Subgroups — #28371998

Order 13952.

412 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13182029	C₂	1	1	yes	maximal, characteristic
2	#20295156	C₂	1	2	yes	maximal, complemented
3	#20295105	C₂	1	2	yes	maximal, complemented
4	#20295097	C₂	1	4	yes	maximal, complemented
5	#20295079	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295077	C₂	1	2	yes	maximal, complemented
7	C₄×#13182012	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13182029	C₄	1	1	yes	characteristic
10	C₂×#13182012	C₂²	1	4	yes	—
11	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
12	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
13	C₄×D₄₃₆	C₄	1	1	yes	characteristic
14	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₁₀₉	C₄	1	2	yes	—
16	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
17	#16812658	C₄	1	1	yes	characteristic
18	#16812656	C₄	1	2	yes	—
19	C₄²×D₁₀₉	C₂²	1	2	yes	—
20	C₄⋊#11724784	C₂²	1	2	yes	—
21	#16812628	C₂²	1	4	yes	—
22	C₂²⋊#11724784	C₂²	1	2	yes	—
23	#16812611	C₂²	1	4	yes	—
24	#16812609	C₂²	1	4	yes	—
25	C₄×#11724784	C₂²	1	2	yes	—
26	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
31	#13182029	—	2	4	no	—
32	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
38	#13182012	—	2	8	no	—
39	C₂×#11724784	C₂³	1	4	yes	—
40	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
42	#13182008	C₂×C₄	1	2	yes	—
43	#13182008	C₂×C₄	1	2	yes	—
44	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
45	#13182006	C₂×C₄	1	2	yes	—
46	#13182006	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₀₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	—	2	4	no	—
54	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
55	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
56	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄×C₁₀₉	—	2	2	no	—
64	C₂×C₄×C₁₀₉	M₁₆	1	4	yes	complemented
65	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₁₀₉	—	2	4	no	—
72	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
75	D₄₃₆	—	2	2	no	—
76	C₄×D₁₀₉	—	2	4	no	—
77	C₄×D₁₀₉	—	2	4	no	—
78	C₄×D₁₀₉	—	2	4	no	—
79	Dic₂₁₈	—	2	2	no	—
80	C₈×C₁₀₉	—	2	2	no	—
81	C₈×C₁₀₉	#41	1	4	yes	complemented
82	#11724784	—	2	4	no	—
83	C₂²×C₁₀₉	—	2	1	no	—
84	C₂²×C₁₀₉	—	2	1	no	—
85	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
86	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₁₀₉	—	2	2	no	—
88	C₂×D₁₀₉	—	2	2	no	—
89	C₂×D₁₀₉	—	2	2	no	—
90	C₄×C₁₀₉	—	2	1	no	—
91	C₄×C₁₀₉	—	2	1	no	—
92	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
93	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
94	C₄×C₁₀₉	#131	1	2	yes	—
95	C₄×C₁₀₉	C₂×#41	1	2	yes	—
96	Dic₁₀₉	—	2	2	no	—
97	Dic₁₀₉	—	2	2	no	—
98	Dic₁₀₉	—	2	2	no	—
99	C₂×C₁₀₉	—	2	2	no	—
100	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
101	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
103	D₁₀₉	—	4	2	no	—
104	#2965	—	109	1	no	maximal, Sylow
105	C₁₀₉	#2965	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	—	109	1	no	—
107	#446	—	109	2	no	—
108	#436	—	109	2	no	—
109	#435	—	109	4	no	—
110	#431	—	109	1	no	—
111	#431	—	109	2	no	—
112	C₄×M₁₆	—	109	2	no	—
113	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic, complemented
114	C₂×#41	—	109	1	no	—
115	C₂×M₁₆	—	109	4	no	—
116	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
117	C₄×Q₈	—	109	1	no	—
118	C₄×Q₈	—	109	1	no	—
119	C₄×D₄	—	109	2	no	—
120	C₄×D₄	—	109	2	no	—
121	C₄×D₄	—	109	1	no	—
122	C₄×D₄	—	109	1	no	—
123	#117	—	109	1	no	—
124	#117	—	109	2	no	—
125	C₂×C₄²	—	109	2	no	—
126	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
127	C₄⋊C₈	—	109	4	no	—
128	C₄⋊C₈	—	109	2	no	—
129	C₂²⋊C₈	—	109	2	no	—
130	C₂²⋊C₈	—	109	4	no	—
131	#97	—	109	4	no	—
132	C₄×C₈	—	109	2	no	—
133	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
134	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
135	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
136	#41	—	218	4	no	—
137	C₂×Q₈	—	109	1	no	—
138	C₂×D₄	—	109	2	no	—
139	C₂×D₄	—	109	1	no	—
140	C₂²×C₄	—	109	2	no	—
141	C₂²×C₄	—	109	2	no	—
142	C₂²×C₄	—	109	2	no	—
143	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
146	M₁₆	—	218	8	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	109	4	no	—
149	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
150	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
151	C₄⋊C₄	—	109	2	no	—
152	C₄⋊C₄	—	109	2	no	—
153	C₄⋊C₄	—	109	1	no	—
154	C₄⋊C₄	—	109	1	no	—
155	C₂²⋊C₄	—	109	2	no	—
156	C₂²⋊C₄	—	109	2	no	—
157	C₂²⋊C₄	—	109	2	no	—
158	C₄²	—	109	2	no	—
159	C₄²	—	109	2	no	—
160	C₄²	—	109	2	no	—
161	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
162	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
163	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
164	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
165	C₂³	—	109	2	no	—
166	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
167	Q₈	—	218	2	no	—
168	D₄	—	218	4	no	—
169	D₄	—	218	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	109	2	no	—
173	C₂×C₄	—	109	2	no	—
174	C₂×C₄	—	109	2	no	—
175	C₂×C₄	—	218	4	no	—
176	C₂×C₄	—	218	4	no	—
177	C₂×C₄	—	218	4	no	—
178	C₂×C₄	#13182012	1	4	yes	complemented
179	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	218	4	no	—
187	C₈	#13182029	1	4	yes	complemented
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	218	2	no	—
191	C₂²	—	218	2	no	—
192	C₂²	—	218	2	no	—
193	C₂²	C₂×#13182012	1	2	yes	—
194	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	218	2	no	—
198	C₄	—	218	2	no	—
199	C₄	—	218	2	no	—
200	C₄	C₂×#13182012	1	2	yes	—
201	C₄	C₂×#13182012	1	2	yes	—
202	C₄	#16812705	1	2	yes	—
203	C₄	C₂×#13182029	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	436	2	no	—
206	C₂	C₄×#13182029	1	1	yes	characteristic
207	C₂	C₂²×#13182012	1	1	yes	characteristic
208	C₂	C₂×#16812705	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
prime radical	C₂×#13182029	C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	#2965
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20295156	C₂	semidirect
3. #20295105	C₂	semidirect
4. #20295097	C₂	semidirect
5. #20295079	C₂	semidirect
6. #20295077	C₂	semidirect
7. C₄×#13182012	C₂	semidirect
8. C₂×C₄×C₈×C₁₀₉	C₂	semidirect
64. C₂×C₄×C₁₀₉	M₁₆	semidirect
81. C₈×C₁₀₉	#41	semidirect
105. C₁₀₉	#2965	semidirect
113. C₂×C₄×C₈	D₁₀₉	semidirect
178. C₂×C₄	#13182012	semidirect
187. C₈	#13182029	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13182029	2
#20295156	2
#20295105	2
#20295097	2
#20295079	2
#20295077	2
C₄×#13182012	2
C₂×C₄×C₈×C₁₀₉	2
#2965	109