Subgroups — #28371697

Order 13952.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 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	#20295086	C₂	1	4	yes	maximal, complemented
3	#20295082	C₂	1	2	yes	maximal, complemented
4	#20295078	C₂	1	2	yes	maximal, complemented
5	#20295050	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294992	C₂	1	2	yes	maximal
7	#20294981	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13182029	C₂²	1	1	yes	characteristic
10	C₂×#13182013	C₂²	1	2	yes	—
11	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
12	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
13	C₄×D₄₃₆	C₄	1	1	yes	characteristic
14	C₄×D₄₃₆	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₁₀₉	C₄	1	2	yes	—
16	C₄×C₂²⋊D₁₀₉	C₄	1	2	yes	—
17	#16812658	C₂²	1	1	yes	characteristic
18	#16812656	C₂²	1	2	yes	—
19	C₄²×D₁₀₉	C₂²	1	2	yes	—
20	C₄⋊#11724784	C₂²	1	2	yes	—
21	C₄⋊#11724784	C₂²	1	2	yes	—
22	#16812611	C₂²	1	4	yes	—
23	#16812611	C₂²	1	4	yes	—
24	#16812609	C₂²	1	4	yes	—
25	C₈×Dic₁₀₉	C₂²	1	4	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	2	no	—
32	#13182029	—	2	2	no	—
33	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
34	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
39	#13182013	—	2	4	no	—
40	C₂×#11724784	C₂³	1	4	yes	—
41	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
43	#13182008	C₂×C₄	1	2	yes	—
44	#13182008	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
46	#13182006	C₂×C₄	1	2	yes	—
47	#13182006	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₀₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₀₉	—	2	4	no	—
55	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
56	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
57	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₀₉	—	2	2	no	—
64	C₂×C₄×C₁₀₉	—	4	1	no	—
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	#41	1	2	yes	—
71	C₂×C₄×C₁₀₉	#41	1	2	yes	—
72	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
73	C₂²⋊D₁₀₉	—	2	4	no	—
74	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
77	D₄₃₆	—	2	2	no	—
78	C₄×D₁₀₉	—	2	4	no	—
79	C₄×D₁₀₉	—	2	2	no	—
80	C₄×D₁₀₉	—	2	2	no	—
81	C₄×D₁₀₉	—	4	2	no	—
82	Dic₂₁₈	—	2	2	no	—
83	C₈×C₁₀₉	—	2	2	no	—
84	C₈×C₁₀₉	—	2	2	no	—
85	#11724784	—	2	4	no	—
86	C₂²×C₁₀₉	—	2	1	no	—
87	C₂²×C₁₀₉	—	2	1	no	—
88	C₂²×C₁₀₉	#117	1	2	yes	—
89	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₁₀₉	—	2	2	no	—
91	C₂×D₁₀₉	—	2	2	no	—
92	C₂×D₁₀₉	—	2	2	no	—
93	C₄×C₁₀₉	—	2	1	no	—
94	C₄×C₁₀₉	—	2	1	no	—
95	C₄×C₁₀₉	—	2	1	no	—
96	C₄×C₁₀₉	#117	1	2	yes	—
97	C₄×C₁₀₉	#131	1	2	yes	—
98	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
99	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
100	Dic₁₀₉	—	2	2	no	—
101	Dic₁₀₉	—	2	2	no	—
102	Dic₁₀₉	—	2	2	no	—
103	C₂×C₁₀₉	—	2	2	no	—
104	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
105	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
106	C₂×C₁₀₉	#568	1	1	yes	characteristic
107	D₁₀₉	—	4	2	no	—
108	#2925	—	109	1	no	maximal, Sylow
109	C₁₀₉	#2925	1	1	yes	characteristic, complemented, Sylow
110	C₄×#41	—	109	1	no	—
111	#433	—	109	4	no	—
112	#432	—	109	2	no	—
113	#431	—	109	2	no	—
114	#423	—	109	1	no	—
115	C₂×C₄⋊C₈	D₁₀₉	1	1	yes	characteristic, complemented
116	#408	—	109	2	no	—
117	#405	—	109	2	no	—
118	C₂×#41	—	109	1	no	—
