Subgroups — #28371776

Order 13952.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812706	C₂	1	2	yes	maximal, complemented
2	#20295104	C₂	1	2	yes	maximal, complemented
3	#20295098	C₂	1	4	yes	maximal, complemented
4	D₄×#11724784	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294993	C₂	1	2	yes	maximal, complemented
7	C₄×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
8	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	#16812706	—	2	8	no	—
10	C₂×#13182013	C₂²	1	1	yes	characteristic
11	C₂×#13182013	C₂²	1	4	yes	—
12	C₂×#13182013	C₂²	1	1	yes	characteristic
13	C₂²×#11724784	C₂²	1	2	yes	—
14	C₂²×#11724784	C₂²	1	4	yes	—
15	C₄⋊#11724784	C₂²	1	1	yes	characteristic
16	C₄⋊#11724784	C₂²	1	2	yes	—
17	C₄⋊#11724784	C₂²	1	1	yes	characteristic
18	C₂²⋊#11724784	C₂²	1	4	yes	—
19	C₂²⋊#11724784	C₂²	1	4	yes	—
20	#16812610	C₂²	1	2	yes	—
21	C₄×#11724784	C₂²	1	2	yes	—
22	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
23	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
24	#127×C₁₀₉	C₄	1	1	yes	characteristic
25	#124×C₁₀₉	C₄	1	2	yes	—
26	#121×C₁₀₉	C₄	1	2	yes	—
27	#121×C₁₀₉	C₄	1	2	yes	—
28	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
29	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
30	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
31	#13182013	—	2	2	no	—
32	#13182013	—	2	8	no	—
33	#13182013	D₄	1	4	yes	complemented
34	C₂×#11724784	—	2	4	no	—
35	C₂×#11724784	—	2	8	no	—
36	C₂×#11724784	C₂³	1	2	yes	—
37	C₂×#11724784	C₂³	1	2	yes	—
38	C₂×#11724784	C₂³	1	2	yes	—
39	C₂×#11724784	C₂³	1	2	yes	—
40	#41×C₁₀₉	—	2	4	no	—
41	#41×C₁₀₉	—	2	4	no	—
42	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
44	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
45	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
46	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
50	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
54	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
55	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
60	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	Q₈×C₁₀₉	—	2	4	no	—
62	D₄×C₁₀₉	—	2	8	no	—
63	D₄×C₁₀₉	—	2	4	no	—
64	D₄×C₁₀₉	—	2	4	no	—
65	D₄×C₁₀₉	—	4	2	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	2	4	no	—
68	C₂×C₄×C₁₀₉	—	2	4	no	—
69	C₂×C₄×C₁₀₉	—	4	1	no	—
70	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₀₉	#41	1	2	yes	—
78	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
79	#11724784	—	2	2	no	—
80	#11724784	—	2	2	no	—
81	#11724784	—	2	2	no	—
82	#11724784	C₂×D₄	1	4	yes	—
83	C₂²×C₁₀₉	—	2	1	no	—
84	C₂²×C₁₀₉	—	2	1	no	—
85	C₂²×C₁₀₉	—	2	4	no	—
86	C₂²×C₁₀₉	—	2	4	no	—
87	C₂²×C₁₀₉	—	2	4	no	—
88	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
89	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₁₀₉	—	2	4	no	—
91	C₄×C₁₀₉	—	2	1	no	—
92	C₄×C₁₀₉	—	2	1	no	—
93	C₄×C₁₀₉	—	2	1	no	—
94	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
95	C₄×C₁₀₉	#131	1	2	yes	complemented
96	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
98	C₂×C₁₀₉	—	2	2	no	—
99	C₂×C₁₀₉	—	4	4	no	—
100	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
101	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
102	C₂×C₁₀₉	#568	1	1	yes	characteristic
103	#2935	—	109	1	no	maximal, Sylow
104	C₁₀₉	#2935	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	—	109	2	no	—
106	#532	D₁₀₉	1	1	yes	characteristic
107	#436	—	109	2	no	—
108	#435	—	109	4	no	—
109	C₈×D₄	—	109	2	no	—
110	C₂×C₄⋊C₈	—	109	1	no	—
111	#408	—	109	2	no	—
112	C₄×M₁₆	—	109	1	no	—
113	C₂×#41	C₂×D₁₀₉	1	2	yes	—
