈	C₂²	1	3	yes	—
27	C₄×Dic₂₁₈	C₂²	1	3	yes	—
28	C₄×Dic₂₁₈	C₂²	1	6	yes	—
29	C₄×Dic₂₁₈	C₂²	1	6	yes	—
30	#16812664	C₂²	1	6	yes	—
31	#16812664	C₂²	1	6	yes	—
32	#16812664	C₂²	1	3	yes	—
33	#16812664	C₂²	1	3	yes	—
34	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₀₉	C₂²	1	6	yes	—
36	C₄⋊C₄×D₁₀₉	C₂²	1	3	yes	—
37	C₄⋊C₄×D₁₀₉	C₂²	1	6	yes	—
38	C₄⋊C₄×D₁₀₉	C₂²	1	6	yes	—
39	C₄²×D₁₀₉	C₂²	1	6	yes	—
40	C₄²×D₁₀₉	C₂²	1	3	yes	—
41	C₄⋊Q₈×C₁₀₉	C₂²	1	3	yes	—
42	#125×C₁₀₉	C₂²	1	6	yes	—
43	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
44	C₄×Q₈×C₁₀₉	C₂²	1	3	yes	—
45	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
46	C₂×Dic₂₁₈	C₂³	1	3	yes	—
47	C₂×Dic₂₁₈	C₂³	1	6	yes	—
48	C₂×Dic₂₁₈	C₂³	1	3	yes	—
49	Q₈×D₁₀₉	—	2	12	no	—
50	Q₈×D₁₀₉	Q₈	1	8	yes	complemented
51	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
52	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
53	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
54	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
55	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
56	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
58	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
59	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
60	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
61	#13182008	C₂³	1	2	yes	—
62	#13182008	C₂³	1	6	yes	—
63	#13182008	C₂³	1	6	yes	—
64	#13182008	C₂³	1	6	yes	—
65	#13182008	C₂³	1	12	yes	—
66	#13182008	C₂³	1	12	yes	—
67	C₄×Dic₁₀₉	C₂³	1	3	yes	—
68	C₄×Dic₁₀₉	C₂³	1	6	yes	—
69	C₄×Dic₁₀₉	C₂³	1	6	yes	—
70	C₄×Dic₁₀₉	C₂³	1	6	yes	—
71	C₄×Dic₁₀₉	C₂³	1	3	yes	—
72	C₄×Dic₁₀₉	C₂³	1	3	yes	—
73	C₂×Q₈×C₁₀₉	C₂³	1	3	yes	—
74	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
77	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
78	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
79	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
80	C₄²×C₁₀₉	C₂³	1	6	yes	—
81	C₄²×C₁₀₉	C₂³	1	3	yes	—
82	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
83	Q₈×C₁₀₉	—	2	6	no	—
84	Q₈×C₁₀₉	C₂×Q₈	1	4	yes	complemented
85	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
86	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
87	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
88	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
89	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
91	C₂×Dic₁₀₉	C₂⁴	1	3	yes	—
92	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
93	C₂×Dic₁₀₉	C₂⁴	1	3	yes	—
94	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
95	C₄×D₁₀₉	—	2	12	no	—
