Subgroups — C₄×#519×C₁₀₉ #42151996

Order 27904.

3402 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#519×C₁₀₉	C₂	1	2	yes	maximal
2	C₂²×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2276	C₂	1	8	yes	maximal, complemented
4	C₁₀₉×#2274	C₂	1	6	yes	maximal
5	C₄²×Q₈×C₁₀₉	C₂	1	8	yes	maximal, complemented
6	C₄×#117×C₁₀₉	C₂	1	6	yes	maximal
7	C₂³×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	#519×C₁₀₉	C₄	1	32	yes	direct product factor
9	C₂×C₄×Q₈×C₁₀₉	C₄	1	16	yes	complemented
10	C₂×C₄×Q₈×C₁₀₉	C₂²	1	4	yes	—
11	C₂×C₄×Q₈×C₁₀₉	C₂²	1	4	yes	—
12	C₂×C₄×Q₈×C₁₀₉	C₂²	1	16	yes	—
13	C₂×#117×C₁₀₉	C₂²	1	12	yes	—
14	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
15	C₂²×C₄²×C₁₀₉	C₂²	1	3	yes	—
16	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	12	yes	—
17	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	12	yes	—
18	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	48	yes	—
19	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
20	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	12	yes	—
21	#376×C₁₀₉	C₂²	1	12	yes	—
22	C₄³×C₁₀₉	C₂²	1	12	yes	—
23	C₂²×Q₈×C₁₀₉	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₁₀₉	C₂×C₄	1	16	yes	—
25	C₂²×Q₈×C₁₀₉	C₂³	1	4	yes	—
26	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₁₀₉	C₂³	1	3	yes	—
28	C₂³×C₄×C₁₀₉	C₂³	1	3	yes	—
29	C₄×Q₈×C₁₀₉	C₂×C₄	1	32	yes	—
30	C₄×Q₈×C₁₀₉	C₂×C₄	1	32	yes	—
31	C₄×Q₈×C₁₀₉	C₂×C₄	1	128	yes	complemented
32	#117×C₁₀₉	C₂×C₄	1	96	yes	—
33	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	24	yes	—
34	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	12	yes	—
35	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	24	yes	—
36	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
37	C₂×C₄²×C₁₀₉	C₂×C₄	1	24	yes	—
38	C₂×C₄²×C₁₀₉	C₂³	1	8	yes	—
39	C₂×C₄²×C₁₀₉	C₂³	1	24	yes	—
40	C₂×C₄²×C₁₀₉	C₂³	1	6	yes	—
41	C₂×C₄²×C₁₀₉	C₂³	1	24	yes	—
42	C₂×C₄²×C₁₀₉	C₂³	1	6	yes	—
43	#95×C₁₀₉	C₂³	1	8	yes	—
44	#95×C₁₀₉	C₂³	1	24	yes	—
45	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
46	C₂×Q₈×C₁₀₉	C₄²	1	64	yes	complemented
47	C₂×Q₈×C₁₀₉	C₂²×C₄	1	16	yes	—
48	C₂×Q₈×C₁₀₉	C₂²×C₄	1	32	yes	—
49	C₂²×C₄×C₁₀₉	—	2	4	no	—
50	C₂²×C₄×C₁₀₉	—	4	6	no	—
51	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₁₀₉	C₂²×C₄	1	12	yes	—
53	C₂²×C₄×C₁₀₉	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₁₀₉	C₂²×C₄	1	24	yes	—
55	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂⁴	1	16	yes	—
57	C₂²×C₄×C₁₀₉	C₂⁴	1	6	yes	—
58	C₂²×C₄×C₁₀₉	C₂⁴	1	6	yes	—
59	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
60	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	48	yes	—
61	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	96	yes	—
62	C₂²⋊C₄×C₁₀₉	C₂²×C₄	1	32	yes	—
63	C₄²×C₁₀₉	—	2	16	no	—
64	C₄²×C₁₀₉	—	4	24	no	—
65	C₄²×C₁₀₉	C₂²×C₄	1	24	yes	—
66	C₄²×C₁₀₉	C₂²×C₄	1	24	yes	—
67	C₄²×C₁₀₉	C₂²×C₄	1	96	yes	—
68	C₂³×C₁₀₉	—	2	2	no	—
69	C₂³×C₁₀₉	—	2	2	no	—
70	C₂³×C₁₀₉	C₂³×C₄	1	4	yes	—
71	C₂³×C₁₀₉	C₂³×C₄	1	2	yes	—
72	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
73	Q₈×C₁₀₉	C₂×C₄²	1	64	yes	—
74	C₂×C₄×C₁₀₉	—	2	16	no	—
