Subgroups — #28372883

Order 13952.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295654	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295333	C₂	1	2	yes	maximal, complemented
3	#20295326	C₂	1	2	yes	maximal, complemented
4	#20295321	C₂	1	2	yes	maximal, complemented
5	#20295306	C₂	1	2	yes	maximal, complemented
6	#20295066	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16812632	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295053	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295027	C₂	1	1	yes	maximal, characteristic
10	#20295022	C₂	1	1	yes	maximal, characteristic
11	#520×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#13182013	C₂²	1	1	yes	characteristic
13	C₂²×#11724784	C₂²	1	1	yes	characteristic
14	#16812699	C₂²	1	2	yes	—
15	#16812698	—	2	1	no	—
16	#16812691	C₂²	1	2	yes	—
17	#16812690	—	2	1	no	—
18	#16812676	—	2	2	no	—
19	#16812658	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
21	#16812632	C₂²	1	2	yes	—
22	#16812632	C₂²	1	2	yes	—
23	#16812632	C₂²	1	2	yes	—
24	#16812630	C₂²	1	2	yes	—
25	C₄⋊#11724784	C₂²	1	2	yes	—
26	C₄⋊#11724784	C₂²	1	2	yes	—
27	#16812623	C₂²	1	1	yes	characteristic
28	#16812623	C₂²	1	1	yes	characteristic
29	#16812623	C₂²	1	2	yes	—
30	#16812620	C₂²	1	1	yes	characteristic
31	#16812620	C₂²	1	1	yes	characteristic
32	#16812620	C₂²	1	2	yes	—
33	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
35	C₄×D₄×C₁₀₉	—	2	2	no	—
36	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
37	#117×C₁₀₉	—	2	1	no	—
38	#117×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₁₈	—	2	2	no	—
42	C₂²×Dic₁₀₉	—	2	1	no	—
43	#13182013	—	2	2	no	—
44	C₂×#11724784	—	2	2	no	—
45	C₂×#11724784	C₂³	1	2	yes	—
46	C₂×#11724784	C₂³	1	1	yes	characteristic
47	C₂×#11724784	C₂³	1	1	yes	characteristic
48	C₄⋊Dic₁₀₉	—	2	1	no	—
49	C₄⋊Dic₁₀₉	—	2	2	no	—
50	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
51	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
52	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
53	#13182008	—	2	2	no	—
54	#13182008	—	2	2	no	—
55	C₂²⋊Dic₁₀₉	—	2	1	no	—
56	C₄×Dic₁₀₉	—	2	1	no	—
57	#41×C₁₀₉	—	2	2	no	—
58	#41×C₁₀₉	Q₈	1	4	yes	complemented
59	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₀₉	—	2	1	no	—
61	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	—	2	1	no	—
63	C₂²×C₄×C₁₀₉	—	2	1	no	—
64	C₂²×C₄×C₁₀₉	—	2	2	no	—
65	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₀₉	—	2	1	no	—
67	C₄⋊C₄×C₁₀₉	—	2	2	no	—
68	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
71	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
72	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
73	C₄²×C₁₀₉	—	2	2	no	—
74	C₄²×C₁₀₉	C₂³	1	2	yes	—
75	C₂³×C₁₀₉	—	2	1	no	—
76	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₁₀₉	—	2	1	no	—
78	Q₈×C₁₀₉	C₂×Q₈	1	2	yes	—
79	D₄×C₁₀₉	—	2	4	no	—
80	D₄×C₁₀₉	—	2	1	no	—
81	D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₁₀₉	—	2	1	no	—
83	C₂×C₄×C₁₀₉	—	2	4	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	8	2	no	—
86	C₂×C₄×C₁₀₉	—	8	1	no	—
87	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₁₀₉	#41	1	2	yes	—
92	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
93	C₂×Dic₁₀₉	—	2	1	no	—
94	C₂×Dic₁₀₉	—	2	1	no	—
95	C₂×Dic₁₀₉	—	2	2	no	—
96	C₂×Dic₁₀₉	—	8	1	no	—
97	Dic₂₁₈	—	8	2	no	—
98	#11724784	—	2	1	no	—
99	#11724784	—	2	1	no	—
100	#11724784	—	2	2	no	—
101	C₂²×C₁₀₉	—	2	1	no	—
102	C₂²×C₁₀₉	—	2	1	no	—
103	C₂²×C₁₀₉	—	2	2	no	—
104	C₂²×C₁₀₉	—	4	1	no	—
105	C₂²×C₁₀₉	—	4	1	no	—
106	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₁₀₉	—	2	2	no	—
109	C₄×C₁₀₉	—	2	2	no	—
110	C₄×C₁₀₉	—	2	2	no	—
111	C₄×C₁₀₉	—	4	2	no	—
112	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
113	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
114	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
115	Dic₁₀₉	—	4	1	no	—
116	Dic₁₀₉	—	4	2	no	—
117	Dic₁₀₉	—	4	1	no	—
118	C₂×C₁₀₉	—	2	2	no	—
119	C₂×C₁₀₉	—	4	2	no	—
