Subgroups — C₁₀₉×#2398 #28359652

Order 13952.

568 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#400×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#397×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#394×C₁₀₉	C₂	1	2	yes	maximal
6	#393×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊C₄×C₁₀₉	C₂	1	4	yes	maximal, complemented
8	#376×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
12	C₄×Q₈×C₁₀₉	—	2	4	no	—
13	C₄×D₄×C₁₀₉	—	2	4	no	—
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	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
20	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
23	#95×C₁₀₉	C₂²	1	2	yes	—
24	#95×C₁₀₉	C₂²	1	2	yes	—
25	#95×C₁₀₉	C₂²	1	4	yes	—
26	#95×C₁₀₉	C₂²	1	4	yes	—
27	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
28	C₂×Q₈×C₁₀₉	—	2	2	no	—
29	C₂×Q₈×C₁₀₉	—	2	4	no	—
30	C₂×D₄×C₁₀₉	—	2	4	no	—
31	C₂×D₄×C₁₀₉	—	2	2	no	—
32	C₂²×C₄×C₁₀₉	—	2	4	no	—
33	C₂²×C₄×C₁₀₉	—	2	8	no	—
34	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
35	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₉	—	2	2	no	—
42	C₄⋊C₄×C₁₀₉	—	2	4	no	—
43	C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
45	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
46	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
47	C₄²×C₁₀₉	—	2	4	no	—
48	C₄²×C₁₀₉	—	2	4	no	—
49	C₄²×C₁₀₉	D₄	1	4	yes	—
50	C₂³×C₁₀₉	—	2	2	no	—
51	C₂³×C₁₀₉	—	2	4	no	—
52	C₂³×C₁₀₉	—	2	4	no	—
53	C₂³×C₁₀₉	—	2	2	no	—
54	C₂³×C₁₀₉	#41	1	4	yes	—
55	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
56	Q₈×C₁₀₉	—	2	8	no	—
57	D₄×C₁₀₉	—	2	8	no	—
58	C₂×C₄×C₁₀₉	—	2	2	no	—
59	C₂×C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₄×C₁₀₉	—	2	2	no	—
62	C₂×C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄×C₁₀₉	—	2	2	no	—
64	C₂×C₄×C₁₀₉	—	2	4	no	—
65	C₂×C₄×C₁₀₉	—	2	4	no	—
66	C₂×C₄×C₁₀₉	—	2	4	no	—
67	C₂×C₄×C₁₀₉	—	2	4	no	—
68	C₂×C₄×C₁₀₉	—	2	4	no	—
69	C₂×C₄×C₁₀₉	—	4	8	no	—
70	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₁₀₉	#41	1	4	yes	—
74	C₂×C₄×C₁₀₉	#41	1	4	yes	—
75	C₂²×C₁₀₉	—	2	4	no	—
76	C₂²×C₁₀₉	—	2	8	no	—
77	C₂²×C₁₀₉	—	4	2	no	—
78	C₂²×C₁₀₉	—	4	2	no	—
79	C₂²×C₁₀₉	—	4	4	no	—
80	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
82	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
83	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
84	C₄×C₁₀₉	—	2	2	no	—
85	C₄×C₁₀₉	—	2	2	no	—
86	C₄×C₁₀₉	—	2	4	no	—
87	C₄×C₁₀₉	—	4	2	no	—
88	C₄×C₁₀₉	—	4	2	no	—
89	C₄×C₁₀₉	—	4	4	no	—
90	C₄×C₁₀₉	#124	1	4	yes	—
91	C₂×C₁₀₉	—	4	4	no	—
92	C₂×C₁₀₉	#525	1	1	yes	characteristic
93	C₂×C₁₀₉	#525	1	1	yes	characteristic
94	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
95	C₂×C₁₀₉	#529	1	4	yes	—
96	#2398	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₁₀₉	#2398	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×C₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
99	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic
100	#400	C₂×C₁₀₉	1	2	yes	complemented
101	#397	C₂×C₁₀₉	1	2	yes	complemented
102	#394	C₂×C₁₀₉	1	2	yes	—
103	#393	C₂×C₁₀₉	1	2	yes	complemented
104	C₄×C₂²⋊C₄	C₂×C₁₀₉	1	4	yes	complemented
105	#376	C₂×C₁₀₉	1	1	yes	characteristic, complemented
106	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
107	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
108	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
109	C₄×Q₈	—	2	4	no	—
110	C₄×D₄	—	2	4	no	—
111	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
114	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
116	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
117	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
118	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
119	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
120	#95	C₂²×C₁₀₉	1	2	yes	—
121	#95	C₂²×C₁₀₉	1	2	yes	—
122	#95	C₂²×C₁₀₉	1	4	yes	—
123	#95	C₂²×C₁₀₉	1	4	yes	—
124	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
125	C₂×Q₈	—	2	2	no	—
126	C₂×Q₈	—	2	4	no	—
127	C₂×D₄	—	2	4	no	—
128	C₂×D₄	—	2	2	no	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	2	8	no	—
131	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
132	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
133	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
134	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
137	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₂²⋊C₄	—	2	8	no	—
142	C₂²⋊C₄	—	2	4	no	—
143	C₂²⋊C₄	—	2	8	no	—
144	C₄²	—	2	4	no	—
145	C₄²	—	2	4	no	—
146	C₄²	D₄×C₁₀₉	1	4	yes	—
147	C₂³	—	2	2	no	—
148	C₂³	—	2	4	no	—
149	C₂³	—	2	4	no	—
150	C₂³	—	2	2	no	—
151	C₂³	#41×C₁₀₉	1	4	yes	—
152	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
153	Q₈	—	2	8	no	—
154	D₄	—	2	8	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	8	no	—
167	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
168	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
169	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
170	C₂×C₄	#41×C₁₀₉	1	4	yes	—
171	C₂×C₄	#41×C₁₀₉	1	4	yes	—
172	C₂²	—	2	4	no	—
173	C₂²	—	2	8	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	4	no	—
177	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
178	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
179	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
180	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	4	no	—
187	C₄	#124×C₁₀₉	1	4	yes	—
188	C₂	—	4	4	no	—
189	C₂	#525×C₁₀₉	1	1	yes	characteristic
190	C₂	#525×C₁₀₉	1	1	yes	characteristic
191	C₂	C₂×#124×C₁₀₉	1	1	yes	characteristic
192	C₂	#529×C₁₀₉	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₈×C₁₀₉	C₂	semidirect
3. #400×C₁₀₉	C₂	semidirect
4. #397×C₁₀₉	C₂	semidirect
6. #393×C₁₀₉	C₂	semidirect
7. C₄×C₂²⋊C₄×C₁₀₉	C₂	semidirect
8. #376×C₁₀₉	C₂	semidirect
96. #2398	C₁₀₉	direct (N × H)
97. C₁₀₉	#2398	direct (N × H)
98. C₂×C₄×Q₈	C₂×C₁₀₉	semidirect
100. #400	C₂×C₁₀₉	semidirect
101. #397	C₂×C₁₀₉	semidirect
103. #393	C₂×C₁₀₉	semidirect
104. C₄×C₂²⋊C₄	C₂×C₁₀₉	semidirect
105. #376	C₂×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×Q₈×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
#400×C₁₀₉	2
#397×C₁₀₉	2
#394×C₁₀₉	2
#393×C₁₀₉	2
C₄×C₂²⋊C₄×C₁₀₉	2
#376×C₁₀₉	2
#2398	109

Subgroups — C109×#2398 #28359652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2398 #28359652