Subgroups — #28376134

Order 13952.

360 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812705	C₂	1	1	yes	maximal, characteristic
2	#20295444	C₂	1	4	yes	maximal, complemented
3	#20295442	C₂	1	4	yes	maximal, complemented
4	C₂×#16812639	C₂	1	1	yes	maximal, characteristic, complemented
5	M₃₂×D₁₀₉	C₂	1	4	yes	maximal, complemented
6	C₂×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13182029	C₄	1	1	yes	characteristic
8	#16812705	C₄	1	2	yes	—
9	#16812705	C₄	1	2	yes	—
10	#16812705	C₂²	1	2	yes	—
11	#16812705	C₂²	1	2	yes	—
12	C₂×#13182013	C₄	1	1	yes	characteristic
13	C₂×#13182012	C₄	1	2	yes	—
14	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
15	#16812639	C₂²	1	2	yes	—
16	#16812639	C₂²	1	2	yes	complemented
17	#16812638	C₂²	1	8	yes	complemented
18	C₂×#13182004	C₂²	1	2	yes	complemented
19	C₁₆×D₁₀₉	C₂²	1	8	yes	complemented
20	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	M₃₂×C₁₀₉	C₂²	1	2	yes	—
22	M₃₂×C₁₀₉	C₂²	1	2	yes	complemented
23	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	complemented
24	#13182029	C₈	1	2	yes	—
25	#13182029	C₈	1	2	yes	—
26	#13182029	C₂×C₄	1	2	yes	—
27	#13182029	C₂×C₄	1	2	yes	—
28	C₂×Dic₂₁₈	C₈	1	1	yes	characteristic
29	C₂×D₄₃₆	C₈	1	1	yes	characteristic
30	C₂×C₂²⋊D₁₀₉	C₈	1	2	yes	—
31	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
32	#13182013	C₂×C₄	1	2	yes	—
33	#13182013	C₂×C₄	1	2	yes	—
34	#13182012	C₂×C₄	1	4	yes	—
35	#13182012	C₂×C₄	1	4	yes	—
36	C₂×#11724784	C₂×C₄	1	2	yes	—
37	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
38	C₈×D₁₀₉	C₂³	1	4	yes	—
39	#13182004	C₂³	1	4	yes	—
40	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₁₆×C₁₀₉	C₂³	1	4	yes	—
48	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
49	C₂²×D₁₀₉	C₂×C₈	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²⋊D₁₀₉	C₂×C₈	1	4	yes	—
57	C₂²⋊D₁₀₉	C₂×C₈	1	4	yes	—
58	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
59	D₄₃₆	C₂×C₈	1	2	yes	—
60	D₄₃₆	C₂×C₈	1	2	yes	—
61	C₄×D₁₀₉	C₂×C₈	1	4	yes	—
62	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
63	Dic₂₁₈	C₂×C₈	1	2	yes	—
64	Dic₂₁₈	C₂×C₈	1	2	yes	—
65	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
66	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
67	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
69	#11724784	C₂²×C₄	1	4	yes	—
70	C₂²×C₁₀₉	—	4	1	no	—
71	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
72	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
73	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×D₁₀₉	—	4	2	no	—
75	C₂×D₁₀₉	C₂²×C₈	1	4	yes	—
76	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
77	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
78	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
79	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
80	Dic₁₀₉	C₂²×C₈	1	4	yes	—
81	C₂×C₁₀₉	—	2	1	no	—
82	C₂×C₁₀₉	—	2	1	no	—
83	C₂×C₁₀₉	—	2	1	no	—
84	C₂×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
85	D₁₀₉	—	2	4	no	—
86	#3407	—	109	1	no	maximal, Sylow
87	C₁₀₉	#3407	1	1	yes	characteristic, complemented, Sylow
88	C₂×#131	—	109	1	no	—
89	#504	—	109	4	no	—
90	#504	—	109	4	no	—
91	C₂×M₃₂	—	109	1	no	—
92	C₂×M₃₂	—	109	4	no	—
93	C₂×M₃₂	D₁₀₉	1	1	yes	characteristic, complemented
94	C₂×#41	—	109	1	no	—
95	#131	—	109	2	no	—