96	C₄×D₁₀₉	—	2	4	no	—
97	C₄×D₁₀₉	—	2	6	no	—
98	C₄×D₁₀₉	C₂×Q₈	1	12	yes	—
99	C₄×D₁₀₉	#41	1	4	yes	—
100	Dic₂₁₈	—	2	6	no	—
101	Dic₂₁₈	—	2	12	no	—
102	Dic₂₁₈	C₂×Q₈	1	12	yes	complemented
103	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
104	C₂×D₁₀₉	C₂²×Q₈	1	2	yes	—
105	C₂×D₁₀₉	C₂×#41	1	2	yes	—
106	C₂×D₁₀₉	2₋¹⁺⁴	1	2	yes	—
107	C₄×C₁₀₉	—	2	6	no	—
108	C₄×C₁₀₉	—	2	2	no	—
109	C₄×C₁₀₉	—	2	3	no	—
110	C₄×C₁₀₉	C₂²×Q₈	1	6	yes	—
111	C₄×C₁₀₉	C₂×#41	1	2	yes	—
112	Dic₁₀₉	—	2	6	no	—
113	Dic₁₀₉	—	2	2	no	—
114	Dic₁₀₉	—	2	3	no	—
115	Dic₁₀₉	C₂²×Q₈	1	6	yes	—
116	Dic₁₀₉	C₂×#41	1	2	yes	—
117	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
118	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
119	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
120	D₁₀₉	#557	1	4	yes	direct product factor
121	C₂×#557	—	109	1	no	maximal, Sylow
122	C₁₀₉	C₂×#557	1	1	yes	characteristic, complemented, Sylow
123	#557	—	109	3	no	—
124	#557	—	109	6	no	—
125	#557	—	109	2	no	—
126	#557	—	109	3	no	—
127	#557	—	109	1	no	—
128	#557	D₁₀₉	1	1	yes	characteristic, direct product factor
129	C₂×C₄⋊Q₈	—	109	3	no	—
130	C₂×#125	—	109	6	no	—
131	C₂×C₄×Q₈	—	109	2	no	—
132	C₂×C₄×Q₈	—	109	3	no	—
133	C₂×C₄×Q₈	—	109	1	no	—
134	C₂²×Q₈	—	109	1	no	—
135	C₂²×Q₈	—	109	3	no	—
136	C₄⋊Q₈	—	109	6	no	—
137	C₄⋊Q₈	—	109	6	no	—
138	C₄⋊Q₈	—	109	6	no	—
139	C₄⋊Q₈	—	109	3	no	—
140	C₄⋊Q₈	C₂×D₁₀₉	1	3	yes	—
141	#125	—	109	12	no	—
142	#125	—	109	12	no	—
143	#125	—	109	6	no	—
144	#125	—	109	6	no	—
145	#125	—	109	6	no	—
146	#125	C₂×D₁₀₉	1	6	yes	—
147	C₄×Q₈	—	109	2	no	—
148	C₄×Q₈	—	109	3	no	—
149	C₄×Q₈	—	109	1	no	—
150	C₄×Q₈	—	109	6	no	—
151	C₄×Q₈	—	109	6	no	—
152	C₄×Q₈	—	109	3	no	—
153	C₄×Q₈	—	109	3	no	—
154	C₄×Q₈	—	109	6	no	—
155	C₄×Q₈	—	109	3	no	—
156	C₄×Q₈	—	109	6	no	—
157	C₄×Q₈	—	109	3	no	—
158	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
159	C₄×Q₈	C₂×D₁₀₉	1	3	yes	—
160	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	109	1	no	—
162	C₂×C₄⋊C₄	—	109	3	no	—
163	C₂×C₄⋊C₄	—	109	6	no	—
164	C₂×C₄⋊C₄	—	109	6	no	—
165	C₂×C₄⋊C₄	—	109	6	no	—
166	C₂×C₄²	—	109	6	no	—
167	C₂×C₄²	—	109	3	no	—
168	C₂×Q₈	—	109	8	no	—
169	C₂×Q₈	—	109	3	no	—
170	C₂×Q₈	—	109	6	no	—
171	C₂×Q₈	—	109	3	no	—
172	C₂×Q₈	—	218	12	no	—
173	C₂×Q₈	C₂²×D₁₀₉	1	3	yes	—
174	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
175	C₂²×C₄	—	109	3	no	—
176	C₂²×C₄	—	109	1	no	—
177	C₂²×C₄	—	109	2	no	—
178	C₂²×C₄	—	109	3	no	—
179	C₂²×C₄	—	109	6	no	—
180	C₄⋊C₄	—	109	2	no	—
181	C₄⋊C₄	—	109	6	no	—
182	C₄⋊C₄	—	109	6	no	—
183	C₄⋊C₄	—	109	6	no	—
184	C₄⋊C₄	—	109	12	no	—
