Subgroups — C₁₀₉×#2962 #28360216

Order 13952.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#446×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#446×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	C₈×Q₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#424×C₁₀₉	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	#405×C₁₀₉	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
14	#125×C₁₀₉	C₄	1	2	yes	—
15	C₂²⋊Q₈×C₁₀₉	C₄	1	2	yes	—
16	C₂²⋊Q₈×C₁₀₉	C₄	1	2	yes	—
17	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
18	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
19	#117×C₁₀₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
24	C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
25	C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
26	#97×C₁₀₉	C₂²	1	2	yes	—
27	#97×C₁₀₉	C₂²	1	4	yes	—
28	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
29	C₄×C₈×C₁₀₉	C₂²	1	4	yes	—
30	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₀₉	—	2	2	no	—
36	M₁₆×C₁₀₉	Q₈	1	4	yes	complemented
37	C₂×C₈×C₁₀₉	—	2	4	no	—
38	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
39	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
40	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
41	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
42	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
45	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
46	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
49	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₄²×C₁₀₉	C₂³	1	4	yes	—
53	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	Q₈×C₁₀₉	—	2	4	no	—
56	Q₈×C₁₀₉	—	4	2	no	—
57	C₂×C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄×C₁₀₉	—	4	1	no	—
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₁₀₉	#41	1	2	yes	—
68	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
69	C₈×C₁₀₉	—	2	2	no	—
70	C₈×C₁₀₉	—	2	2	no	—
71	C₈×C₁₀₉	—	2	2	no	—
72	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
73	C₂²×C₁₀₉	—	2	1	no	—
74	C₂²×C₁₀₉	—	2	1	no	—
75	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
76	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₁₀₉	—	2	1	no	—
78	C₄×C₁₀₉	—	2	4	no	—
79	C₄×C₁₀₉	—	2	1	no	—
80	C₄×C₁₀₉	—	2	1	no	—
81	C₄×C₁₀₉	—	2	4	no	—
82	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
83	C₄×C₁₀₉	#131	1	2	yes	—
84	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
85	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
86	C₂×C₁₀₉	—	2	2	no	—
87	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
89	C₂×C₁₀₉	#568	1	1	yes	characteristic
90	#2962	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₁₀₉	#2962	1	1	yes	characteristic, direct product factor, Sylow
92	#533	C₂×C₁₀₉	1	1	yes	characteristic
93	#446	C₂×C₁₀₉	1	2	yes	complemented
94	#446	C₂×C₁₀₉	1	4	yes	complemented
95	C₈×Q₈	C₂×C₁₀₉	1	2	yes	complemented
96	#424	C₂×C₁₀₉	1	2	yes	—
97	C₂×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
98	#405	C₂×C₁₀₉	1	2	yes	—
99	C₄×M₁₆	C₂×C₁₀₉	1	1	yes	characteristic
100	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
101	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
103	C₄⋊Q₈	C₄×C₁₀₉	1	1	yes	characteristic
104	C₄⋊Q₈	C₄×C₁₀₉	1	1	yes	characteristic
105	#125	C₄×C₁₀₉	1	2	yes	—
106	C₂²⋊Q₈	C₄×C₁₀₉	1	2	yes	—
107	C₂²⋊Q₈	C₄×C₁₀₉	1	2	yes	—
108	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
109	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
110	#117	C₂²×C₁₀₉	1	2	yes	—
111	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
112	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
113	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
114	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
115	C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
116	C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
117	#97	C₂²×C₁₀₉	1	2	yes	—
118	#97	C₂²×C₁₀₉	1	4	yes	—
119	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
120	C₄×C₈	C₂²×C₁₀₉	1	4	yes	—
121	C₂×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
122	C₂×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
123	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Q₈×C₁₀₉	1	4	yes	complemented
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
130	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
131	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
132	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
133	C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
134	C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
135	C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
136	C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
137	C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
138	C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
139	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
140	C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
141	C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
142	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
143	C₄²	C₂³×C₁₀₉	1	4	yes	—
144	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
145	C₂³	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
146	Q₈	—	2	4	no	—
147	Q₈	—	4	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
151	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
152	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
153	C₂×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
154	C₂×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
155	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
157	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
158	C₂×C₄	#41×C₁₀₉	1	2	yes	—
159	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Q₈×C₁₀₉	1	2	yes	—
167	C₂²	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	4	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	4	no	—
173	C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
174	C₄	#131×C₁₀₉	1	2	yes	—
175	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
176	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Q₈×C₁₀₉	1	1	yes	characteristic
179	C₂	C₂×#131×C₁₀₉	1	1	yes	characteristic
180	C₂	#568×C₁₀₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. #446×C₁₀₉	C₂	semidirect
3. #446×C₁₀₉	C₂	semidirect
4. C₈×Q₈×C₁₀₉	C₂	semidirect
36. M₁₆×C₁₀₉	Q₈	semidirect
90. #2962	C₁₀₉	direct (N × H)
91. C₁₀₉	#2962	direct (N × H)
93. #446	C₂×C₁₀₉	semidirect
94. #446	C₂×C₁₀₉	semidirect
95. C₈×Q₈	C₂×C₁₀₉	semidirect
127. M₁₆	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#533×C₁₀₉	2
#446×C₁₀₉	2
C₈×Q₈×C₁₀₉	2
#424×C₁₀₉	2
C₂×C₄⋊C₈×C₁₀₉	2
#405×C₁₀₉	2
C₄×M₁₆×C₁₀₉	2
#2962	109

Subgroups — C109×#2962 #28360216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2962 #28360216