96	#131	—	109	2	no	—
97	#131	—	109	2	no	—
98	#131	—	109	2	no	—
99	C₂×M₁₆	—	109	1	no	—
100	C₂×M₁₆	—	109	2	no	—
101	C₂²×C₈	—	109	2	no	—
102	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
103	M₃₂	—	109	2	no	—
104	M₃₂	—	109	2	no	—
105	M₃₂	—	109	8	no	—
106	M₃₂	C₂×D₁₀₉	1	2	yes	—
107	M₃₂	C₂×D₁₀₉	1	2	yes	complemented
108	C₂×C₁₆	—	109	2	no	—
109	C₂×C₁₆	—	109	8	no	—
110	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	complemented
111	#41	—	109	2	no	—
112	#41	—	109	2	no	—
113	#41	—	109	2	no	—
114	#41	—	109	2	no	—
115	C₂×Q₈	—	109	1	no	—
116	C₂×D₄	—	109	2	no	—
117	C₂×D₄	—	109	1	no	—
118	C₂²×C₄	—	109	2	no	—
119	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
120	M₁₆	—	109	2	no	—
121	M₁₆	—	109	2	no	—
122	M₁₆	—	109	4	no	—
123	M₁₆	—	109	4	no	—
124	C₂×C₈	—	109	2	no	—
125	C₂×C₈	—	109	4	no	—
126	C₂×C₈	—	109	4	no	—
127	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
128	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
129	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
130	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
131	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
132	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
133	C₁₆	—	109	4	no	—
134	C₁₆	C₂²×D₁₀₉	1	4	yes	—
135	C₂³	—	109	2	no	—
136	C₂³	C₈×D₁₀₉	1	1	yes	characteristic
137	Q₈	—	109	2	no	—
138	Q₈	—	109	2	no	—
139	D₄	—	109	4	no	—
140	D₄	—	109	4	no	—
141	D₄	—	109	2	no	—
142	D₄	—	109	2	no	—
143	C₂×C₄	—	109	2	no	—
144	C₂×C₄	—	109	4	no	—
145	C₂×C₄	—	109	4	no	—
146	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
147	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
148	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
152	C₈	—	109	4	no	—
153	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
154	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
155	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
156	C₈	C₂³×D₁₀₉	1	1	yes	characteristic
157	C₂²	—	4	1	no	—
158	C₂²	—	109	4	no	—
159	C₂²	—	436	2	no	—
160	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
161	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
162	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
163	C₄	—	109	4	no	—
164	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
165	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
166	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
167	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	218	4	no	—
172	C₂	C₂²×C₈×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₁₀₉
commutator, socle	C₂×C₁₀₉	C₂³×C₈
Fitting	C₂×M₃₂×C₁₀₉	C₂
prime radical	C₂×#13182029	C₄
cosocle	C₈×C₁₀₉	C₂⁴
hypercenter	C₂×M₃₂	D₁₀₉
hypocenter	C₁₀₉	#3407
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20295444	C₂	semidirect
3. #20295442	C₂	semidirect
4. C₂×#16812639	C₂	semidirect
5. M₃₂×D₁₀₉	C₂	semidirect
6. C₂×M₃₂×C₁₀₉	C₂	semidirect
16. #16812639	C₂²	semidirect
17. #16812638	C₂²	semidirect
18. C₂×#13182004	C₂²	semidirect
19. C₁₆×D₁₀₉	C₂²	semidirect
22. M₃₂×C₁₀₉	C₂²	semidirect
23. C₂×C₁₆×C₁₀₉	C₂²	semidirect
87. C₁₀₉	#3407	semidirect
93. C₂×M₃₂	D₁₀₉	semidirect
107. M₃₂	C₂×D₁₀₉	semidirect
110. C₂×C₁₆	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812705	2
#20295444	2
#20295442	2
C₂×#16812639	2
M₃₂×D₁₀₉	2
C₂×M₃₂×C₁₀₉	2
#3407	109