185	C₄⋊C₄	—	109	12	no	—
186	C₄⋊C₄	—	109	1	no	—
187	C₄⋊C₄	—	109	6	no	—
188	C₄⋊C₄	—	109	3	no	—
189	C₄⋊C₄	—	109	6	no	—
190	C₄⋊C₄	—	109	6	no	—
191	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
192	C₄⋊C₄	C₂²×D₁₀₉	1	6	yes	—
193	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
194	C₄⋊C₄	C₂²×D₁₀₉	1	6	yes	—
195	C₄⋊C₄	C₂²×D₁₀₉	1	6	yes	—
196	C₄²	—	109	6	no	—
197	C₄²	—	109	6	no	—
198	C₄²	—	109	6	no	—
199	C₄²	—	109	3	no	—
200	C₄²	—	109	3	no	—
201	C₄²	—	109	3	no	—
202	C₄²	C₂²×D₁₀₉	1	6	yes	—
203	C₄²	C₂²×D₁₀₉	1	3	yes	—
204	C₂³	—	109	1	no	—
205	Q₈	—	2	6	no	—
206	Q₈	—	109	12	no	—
207	Q₈	—	218	12	no	—
208	Q₈	—	218	6	no	—
209	Q₈	Q₈×D₁₀₉	1	4	yes	complemented
210	C₂×C₄	—	109	3	no	—
211	C₂×C₄	—	109	1	no	—
212	C₂×C₄	—	109	12	no	—
213	C₂×C₄	—	109	4	no	—
214	C₂×C₄	—	109	2	no	—
215	C₂×C₄	—	109	3	no	—
216	C₂×C₄	—	109	6	no	—
217	C₂×C₄	—	218	12	no	—
218	C₂×C₄	—	218	4	no	—
219	C₂×C₄	—	218	6	no	—
220	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
221	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
222	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
223	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
224	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
225	C₂²	—	109	2	no	—
226	C₂²	—	109	2	no	—
227	C₂²	—	109	2	no	—
228	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
229	C₄	—	2	6	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	3	no	—
232	C₄	—	109	6	no	—
233	C₄	—	109	2	no	—
234	C₄	—	218	6	no	—
235	C₄	—	218	2	no	—
236	C₄	—	218	3	no	—
237	C₄	C₂×Q₈×D₁₀₉	1	6	yes	—
238	C₄	#41×D₁₀₉	1	2	yes	—
239	C₂	—	109	4	no	—
240	C₂	C₂²×Q₈×D₁₀₉	1	1	yes	characteristic
241	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
242	C₂	2₋¹⁺⁴×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#557×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₂⁴
hypercenter	#557	D₁₀₉
hypocenter	C₁₀₉	C₂×#557
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. #20295755	C₂	semidirect
2. #20295754	C₂	semidirect
3. #20295753	C₂	semidirect
4. #20295752	C₂	semidirect
5. #20295751	C₂	semidirect
11. #557×C₁₀₉	C₂	semidirect
50. Q₈×D₁₀₉	Q₈	semidirect
84. Q₈×C₁₀₉	C₂×Q₈	semidirect
102. Dic₂₁₈	C₂×Q₈	semidirect
120. D₁₀₉	#557	direct (N × H)
122. C₁₀₉	C₂×#557	semidirect
128. #557	D₁₀₉	direct (N × H)
209. Q₈	Q₈×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295755	2
#20295754	2
#20295753	2
#20295752	2
#20295751	2
C₄⋊Q₈×D₁₀₉	2
#125×D₁₀₉	2
C₄×Q₈×D₁₀₉	2
#557×C₁₀₉	2
C₂×#557	109

Subgroups — #557×D109 #28376600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #557×D₁₀₉ #28376600