120	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₁₀₉	#576	1	1	yes	characteristic
122	C₂×C₁₀₉	#578	1	1	yes	characteristic
123	#3083	—	109	1	no	maximal, Sylow
124	C₁₀₉	#3083	1	1	yes	characteristic, complemented, Sylow
125	#544	—	109	1	no	—
126	#520	D₁₀₉	1	1	yes	characteristic
127	#479	—	109	2	no	—
128	#478	—	109	2	no	—
129	#477	—	109	2	no	—
130	#474	—	109	2	no	—
131	#428	—	109	1	no	—
132	C₂×#107	—	109	1	no	—
133	#424	—	109	1	no	—
134	#417	—	109	1	no	—
135	#416	—	109	1	no	—
136	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
137	C₂×M₁₆	—	109	1	no	—
138	C₂²×C₈	—	109	1	no	—
139	C₄⋊Q₈	—	109	2	no	—
140	C₄⋊Q₈	—	218	1	no	—
141	#125	—	109	2	no	—
142	#125	—	218	1	no	—
143	C₂²⋊Q₈	—	218	2	no	—
144	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
147	#117	—	2	1	no	—
148	#117	—	109	1	no	—
149	#117	C₂×D₁₀₉	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	109	1	no	—
152	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
153	#107	—	109	2	no	—
154	#107	—	109	2	no	—
155	#107	—	109	2	no	—
156	#106	—	109	2	no	—
157	C₄⋊C₈	—	109	2	no	—
158	C₄⋊C₈	—	109	2	no	—
159	#103	—	109	1	no	—
160	#103	—	109	1	no	—
161	#103	—	109	2	no	—
162	#102	—	109	1	no	—
163	#102	—	109	1	no	—
164	#102	—	109	2	no	—
165	#41	—	2	2	no	—
166	#41	Dic₂₁₈	1	4	yes	complemented
167	C₂×Q₈	—	218	2	no	—
168	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	218	1	no	—
175	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
176	M₁₆	—	218	2	no	—
177	C₂×C₈	—	109	2	no	—
178	C₂×C₈	—	109	1	no	—
179	C₂×C₈	—	109	1	no	—
180	C₂×C₈	—	218	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	109	1	no	—
184	C₄⋊C₄	—	109	2	no	—
185	C₄⋊C₄	—	109	1	no	—
186	C₄⋊C₄	—	218	2	no	—
187	C₄⋊C₄	—	218	2	no	—
188	C₄⋊C₄	—	218	1	no	—
189	C₄⋊C₄	—	218	2	no	—
190	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
191	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
192	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	218	1	no	—
195	C₂²⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	218	1	no	—
198	C₄²	C₂²×D₁₀₉	1	2	yes	—
199	C₂³	—	2	1	no	—
200	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	872	2	no	—
203	Q₈	C₂×Dic₂₁₈	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Dic₂₁₈	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	1	no	—
212	C₂×C₄	—	872	1	no	—
213	C₂×C₄	—	218	1	no	—
214	C₂×C₄	—	218	1	no	—
215	C₂×C₄	—	218	2	no	—
216	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
217	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
219	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
220	C₂×C₄	#13182029	1	2	yes	—
221	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
222	C₈	—	218	1	no	—
223	C₈	—	218	1	no	—
224	C₈	—	218	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	#16812678	1	2	yes	—
231	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	436	2	no	—
237	C₄	—	436	1	no	—
238	C₄	—	436	1	no	—
239	C₄	#16812678	1	2	yes	—
240	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
241	C₄	C₂×#13182029	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×#16812678	1	1	yes	characteristic
245	C₂	#20295829	1	1	yes	characteristic
246	C₂	#20295841	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#520×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#41×C₁₀₉	C₂²
hypercenter	#520	D₁₀₉
hypocenter	C₁₀₉	#3083
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. #20295654	C₂	semidirect
2. #20295333	C₂	semidirect
3. #20295326	C₂	semidirect
4. #20295321	C₂	semidirect
5. #20295306	C₂	semidirect
6. #20295066	C₂	semidirect
7. C₂×#16812632	C₂	semidirect
8. #20295053	C₂	semidirect
58. #41×C₁₀₉	Q₈	semidirect
124. C₁₀₉	#3083	semidirect
166. #41	Dic₂₁₈	semidirect

Maximal subgroups

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

subgroup	index
#20295654	2
#20295333	2
#20295326	2
#20295321	2
#20295306	2
#20295066	2
C₂×#16812632	2
#20295053	2
#20295027	2
#20295022	2
#520×C₁₀₉	2
#3083	109