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	—	2	16	no	—
77	C₂×C₄×C₁₀₉	—	2	12	no	—
78	C₂×C₄×C₁₀₉	—	2	8	no	—
79	C₂×C₄×C₁₀₉	—	4	12	no	—
80	C₂×C₄×C₁₀₉	C₂×C₄²	1	48	yes	—
81	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
82	C₂×C₄×C₁₀₉	C₂³×C₄	1	32	yes	—
83	C₂×C₄×C₁₀₉	C₂³×C₄	1	12	yes	—
84	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
85	C₂×C₄×C₁₀₉	C₂³×C₄	1	12	yes	—
86	C₂×C₄×C₁₀₉	C₂³×C₄	1	24	yes	—
87	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	4	yes	—
88	C₂²×C₁₀₉	—	2	2	no	—
89	C₂²×C₁₀₉	—	2	2	no	—
90	C₂²×C₁₀₉	—	2	8	no	—
91	C₂²×C₁₀₉	C₂²×C₄²	1	4	yes	—
92	C₂²×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
93	C₂²×C₁₀₉	C₂⁴×C₄	1	2	yes	—
94	C₂²×C₁₀₉	C₂×2₋¹⁺⁴	1	2	yes	—
95	C₂²×C₁₀₉	#585	1	2	yes	—
96	C₄×C₁₀₉	—	2	2	no	—
97	C₄×C₁₀₉	—	2	12	no	—
98	C₄×C₁₀₉	—	2	32	no	—
99	C₄×C₁₀₉	C₂²×C₄²	1	24	yes	—
100	C₄×C₁₀₉	#519	1	4	yes	direct product factor
101	C₄×#519	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₁₀₉	—	2	4	no	—
103	C₂×C₁₀₉	C₂³×C₄²	1	1	yes	characteristic
104	C₂×C₁₀₉	C₂×#519	1	1	yes	characteristic
105	C₂×C₁₀₉	#3428	1	1	yes	characteristic
106	C₂×C₁₀₉	C₄×2₋¹⁺⁴	1	4	yes	—
107	C₂×#519	C₂×C₁₀₉	1	2	yes	—
108	C₂²×C₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic
109	#2276	C₂×C₁₀₉	1	8	yes	complemented
110	#2274	C₂×C₁₀₉	1	6	yes	—
111	C₄²×Q₈	C₂×C₁₀₉	1	8	yes	complemented
112	C₄×#117	C₂×C₁₀₉	1	6	yes	—
113	C₁₀₉	C₄×#519	1	1	yes	characteristic, direct product factor, Sylow
114	C₂³×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
115	#519	C₄×C₁₀₉	1	32	yes	direct product factor
116	C₂×C₄×Q₈	C₄×C₁₀₉	1	16	yes	complemented
117	C₂×C₄×Q₈	C₂²×C₁₀₉	1	4	yes	—
118	C₂×C₄×Q₈	C₂²×C₁₀₉	1	4	yes	—
119	C₂×C₄×Q₈	C₂²×C₁₀₉	1	16	yes	—
120	C₂×#117	C₂²×C₁₀₉	1	12	yes	—
121	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	3	yes	—
122	C₂²×C₄²	C₂²×C₁₀₉	1	3	yes	—
123	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	12	yes	—
124	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	12	yes	—
125	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	48	yes	—
126	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
127	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	12	yes	—
128	#376	C₂²×C₁₀₉	1	12	yes	—
129	C₄³	C₂²×C₁₀₉	1	12	yes	—
130	C₂²×Q₈	C₂×C₄×C₁₀₉	1	8	yes	—
131	C₂²×Q₈	C₂×C₄×C₁₀₉	1	16	yes	—
132	C₂²×Q₈	C₂³×C₁₀₉	1	4	yes	—
133	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
134	C₂³×C₄	C₂³×C₁₀₉	1	3	yes	—
135	C₂³×C₄	C₂³×C₁₀₉	1	3	yes	—
136	C₄×Q₈	C₂×C₄×C₁₀₉	1	32	yes	—
137	C₄×Q₈	C₂×C₄×C₁₀₉	1	32	yes	—
138	C₄×Q₈	C₂×C₄×C₁₀₉	1	128	yes	complemented
139	#117	C₂×C₄×C₁₀₉	1	96	yes	—
140	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	24	yes	—
141	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	12	yes	—
142	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	24	yes	—
143	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
144	C₂×C₄²	C₂×C₄×C₁₀₉	1	24	yes	—
145	C₂×C₄²	C₂³×C₁₀₉	1	8	yes	—
146	C₂×C₄²	C₂³×C₁₀₉	1	24	yes	—
147	C₂×C₄²	C₂³×C₁₀₉	1	24	yes	—
148	C₂×C₄²	C₂³×C₁₀₉	1	6	yes	—
149	C₂×C₄²	C₂³×C₁₀₉	1	6	yes	—
150	#95	C₂³×C₁₀₉	1	8	yes	—
151	#95	C₂³×C₁₀₉	1	24	yes	—
152	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
