Subgroups — C₁₀₉×#2354 #28359608

Order 13952.

632 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#390×C₁₀₉	C₂	1	2	yes	maximal
4	#388×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	#384×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#380×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
11	C₄×D₄×C₁₀₉	C₄	1	8	yes	complemented
12	#117×C₁₀₉	—	2	8	no	—
13	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₀₉	C₂²	1	4	yes	—
23	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
25	#95×C₁₀₉	C₂²	1	4	yes	—
26	#95×C₁₀₉	C₂²	1	2	yes	—
27	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
28	C₂×D₄×C₁₀₉	—	2	2	no	—
29	C₂×D₄×C₁₀₉	—	2	2	no	—
30	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₁₀₉	—	2	4	no	—
32	C₂²×C₄×C₁₀₉	—	2	4	no	—
33	C₂²×C₄×C₁₀₉	C₂×C₄	1	8	yes	—
34	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₉	—	2	4	no	—
42	C₄⋊C₄×C₁₀₉	—	2	4	no	—
43	C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₄⋊C₄×C₁₀₉	—	4	2	no	—
45	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
46	C₄⋊C₄×C₁₀₉	D₄	1	4	yes	complemented
47	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
48	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
49	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
50	C₄²×C₁₀₉	—	2	8	no	—
51	C₄²×C₁₀₉	—	2	4	no	—
52	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
53	C₂³×C₁₀₉	—	2	2	no	—
54	C₂³×C₁₀₉	—	2	2	no	—
55	C₂³×C₁₀₉	—	2	2	no	—
56	C₂³×C₁₀₉	—	2	2	no	—
57	C₂³×C₁₀₉	—	2	2	no	—
58	C₂³×C₁₀₉	—	2	2	no	—
59	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
60	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
61	D₄×C₁₀₉	—	2	8	no	—
62	C₂×C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄×C₁₀₉	—	2	2	no	—
64	C₂×C₄×C₁₀₉	—	2	2	no	—
65	C₂×C₄×C₁₀₉	—	2	1	no	—
66	C₂×C₄×C₁₀₉	—	2	1	no	—
67	C₂×C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄×C₁₀₉	—	2	16	no	—
69	C₂×C₄×C₁₀₉	—	2	4	no	—
70	C₂×C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₄×C₁₀₉	—	2	4	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	—	2	4	no	—
75	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₀₉	#41	1	2	yes	—
82	C₂×C₄×C₁₀₉	#41	1	2	yes	—
83	C₂×C₄×C₁₀₉	#41	1	2	yes	—
84	C₂²×C₁₀₉	—	2	4	no	—
85	C₂²×C₁₀₉	—	2	4	no	—
86	C₂²×C₁₀₉	—	2	4	no	—
87	C₂²×C₁₀₉	—	4	2	no	—
88	C₂²×C₁₀₉	—	4	2	no	—
89	C₂²×C₁₀₉	—	4	2	no	—
90	C₂²×C₁₀₉	—	4	2	no	—
91	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
92	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
94	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
95	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
96	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
97	C₂²×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
98	C₄×C₁₀₉	—	2	4	no	—
99	C₄×C₁₀₉	—	2	2	no	—
100	C₄×C₁₀₉	—	2	4	no	—
101	C₄×C₁₀₉	—	4	4	no	—
102	C₄×C₁₀₉	—	4	2	no	—
103	C₄×C₁₀₉	#117	1	4	yes	complemented
104	C₄×C₁₀₉	C₄×D₄	1	4	yes	—
105	C₂×C₁₀₉	—	4	4	no	—
106	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
107	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
108	C₂×C₁₀₉	#520	1	1	yes	characteristic
109	C₂×C₁₀₉	#547	1	1	yes	characteristic
110	C₂×C₁₀₉	#550	1	1	yes	characteristic
111	C₂×C₁₀₉	#555	1	1	yes	characteristic
112	C₂×C₁₀₉	#556	1	1	yes	characteristic
113	#2354	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₁₀₉	#2354	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic
116	C₂×#117	C₂×C₁₀₉	1	2	yes	complemented
117	#390	C₂×C₁₀₉	1	2	yes	—
118	#388	C₂×C₁₀₉	1	4	yes	complemented
119	#384	C₂×C₁₀₉	1	1	yes	characteristic, complemented
120	#380	C₂×C₁₀₉	1	2	yes	complemented
121	C₄×C₄⋊C₄	C₂×C₁₀₉	1	2	yes	complemented
122	C₄×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
