Subgroups — #28362748

Order 13952.

424 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	C₂³×#11724784	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812658	C₂	1	1	yes	maximal, characteristic
3	#20294788	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	—	2	2	no	—
6	C₂²×#11724784	—	2	2	no	—
7	C₂²×#11724784	C₄	1	4	yes	complemented
8	C₂²×#11724784	C₂²	1	2	yes	—
9	#16812658	—	2	2	no	—
10	#16812658	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₁₀₉	—	2	1	no	—
13	C₂×C₄×Dic₁₀₉	—	2	1	no	—
14	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	#117×C₁₀₉	—	2	2	no	—
16	#117×C₁₀₉	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
19	C₂×C₄²×C₁₀₉	—	2	1	no	—
20	C₂²×Dic₁₀₉	—	2	2	no	—
21	C₂×#11724784	—	2	8	no	—
22	C₂×#11724784	—	2	2	no	—
23	C₂×#11724784	—	2	2	no	—
24	C₂×#11724784	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₀₉	—	2	2	no	—
26	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
27	C₂²⋊Dic₁₀₉	—	2	4	no	—
28	C₄×Dic₁₀₉	—	2	4	no	—
29	C₂⁴×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
35	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
36	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
42	C₄²×C₁₀₉	—	2	4	no	—
43	C₂³×C₁₀₉	—	2	4	no	—
44	C₂³×C₁₀₉	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₁₀₉	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₁₀₉	C₄⋊C₄	1	2	yes	—
47	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	—	2	4	no	—
49	C₂×C₄×C₁₀₉	—	4	4	no	—
50	C₂×C₄×C₁₀₉	C₄²	1	4	yes	—
51	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
57	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×Dic₁₀₉	—	2	4	no	—
64	C₂×Dic₁₀₉	—	4	4	no	—
65	#11724784	—	2	4	no	—
66	C₂²×C₁₀₉	—	2	4	no	—
67	C₂²×C₁₀₉	—	2	4	no	—
68	C₂²×C₁₀₉	—	2	4	no	—
69	C₂²×C₁₀₉	#95	1	4	yes	—
70	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₁₀₉	#135	1	4	yes	—
74	C₄×C₁₀₉	—	4	4	no	—
75	C₄×C₁₀₉	#95	1	4	yes	—
76	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
77	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	Dic₁₀₉	—	4	4	no	—
81	C₂×C₁₀₉	—	2	4	no	—
82	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
83	C₂×C₁₀₉	#416	1	2	yes	—
84	C₂×C₁₀₉	#416	1	2	yes	—
85	C₂×C₁₀₉	#427	1	2	yes	—
86	#1735	—	109	1	no	maximal, Sylow
87	C₁₀₉	#1735	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₈	—	109	1	no	—
89	C₂×#117	—	109	1	no	—
90	C₂×#117	D₁₀₉	1	1	yes	characteristic
91	#340	—	109	4	no	—
92	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
93	C₂²×C₈	—	109	4	no	—
94	C₂²×C₈	—	109	2	no	—
95	C₂²×C₈	—	218	2	no	—
96	C₂²×C₈	—	218	2	no	—
97	#117	—	2	2	no	—
98	#117	—	109	4	no	—
99	#117	—	218	2	no	—
100	#117	Dic₁₀₉	1	4	yes	complemented
101	C₂×C₄⋊C₄	—	109	2	no	—
102	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
103	C₂×C₂²⋊C₄	—	2	1	no	—
104	C₂×C₂²⋊C₄	—	218	1	no	—
105	C₂×C₄²	—	2	1	no	—
106	C₂×C₄²	—	218	1	no	—
107	C₂⁴	C₂²⋊D₁₀₉	1	1	yes	characteristic
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	218	2	no	—
110	C₂²×C₄	Dic₂₁₈	1	2	yes	—
111	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
112	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
113	C₂²×C₄	D₄₃₆	1	2	yes	—
114	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
115	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
116	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
118	C₂×C₈	—	109	4	no	—
119	C₂×C₈	—	218	8	no	—
120	C₂×C₈	—	218	2	no	—
121	C₂×C₈	—	218	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	109	4	no	—
124	C₄⋊C₄	—	218	2	no	—
125	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
126	C₂²⋊C₄	—	2	4	no	—
127	C₂²⋊C₄	—	218	4	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	218	4	no	—
130	C₂³	—	2	4	no	—
131	C₂³	#13182006	1	2	yes	—
132	C₂³	C₂²⋊Dic₁₀₉	1	2	yes	—
133	C₂³	#13182008	1	2	yes	—
134	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	436	4	no	—
138	C₂×C₄	—	218	4	no	—
139	C₂×C₄	C₄×Dic₁₀₉	1	4	yes	—
140	C₂×C₄	#13182006	1	2	yes	—
141	C₂×C₄	#13182006	1	4	yes	—
142	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
143	C₂×C₄	#13182008	1	2	yes	—
144	C₂×C₄	#13182008	1	4	yes	—
145	C₂×C₄	C₄⋊Dic₁₀₉	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
148	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
150	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
152	C₈	—	218	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	#16812606	1	4	yes	—
157	C₂²	C₂×#13182006	1	1	yes	characteristic
158	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
159	C₂²	C₂×#13182008	1	1	yes	characteristic
160	C₂²	#16812719	1	4	yes	—
161	C₄	—	4	4	no	—
162	C₄	—	436	4	no	—
163	C₄	#16812606	1	4	yes	—
164	C₄	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
165	C₄	C₂×#13182006	1	1	yes	characteristic
166	C₄	C₂×#13182008	1	1	yes	characteristic
167	C₄	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
168	C₂	—	2	4	no	—
169	C₂	C₂×#16812606	1	1	yes	characteristic
170	C₂	#20295020	1	2	yes	—
171	C₂	#20295022	1	2	yes	—
172	C₂	#20295062	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#117×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
prime radical	C₂⁴×C₁₀₉	D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#117	D₁₀₉
hypocenter	C₁₀₉	#1735
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #20294788	C₂	semidirect
7. C₂²×#11724784	C₄	semidirect
10. #16812658	C₄	semidirect
16. #117×C₁₀₉	C₄	semidirect
87. C₁₀₉	#1735	semidirect
100. #117	Dic₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#11724784	2
C₂×#16812658	2
#20294788	2
C₂×#117×C₁₀₉	2
#1735	109