153	C₂×Q₈	C₄²×C₁₀₉	1	64	yes	complemented
154	C₂×Q₈	C₂²×C₄×C₁₀₉	1	16	yes	—
155	C₂×Q₈	C₂²×C₄×C₁₀₉	1	32	yes	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	4	6	no	—
158	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
159	C₂²×C₄	C₂²×C₄×C₁₀₉	1	12	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₀₉	1	12	yes	—
161	C₂²×C₄	C₂²×C₄×C₁₀₉	1	24	yes	—
162	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
163	C₂²×C₄	C₂⁴×C₁₀₉	1	16	yes	—
164	C₂²×C₄	C₂⁴×C₁₀₉	1	6	yes	—
165	C₂²×C₄	C₂⁴×C₁₀₉	1	6	yes	—
166	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
167	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	48	yes	—
168	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	96	yes	—
169	C₂²⋊C₄	C₂²×C₄×C₁₀₉	1	32	yes	—
170	C₄²	—	2	16	no	—
171	C₄²	—	4	24	no	—
172	C₄²	C₂²×C₄×C₁₀₉	1	24	yes	—
173	C₄²	C₂²×C₄×C₁₀₉	1	24	yes	—
174	C₄²	C₂²×C₄×C₁₀₉	1	96	yes	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	C₂³×C₄×C₁₀₉	1	4	yes	—
178	C₂³	C₂³×C₄×C₁₀₉	1	2	yes	—
179	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
180	Q₈	C₂×C₄²×C₁₀₉	1	64	yes	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	2	12	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	4	12	no	—
187	C₂×C₄	C₂×C₄²×C₁₀₉	1	48	yes	—
188	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
189	C₂×C₄	C₂³×C₄×C₁₀₉	1	32	yes	—
190	C₂×C₄	C₂³×C₄×C₁₀₉	1	12	yes	—
191	C₂×C₄	C₂³×C₄×C₁₀₉	1	12	yes	—
192	C₂×C₄	C₂³×C₄×C₁₀₉	1	24	yes	—
193	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
194	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	4	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	C₂²×C₄²×C₁₀₉	1	4	yes	—
199	C₂²	C₂⁴×C₄×C₁₀₉	1	1	yes	characteristic
200	C₂²	C₂⁴×C₄×C₁₀₉	1	2	yes	—
201	C₂²	C₂×2₋¹⁺⁴×C₁₀₉	1	2	yes	—
202	C₂²	#585×C₁₀₉	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	12	no	—
205	C₄	—	2	32	no	—
206	C₄	C₂²×C₄²×C₁₀₉	1	24	yes	—
207	C₄	#519×C₁₀₉	1	4	yes	direct product factor
208	C₂	—	2	4	no	—
209	C₂	C₂³×C₄²×C₁₀₉	1	1	yes	characteristic
210	C₂	C₂×#519×C₁₀₉	1	1	yes	characteristic
211	C₂	C₁₀₉×#3428	1	1	yes	characteristic
212	C₂	C₄×2₋¹⁺⁴×C₁₀₉	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂⁴
commutator	C₂	C₂³×C₄²×C₁₀₉
Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
socle	C₂³×C₁₀₉	C₂⁵
prime radical	C₂⁴×C₁₀₉	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#2276	C₂	semidirect
5. C₄²×Q₈×C₁₀₉	C₂	semidirect
8. #519×C₁₀₉	C₄	direct (N × H)
9. C₂×C₄×Q₈×C₁₀₉	C₄	semidirect
31. C₄×Q₈×C₁₀₉	C₂×C₄	semidirect
46. C₂×Q₈×C₁₀₉	C₄²	semidirect
100. C₄×C₁₀₉	#519	direct (N × H)
101. C₄×#519	C₁₀₉	direct (N × H)
109. #2276	C₂×C₁₀₉	semidirect
111. C₄²×Q₈	C₂×C₁₀₉	semidirect
113. C₁₀₉	C₄×#519	direct (N × H)
115. #519	C₄×C₁₀₉	direct (N × H)
116. C₂×C₄×Q₈	C₄×C₁₀₉	semidirect
138. C₄×Q₈	C₂×C₄×C₁₀₉	semidirect
153. C₂×Q₈	C₄²×C₁₀₉	semidirect
207. C₄	#519×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×#519×C₁₀₉	2
C₂²×C₄×Q₈×C₁₀₉	2
C₁₀₉×#2276	2
C₁₀₉×#2274	2
C₄²×Q₈×C₁₀₉	2
C₄×#117×C₁₀₉	2
C₄×#519	109

Subgroups — C4×#519×C109 #42151996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#519×C₁₀₉ #42151996