Subgroups — C₄×#364×C₁₀₉ #42130649

Order 27904.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364×C₁₀₉	C₂	1	2	yes	maximal
2	C₄×M₃₂×C₁₀₉	C₂	1	2	yes	maximal
3	C₄²×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1376	C₂	1	2	yes	maximal
5	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄³×C₁₀₉	C₄	1	1	yes	characteristic
13	#364×C₁₀₉	C₄	1	8	yes	direct product factor
14	#346×C₁₀₉	—	2	2	no	—
15	C₄×C₁₆×C₁₀₉	—	2	2	no	—
16	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₀₉	—	2	2	no	—
21	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	M₃₂×C₁₀₉	—	2	4	no	—
25	M₃₂×C₁₀₉	C₂×C₄	1	8	yes	—
26	C₂×C₁₆×C₁₀₉	—	2	4	no	—
27	C₄×C₈×C₁₀₉	—	2	2	no	—
28	C₄×C₈×C₁₀₉	—	2	2	no	—
29	C₄×C₈×C₁₀₉	C₈	1	4	yes	—
30	C₄×C₈×C₁₀₉	C₈	1	4	yes	—
31	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	—	2	1	no	—
37	C₂²×C₄×C₁₀₉	—	2	1	no	—
38	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	—	2	1	no	—
41	C₂×C₈×C₁₀₉	—	2	1	no	—
42	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
43	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
50	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	#41	1	2	yes	—
54	C₄²×C₁₀₉	—	2	8	no	—
55	C₄²×C₁₀₉	—	2	1	no	—
56	C₄²×C₁₀₉	—	2	1	no	—
57	C₄²×C₁₀₉	C₄²	1	2	yes	—
58	C₄²×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₀₉	C₂×C₈	1	2	yes	—
61	C₄²×C₁₀₉	C₂×C₈	1	2	yes	—
62	C₁₆×C₁₀₉	—	2	8	no	—
63	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₁₀₉	—	2	4	no	—
65	C₂×C₄×C₁₀₉	—	2	4	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄×C₁₀₉	—	2	4	no	—
69	C₂×C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
70	C₂×C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
71	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
74	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
77	C₈×C₁₀₉	—	2	2	no	—
78	C₈×C₁₀₉	C₄×C₈	1	4	yes	—
79	C₈×C₁₀₉	C₄⋊C₈	1	2	yes	—
80	C₈×C₁₀₉	C₄⋊C₈	1	2	yes	—
81	C₈×C₁₀₉	C₄×D₄	1	2	yes	—
82	C₈×C₁₀₉	C₄×Q₈	1	2	yes	—
83	C₂²×C₁₀₉	—	2	2	no	—
84	C₂²×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
85	C₂²×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₁₀₉	#431	1	1	yes	characteristic
87	C₄×C₁₀₉	—	2	4	no	—
88	C₄×C₁₀₉	—	2	4	no	—
89	C₄×C₁₀₉	—	2	2	no	—
90	C₄×C₁₀₉	#364	1	4	yes	direct product factor
91	C₄×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₄×C₁₀₉	#431	1	1	yes	characteristic
95	C₄×#364	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₁₀₉	—	2	1	no	—
97	C₂×C₁₀₉	—	2	1	no	—
98	C₂×C₁₀₉	C₄×C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₁₀₉	C₂×#364	1	2	yes	—
100	C₂×#364	C₂×C₁₀₉	1	2	yes	—
101	C₄×M₃₂	C₂×C₁₀₉	1	2	yes	—
102	C₄²×C₈	C₂×C₁₀₉	1	1	yes	characteristic
103	#1376	C₂×C₁₀₉	1	2	yes	—
104	C₁₀₉	C₄×#364	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
106	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
107	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
108	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
112	C₄³	C₄×C₁₀₉	1	1	yes	characteristic
113	#364	C₄×C₁₀₉	1	8	yes	direct product factor
114	#346	—	2	2	no	—
115	C₄×C₁₆	—	2	2	no	—
116	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
118	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
122	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
123	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
124	M₃₂	—	2	4	no	—
125	M₃₂	C₂×C₄×C₁₀₉	1	8	yes	—
126	C₂×C₁₆	—	2	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	C₈×C₁₀₉	1	4	yes	—
130	C₄×C₈	C₈×C₁₀₉	1	4	yes	—
131	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
132	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
133	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
134	C₄×C₈	Q₈×C₁₀₉	1	1	yes	characteristic
135	C₄×C₈	Q₈×C₁₀₉	1	1	yes	characteristic
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
139	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
143	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
144	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
145	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
146	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
147	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
148	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
149	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
150	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
151	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
152	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
153	C₂×C₈	#41×C₁₀₉	1	2	yes	—
154	C₄²	—	2	8	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	C₄²×C₁₀₉	1	2	yes	—
158	C₄²	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
159	C₄²	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
160	C₄²	C₂×C₈×C₁₀₉	1	2	yes	—
161	C₄²	C₂×C₈×C₁₀₉	1	2	yes	—
162	C₁₆	—	2	8	no	—
163	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	C₄⋊C₈×C₁₀₉	1	2	yes	—
170	C₂×C₄	C₄⋊C₈×C₁₀₉	1	2	yes	—
171	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
173	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
176	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	C₄×C₈×C₁₀₉	1	4	yes	—
179	C₈	C₄⋊C₈×C₁₀₉	1	2	yes	—
180	C₈	C₄⋊C₈×C₁₀₉	1	2	yes	—
181	C₈	C₄×D₄×C₁₀₉	1	2	yes	—
182	C₈	C₄×Q₈×C₁₀₉	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₄×M₁₆×C₁₀₉	1	1	yes	characteristic
185	C₂²	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
186	C₂²	#431×C₁₀₉	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	#364×C₁₀₉	1	4	yes	direct product factor
191	C₄	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
192	C₄	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
193	C₄	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
194	C₄	#431×C₁₀₉	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	C₄×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
198	C₂	C₂×#364×C₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₁₀₉	D₄
commutator	C₄	C₂×C₄×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₄⋊C₈
prime radical	C₂³×C₁₀₉	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. #364×C₁₀₉	C₄	direct (N × H)
90. C₄×C₁₀₉	#364	direct (N × H)
95. C₄×#364	C₁₀₉	direct (N × H)
104. C₁₀₉	C₄×#364	direct (N × H)
113. #364	C₄×C₁₀₉	direct (N × H)
190. C₄	#364×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×#364×C₁₀₉	2
C₄×M₃₂×C₁₀₉	2
C₄²×C₈×C₁₀₉	2
C₁₀₉×#1376	2
C₄×#364	109

Subgroups — C4×#364×C109 #42130649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#364×C₁₀₉ #42130649