123	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
125	C₄×D₄	C₄×C₁₀₉	1	8	yes	complemented
126	#117	—	2	8	no	—
127	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
128	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
130	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
131	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
136	C₂×C₄²	C₂²×C₁₀₉	1	4	yes	—
137	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
138	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
139	#95	C₂²×C₁₀₉	1	4	yes	—
140	#95	C₂²×C₁₀₉	1	2	yes	—
141	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂×C₄×C₁₀₉	1	4	yes	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	C₂×C₄×C₁₀₉	1	8	yes	—
148	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
149	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
150	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
151	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
152	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
153	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
160	C₄⋊C₄	D₄×C₁₀₉	1	4	yes	complemented
161	C₂²⋊C₄	—	2	8	no	—
162	C₂²⋊C₄	—	4	4	no	—
163	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	—
164	C₄²	—	2	8	no	—
165	C₄²	—	2	4	no	—
166	C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	C₂²×C₄×C₁₀₉	1	4	yes	—
174	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
175	D₄	—	2	8	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
190	C₂×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
191	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
192	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
193	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
194	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
195	C₂×C₄	#41×C₁₀₉	1	2	yes	—
196	C₂×C₄	#41×C₁₀₉	1	2	yes	—
197	C₂×C₄	#41×C₁₀₉	1	2	yes	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
206	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
207	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
208	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
209	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
210	C₂²	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
211	C₂²	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	4	no	—
215	C₄	—	4	4	no	—
216	C₄	—	4	2	no	—
217	C₄	#117×C₁₀₉	1	4	yes	complemented
218	C₄	C₄×D₄×C₁₀₉	1	4	yes	—
219	C₂	—	4	4	no	—
220	C₂	C₂×#117×C₁₀₉	1	1	yes	characteristic
221	C₂	C₂×C₄×D₄×C₁₀₉	1	1	yes	characteristic
222	C₂	#520×C₁₀₉	1	1	yes	characteristic
223	C₂	#547×C₁₀₉	1	1	yes	characteristic
224	C₂	#550×C₁₀₉	1	1	yes	characteristic
225	C₂	#555×C₁₀₉	1	1	yes	characteristic
226	C₂	#556×C₁₀₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#117×C₁₀₉	C₂	semidirect
4. #388×C₁₀₉	C₂	semidirect
5. #384×C₁₀₉	C₂	semidirect
6. #380×C₁₀₉	C₂	semidirect
7. C₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
8. C₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
11. C₄×D₄×C₁₀₉	C₄	semidirect
46. C₄⋊C₄×C₁₀₉	D₄	semidirect
103. C₄×C₁₀₉	#117	semidirect
113. #2354	C₁₀₉	direct (N × H)
114. C₁₀₉	#2354	direct (N × H)
116. C₂×#117	C₂×C₁₀₉	semidirect
118. #388	C₂×C₁₀₉	semidirect
119. #384	C₂×C₁₀₉	semidirect
120. #380	C₂×C₁₀₉	semidirect
121. C₄×C₄⋊C₄	C₂×C₁₀₉	semidirect
122. C₄×C₄⋊C₄	C₂×C₁₀₉	semidirect
125. C₄×D₄	C₄×C₁₀₉	semidirect
160. C₄⋊C₄	D₄×C₁₀₉	semidirect
217. C₄	#117×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₁₀₉	2
C₂×#117×C₁₀₉	2
#390×C₁₀₉	2
#388×C₁₀₉	2
#384×C₁₀₉	2
#380×C₁₀₉	2
C₄×C₄⋊C₄×C₁₀₉	2
#2354	109

Subgroups — C109×#2354 #28359608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2354 #28359608