119	C₂×M₁₆	—	109	2	no	—
120	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
121	C₄×Q₈	—	109	1	no	—
122	C₄×Q₈	—	109	1	no	—
123	C₄×D₄	—	109	2	no	—
124	C₄×D₄	—	109	2	no	—
125	C₄×D₄	—	109	1	no	—
126	C₄×D₄	—	109	1	no	—
127	#117	—	109	1	no	—
128	#117	—	109	2	no	—
129	C₂×C₄²	—	109	2	no	—
130	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
131	C₄⋊C₈	—	109	2	no	—
132	C₄⋊C₈	—	109	2	no	—
133	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
134	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
135	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
136	C₂²⋊C₈	—	109	4	no	—
137	C₂²⋊C₈	—	109	4	no	—
138	#97	—	109	4	no	—
139	C₄×C₈	—	109	4	no	—
140	#41	—	218	2	no	—
141	#41	—	218	2	no	—
142	C₂×Q₈	—	109	1	no	—
143	C₂×D₄	—	109	2	no	—
144	C₂×D₄	—	109	1	no	—
145	C₂²×C₄	—	109	2	no	—
146	C₂²×C₄	—	109	2	no	—
147	C₂²×C₄	—	109	2	no	—
148	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
149	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
151	M₁₆	—	218	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	109	4	no	—
154	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
155	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
156	C₄⋊C₄	—	109	2	no	—
157	C₄⋊C₄	—	109	2	no	—
158	C₄⋊C₄	—	109	1	no	—
159	C₄⋊C₄	—	109	1	no	—
160	C₂²⋊C₄	—	109	2	no	—
161	C₂²⋊C₄	—	109	2	no	—
162	C₂²⋊C₄	—	109	2	no	—
163	C₄²	—	109	2	no	—
164	C₄²	—	109	2	no	—
165	C₄²	—	109	2	no	—
166	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
167	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
168	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
169	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
170	C₂³	—	109	2	no	—
171	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
172	Q₈	—	218	2	no	—
173	D₄	—	218	4	no	—
174	D₄	—	218	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	109	2	no	—
178	C₂×C₄	—	109	2	no	—
179	C₂×C₄	—	109	2	no	—
180	C₂×C₄	—	436	2	no	—
181	C₂×C₄	—	218	4	no	—
182	C₂×C₄	—	218	2	no	—
183	C₂×C₄	—	218	2	no	—
184	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
189	C₂×C₄	#13182028	1	2	yes	—
190	C₂×C₄	#13182030	1	2	yes	—
191	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	218	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	218	2	no	—
198	C₂²	—	218	2	no	—
199	C₂²	—	218	2	no	—
200	C₂²	#16812657	1	2	yes	—
201	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	218	2	no	—
206	C₄	—	218	2	no	—
207	C₄	—	218	2	no	—
208	C₄	#16812657	1	2	yes	—
209	C₄	#16812705	1	2	yes	—
210	C₄	C₂×#13182028	1	1	yes	characteristic
211	C₄	C₂×#13182030	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	436	2	no	—
214	C₂	C₂×#16812657	1	1	yes	characteristic
215	C₂	C₂×#16812705	1	1	yes	characteristic
216	C₂	#20295792	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₁₀₉	C₂
prime radical	C₂×#13182029	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#2925
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20295086	C₂	semidirect
3. #20295082	C₂	semidirect
4. #20295078	C₂	semidirect
5. #20295050	C₂	semidirect
7. #20294981	C₂	semidirect
8. C₂×C₄⋊C₈×C₁₀₉	C₂	semidirect
109. C₁₀₉	#2925	semidirect
115. C₂×C₄⋊C₈	D₁₀₉	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
#20295086	2
#20295082	2
#20295078	2
#20295050	2
#20294992	2
#20294981	2
C₂×C₄⋊C₈×C₁₀₉	2
#2925	109