Subgroups — #28371640

Order 13952.

440 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295792	C₂	1	2	yes	maximal
2	C₂×#16812705	C₂	1	1	yes	maximal, characteristic
3	#20295074	C₂	1	2	yes	maximal, complemented
4	#20295073	C₂	1	4	yes	maximal, complemented
5	C₂×#16812635	C₂	1	1	yes	maximal, characteristic, complemented
6	#108×D₁₀₉	C₂	1	4	yes	maximal, complemented
7	C₂×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13182029	C₂²	1	1	yes	characteristic
9	#16812707	—	2	4	no	—
10	#16812705	—	2	4	no	—
11	#16812705	C₄	1	4	yes	—
12	#16812705	C₄	1	4	yes	—
13	C₂×#13182013	C₂²	1	2	yes	—
14	C₂×#13182013	C₂²	1	1	yes	characteristic
15	C₂×#13182012	C₂²	1	2	yes	—
16	M₁₆×D₁₀₉	C₂²	1	8	yes	—
17	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
18	#16812635	C₂²	1	2	yes	—
19	#16812635	C₂²	1	2	yes	—
20	#16812634	C₂²	1	8	yes	—
21	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
22	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	#108×C₁₀₉	C₂²	1	2	yes	—
24	#108×C₁₀₉	C₂²	1	2	yes	—
25	#13182029	C₂×C₄	1	2	yes	—
26	#13182029	C₂×C₄	1	2	yes	—
27	#13182029	D₄	1	2	yes	—
28	#13182029	D₄	1	2	yes	—
29	C₂×Dic₂₁₈	Q₈	1	1	yes	characteristic
30	C₂×D₄₃₆	Q₈	1	1	yes	characteristic
31	C₂×C₂²⋊D₁₀₉	Q₈	1	2	yes	—
32	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
33	#13182013	—	2	2	no	—
34	#13182013	C₂×C₄	1	2	yes	—
35	#13182013	C₂×C₄	1	2	yes	—
36	#13182013	C₂³	1	4	yes	—
37	#13182012	—	2	8	no	—
38	#13182012	C₂×C₄	1	8	yes	—
39	C₂×#11724784	—	2	2	no	—
40	C₂×#11724784	C₂³	1	2	yes	—
41	C₈×D₁₀₉	—	2	8	no	—
42	C₈×D₁₀₉	C₂×C₄	1	8	yes	—
43	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
44	M₁₆×C₁₀₉	—	2	2	no	—
45	M₁₆×C₁₀₉	C₂³	1	4	yes	—
46	C₂×C₈×C₁₀₉	—	2	2	no	—
47	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
48	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
52	C₂²×D₁₀₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₁₀₉	C₄⋊C₄	1	4	yes	—
60	C₂²⋊D₁₀₉	C₄⋊C₄	1	4	yes	—
61	C₂×Dic₁₀₉	C₂×Q₈	1	2	yes	—
62	D₄₃₆	C₄⋊C₄	1	2	yes	—
63	D₄₃₆	C₄⋊C₄	1	2	yes	—
64	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
65	C₄×D₁₀₉	C₂×D₄	1	4	yes	—
66	Dic₂₁₈	C₄⋊C₄	1	2	yes	—
67	Dic₂₁₈	C₄⋊C₄	1	2	yes	—
68	C₈×C₁₀₉	—	2	4	no	—
69	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
70	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
71	#11724784	—	2	4	no	—
72	#11724784	C₂²×C₄	1	4	yes	—
73	C₂²×C₁₀₉	—	4	1	no	—
74	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
77	C₂×D₁₀₉	—	4	2	no	—
78	C₂×D₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
79	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
82	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
83	Dic₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₁₀₉	—	2	1	no	—
85	C₂×C₁₀₉	—	2	1	no	—
86	C₂×C₁₀₉	—	2	1	no	—
87	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	D₁₀₉	—	2	4	no	—
89	#2916	—	109	1	no	maximal, Sylow
90	C₁₀₉	#2916	1	1	yes	characteristic, complemented, Sylow
91	#568	—	109	2	no	—
92	C₂×#131	—	109	1	no	—
93	#430	—	109	2	no	—
94	#430	—	109	4	no	—
95	C₂×#108	—	109	1	no	—
96	C₂×#108	—	109	4	no	—
97	C₂×#108	D₁₀₉	1	1	yes	characteristic, complemented
98	C₂×#41	—	109	1	no	—
99	#131	—	109	4	no	—
100	#131	—	109	4	no	—
101	#131	—	218	4	no	—
102	#131	—	218	4	no	—
103	C₂×M₁₆	—	109	1	no	—
104	C₂×M₁₆	—	109	2	no	—
105	C₂×M₁₆	—	109	2	no	—
106	C₂×M₁₆	—	109	8	no	—
107	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
108	C₂²×C₈	—	109	2	no	—
109	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
110	#108	—	109	8	no	—
111	#108	—	109	2	no	—
112	#108	—	109	2	no	—
113	#108	C₂×D₁₀₉	1	2	yes	—
114	#108	C₂×D₁₀₉	1	2	yes	—
115	#41	—	109	2	no	—
116	#41	—	109	2	no	—
117	#41	—	109	2	no	—
118	#41	—	109	2	no	—
119	C₂×Q₈	—	109	1	no	—
120	C₂×D₄	—	109	2	no	—
121	C₂×D₄	—	109	1	no	—
122	C₂²×C₄	—	109	2	no	—
123	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
124	M₁₆	—	2	2	no	—
125	M₁₆	—	109	2	no	—
126	M₁₆	—	109	2	no	—
127	M₁₆	—	109	8	no	—
128	M₁₆	—	109	4	no	—
129	M₁₆	—	218	8	no	—
130	M₁₆	—	218	2	no	—
131	M₁₆	C₂²×D₁₀₉	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	109	2	no	—
134	C₂×C₈	—	109	8	no	—
135	C₂×C₈	—	218	8	no	—
136	C₂×C₈	—	218	2	no	—
137	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
138	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
139	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
140	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
141	C₂³	—	109	2	no	—
142	C₂³	Q₈×D₁₀₉	1	1	yes	characteristic
143	Q₈	—	109	2	no	—
144	Q₈	—	109	2	no	—
145	D₄	—	109	4	no	—
146	D₄	—	109	4	no	—
147	D₄	—	109	2	no	—
148	D₄	—	109	2	no	—
149	C₂×C₄	—	109	2	no	—
150	C₂×C₄	—	109	4	no	—
151	C₂×C₄	—	109	4	no	—
152	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
154	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
155	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
156	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	109	4	no	—
160	C₈	—	218	4	no	—
161	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
162	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
163	C₂²	—	4	1	no	—
164	C₂²	—	109	4	no	—
165	C₂²	—	436	2	no	—
166	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
167	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
169	C₄	—	109	4	no	—
170	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
171	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
172	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
173	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	218	4	no	—
178	C₂	C₂×C₄⋊C₄×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#108×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂²×C₄⋊C₄
prime radical	C₂×#13182029	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×#108	D₁₀₉
hypocenter	C₁₀₉	#2916
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. #20295074	C₂	semidirect
4. #20295073	C₂	semidirect
5. C₂×#16812635	C₂	semidirect
6. #108×D₁₀₉	C₂	semidirect
7. C₂×#108×C₁₀₉	C₂	semidirect
90. C₁₀₉	#2916	semidirect
97. C₂×#108	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295792	2
C₂×#16812705	2
#20295074	2
#20295073	2
C₂×#16812635	2
#108×D₁₀₉	2
C₂×#108×C₁₀₉	2
#2916	109