Subgroups — C₄×#124×C₁₀₉ #28359557

Order 13952.

652 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#124×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#391×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#388×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	#382×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊C₄×C₁₀₉	C₂	1	4	yes	maximal, complemented
9	C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
13	#124×C₁₀₉	C₄	1	8	yes	direct product factor
14	C₄×Q₈×C₁₀₉	—	2	4	no	—
15	C₄×D₄×C₁₀₉	—	2	4	no	—
16	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
22	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
28	#95×C₁₀₉	C₂²	1	4	yes	—
29	#95×C₁₀₉	C₂²	1	4	yes	—
30	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
31	C₂×Q₈×C₁₀₉	—	2	2	no	—
32	C₂×Q₈×C₁₀₉	—	2	2	no	—
33	C₂×Q₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
34	C₂×D₄×C₁₀₉	—	2	2	no	—
35	C₂×D₄×C₁₀₉	—	2	2	no	—
36	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₁₀₉	—	2	4	no	—
38	C₂²×C₄×C₁₀₉	—	2	8	no	—
39	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₀₉	—	2	2	no	—
48	C₄⋊C₄×C₁₀₉	—	2	4	no	—
49	C₄⋊C₄×C₁₀₉	—	2	2	no	—
50	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
51	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
52	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	16	yes	complemented
53	C₄²×C₁₀₉	—	2	4	no	—
54	C₄²×C₁₀₉	—	2	4	no	—
55	C₄²×C₁₀₉	—	2	2	no	—
56	C₄²×C₁₀₉	—	2	4	no	—
57	C₄²×C₁₀₉	C₂×C₄	1	4	yes	complemented
58	C₄²×C₁₀₉	D₄	1	4	yes	—
59	C₂³×C₁₀₉	—	2	2	no	—
60	C₂³×C₁₀₉	—	2	4	no	—
61	C₂³×C₁₀₉	—	2	4	no	—
62	C₂³×C₁₀₉	—	2	2	no	—
63	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
64	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
65	Q₈×C₁₀₉	—	2	8	no	—
66	D₄×C₁₀₉	—	2	8	no	—
67	C₂×C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄×C₁₀₉	—	2	2	no	—
70	C₂×C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	—	2	4	no	—
77	C₂×C₄×C₁₀₉	—	2	4	no	—
78	C₂×C₄×C₁₀₉	—	4	8	no	—
79	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
80	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₀₉	#41	1	4	yes	—
86	C₂×C₄×C₁₀₉	#41	1	2	yes	—
87	C₂×C₄×C₁₀₉	#41	1	4	yes	—
88	C₂²×C₁₀₉	—	2	4	no	—
89	C₂²×C₁₀₉	—	2	8	no	—
90	C₂²×C₁₀₉	—	4	2	no	—
91	C₂²×C₁₀₉	—	4	4	no	—
92	C₂²×C₁₀₉	—	4	2	no	—
93	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
94	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
95	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
96	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
97	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
98	C₄×C₁₀₉	—	2	4	no	—
99	C₄×C₁₀₉	—	2	4	no	—
100	C₄×C₁₀₉	—	2	4	no	—
101	C₄×C₁₀₉	—	2	2	no	—
102	C₄×C₁₀₉	—	4	2	no	—
103	C₄×C₁₀₉	—	4	2	no	—
104	C₄×C₁₀₉	C₄×D₄	1	4	yes	—
105	C₄×C₁₀₉	#124	1	4	yes	direct product factor
106	C₂×C₁₀₉	—	4	4	no	—
107	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
108	C₂×C₁₀₉	C₄×#41	1	2	yes	—
109	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
110	C₂×C₁₀₉	#529	1	2	yes	—
111	C₂×C₁₀₉	#532	1	1	yes	characteristic
112	C₄×#124	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₀₉	C₄×#124	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×#124	C₂×C₁₀₉	1	1	yes	characteristic
115	C₂×C₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
116	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic
117	#391	C₂×C₁₀₉	1	1	yes	characteristic, complemented
118	#390	C₂×C₁₀₉	1	1	yes	characteristic
119	#388	C₂×C₁₀₉	1	4	yes	complemented
120	#382	C₂×C₁₀₉	1	1	yes	characteristic, complemented
121	C₄×C₂²⋊C₄	C₂×C₁₀₉	1	4	yes	complemented
122	C₄³	C₂×C₁₀₉	1	1	yes	characteristic, complemented
123	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
125	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
126	#124	C₄×C₁₀₉	1	8	yes	direct product factor
127	C₄×Q₈	—	2	4	no	—
128	C₄×D₄	—	2	4	no	—