114	#131	—	218	8	no	—
115	C₂×M₁₆	—	109	1	no	—
116	C₂×M₁₆	—	109	4	no	—
117	C₂×M₁₆	—	109	1	no	—
118	C₂²×C₈	—	109	2	no	—
119	C₂²×C₈	—	109	4	no	—
120	C₄⋊Q₈	Dic₁₀₉	1	1	yes	characteristic
121	#127	Dic₁₀₉	1	1	yes	characteristic
122	#124	Dic₁₀₉	1	2	yes	—
123	#121	Dic₁₀₉	1	2	yes	—
124	#121	Dic₁₀₉	1	2	yes	—
125	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
126	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
127	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
128	C₄⋊C₈	—	109	1	no	—
129	C₄⋊C₈	—	109	2	no	—
130	C₄⋊C₈	—	109	1	no	—
131	C₂²⋊C₈	—	109	4	no	—
132	C₂²⋊C₈	—	109	4	no	—
133	#97	—	109	2	no	—
134	C₄×C₈	—	109	2	no	—
135	#41	—	2	4	no	—
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×Dic₁₀₉	1	2	yes	—
138	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
139	C₂×D₄	C₂×Dic₁₀₉	1	4	yes	—
140	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
141	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
142	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
145	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
146	M₁₆	—	109	4	no	—
147	M₁₆	—	218	8	no	—
148	M₁₆	—	218	2	no	—
149	C₂×C₈	—	109	2	no	—
150	C₂×C₈	—	109	2	no	—
151	C₂×C₈	—	109	2	no	—
152	C₂×C₈	—	109	2	no	—
153	C₂×C₈	—	218	8	no	—
154	C₂×C₈	—	218	4	no	—
155	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
156	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
157	C₂²⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
158	C₂²⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
159	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
160	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
161	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
162	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
163	C₂³	C₂²×Dic₁₀₉	1	4	yes	—
164	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
165	Q₈	—	2	4	no	—
166	D₄	—	2	8	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	4	no	—
169	D₄	—	4	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
175	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
176	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
178	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
180	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
181	C₂×C₄	#13182028	1	2	yes	—
182	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
183	C₈	—	109	4	no	—
184	C₈	—	218	2	no	—
185	C₈	—	218	2	no	—
186	C₈	—	218	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	D₄×Dic₁₀₉	1	2	yes	—
193	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	D₄×Dic₁₀₉	1	2	yes	—
199	C₄	#16812706	1	2	yes	complemented
200	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
201	C₄	C₂×#13182028	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	4	4	no	—
204	C₂	C₂×D₄×Dic₁₀₉	1	1	yes	characteristic
205	C₂	C₂×#16812706	1	1	yes	characteristic
206	C₂	#20295793	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting, prime radical	#532×C₁₀₉	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#532	D₁₀₉
hypocenter	C₁₀₉	#2935
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#16812706	C₂	semidirect
2. #20295104	C₂	semidirect
3. #20295098	C₂	semidirect
4. D₄×#11724784	C₂	semidirect
5. C₂×C₄⋊#11724784	C₂	semidirect
6. #20294993	C₂	semidirect
7. C₄×#13182013	C₂	semidirect
33. #13182013	D₄	semidirect
95. C₄×C₁₀₉	#131	semidirect
104. C₁₀₉	#2935	semidirect
199. C₄	#16812706	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812706	2
#20295104	2
#20295098	2
D₄×#11724784	2
C₂×C₄⋊#11724784	2
#20294993	2
C₄×#13182013	2
#532×C₁₀₉	2
#2935	109