129	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
135	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
136	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
137	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
138	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
139	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
140	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
141	#95	C₂²×C₁₀₉	1	4	yes	—
142	#95	C₂²×C₁₀₉	1	4	yes	—
143	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	C₂×C₄×C₁₀₉	1	4	yes	complemented
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	C₂×C₄×C₁₀₉	1	4	yes	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
153	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
154	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
155	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
156	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
158	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	2	8	no	—
165	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	16	yes	complemented
166	C₄²	—	2	4	no	—
167	C₄²	—	2	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	4	no	—
170	C₄²	C₂×C₄×C₁₀₉	1	4	yes	complemented
171	C₄²	D₄×C₁₀₉	1	4	yes	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	4	no	—
174	C₂³	—	2	4	no	—
175	C₂³	—	2	2	no	—
176	C₂³	C₂²×C₄×C₁₀₉	1	4	yes	—
177	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
178	Q₈	—	2	8	no	—
179	D₄	—	2	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
193	C₂×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
194	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
195	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
196	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
197	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
198	C₂×C₄	#41×C₁₀₉	1	4	yes	—
199	C₂×C₄	#41×C₁₀₉	1	2	yes	—
200	C₂×C₄	#41×C₁₀₉	1	4	yes	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	8	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	4	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
207	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
208	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
209	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
210	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	C₄×D₄×C₁₀₉	1	4	yes	—
218	C₄	#124×C₁₀₉	1	4	yes	direct product factor
219	C₂	—	4	4	no	—
220	C₂	C₂×C₄×D₄×C₁₀₉	1	1	yes	characteristic
221	C₂	C₄×#41×C₁₀₉	1	2	yes	—
222	C₂	C₂×#124×C₁₀₉	1	1	yes	characteristic
223	C₂	#529×C₁₀₉	1	2	yes	—
224	C₂	#532×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂³
commutator	C₂²	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂²×D₄×C₁₀₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₁₀₉	C₂	semidirect
4. #391×C₁₀₉	C₂	semidirect
6. #388×C₁₀₉	C₂	semidirect
7. #382×C₁₀₉	C₂	semidirect
8. C₄×C₂²⋊C₄×C₁₀₉	C₂	semidirect
9. C₄³×C₁₀₉	C₂	semidirect
13. #124×C₁₀₉	C₄	direct (N × H)
33. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
52. C₂²⋊C₄×C₁₀₉	C₂×C₄	semidirect
57. C₄²×C₁₀₉	C₂×C₄	semidirect
105. C₄×C₁₀₉	#124	direct (N × H)
112. C₄×#124	C₁₀₉	direct (N × H)
113. C₁₀₉	C₄×#124	direct (N × H)
115. C₂×C₄×Q₈	C₂×C₁₀₉	semidirect
117. #391	C₂×C₁₀₉	semidirect
119. #388	C₂×C₁₀₉	semidirect
120. #382	C₂×C₁₀₉	semidirect
121. C₄×C₂²⋊C₄	C₂×C₁₀₉	semidirect
122. C₄³	C₂×C₁₀₉	semidirect
126. #124	C₄×C₁₀₉	direct (N × H)
146. C₂×Q₈	C₂×C₄×C₁₀₉	semidirect
165. C₂²⋊C₄	C₂×C₄×C₁₀₉	semidirect
170. C₄²	C₂×C₄×C₁₀₉	semidirect
218. C₄	#124×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#124×C₁₀₉	2
C₂×C₄×Q₈×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
#391×C₁₀₉	2
#390×C₁₀₉	2
#388×C₁₀₉	2
#382×C₁₀₉	2
C₄×C₂²⋊C₄×C₁₀₉	2
C₄³×C₁₀₉	2
C₄×#124	109

Subgroups — C4×#124×C109 #28359557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#124×C₁₀₉ #28359557