Subgroups — #28378730

Order 13952.

420 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812778	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812778	C₂	1	1	yes	maximal, characteristic
3	#20295903	C₂	1	4	yes	maximal
4	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#16812778	—	2	1	no	—
6	#16812778	—	2	1	no	—
7	#16812778	—	2	1	no	—
8	#16812778	—	2	1	no	—
9	#16812778	—	2	1	no	—
10	#16812778	—	2	1	no	—
11	#16812778	C₄	1	2	yes	complemented
12	#16812778	C₄	1	2	yes	complemented
13	C₂×C₄⋊#89137	—	2	1	no	—
14	C₂×C₄⋊#89137	C₄	1	2	yes	complemented
15	C₂×C₂²⋊#89137	C₂²	1	2	yes	—
16	C₂×C₂²⋊#89137	C₂²	1	2	yes	—
17	C₂×C₄×#89137	—	2	1	no	—
18	C₂×C₄×#89137	C₄	1	2	yes	complemented
19	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
20	#16812658	C₄	1	1	yes	characteristic, complemented
21	#16812657	—	2	1	no	—
22	#16812656	—	2	1	no	—
23	#16812655	C₄	1	2	yes	complemented
24	C₄⋊C₄×D₁₀₉	—	2	1	no	—
25	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
26	C₄²×D₁₀₉	—	2	1	no	—
27	#117×C₁₀₉	C₄	1	1	yes	characteristic, complemented
28	C₂²×#89137	—	2	2	no	—
29	C₂²×#89137	C₂×C₄	1	4	yes	—
30	C₄⋊#89137	—	2	2	no	—
31	C₄⋊#89137	—	2	2	no	—
32	C₄⋊#89137	—	2	2	no	—
33	C₄⋊#89137	—	2	2	no	—
34	C₂²⋊#89137	—	2	2	no	—
35	C₂²⋊#89137	—	2	2	no	—
36	C₂²⋊#89137	—	2	2	no	—
37	C₂²⋊#89137	C₂×C₄	1	4	yes	—
38	C₄×#89137	—	2	2	no	—
39	C₄×#89137	—	2	2	no	—
40	C₄×#89137	—	2	2	no	—
41	C₄×#89137	—	2	2	no	—
42	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₀₉	—	2	2	no	—
45	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
50	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
51	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
53	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
55	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
56	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
57	C₄⋊Dic₁₀₉	—	2	1	no	—
58	#13182008	—	2	2	no	—
59	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
60	#13182006	—	2	2	no	—
61	C₄×Dic₁₀₉	—	2	1	no	—
62	C₄×Dic₁₀₉	—	2	2	no	—
63	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₀₉	—	2	1	no	—
65	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
66	C₄²×C₁₀₉	—	2	1	no	—
67	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₁₀₉	—	4	2	no	—
69	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
75	C₂×#89137	—	2	4	no	—
76	C₂×#89137	—	2	4	no	—
77	C₂×#89137	—	4	4	no	—
78	C₂×#89137	—	4	4	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
86	C₂×Dic₁₀₉	—	2	2	no	—
87	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
88	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
89	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
92	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
93	C₄×D₁₀₉	—	4	4	no	—
94	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
100	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
102	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
103	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
104	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
105	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
106	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
107	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
108	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
109	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₁₀₉	—	4	1	no	—
111	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
112	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×D₁₀₉	—	2	2	no	—
115	C₂×D₁₀₉	—	2	2	no	—
116	C₂×D₁₀₉	—	2	2	no	—
117	C₂×D₁₀₉	—	4	1	no	—
118	C₂×D₁₀₉	—	4	1	no	—
119	C₂×D₁₀₉	—	4	1	no	—
120	C₂×D₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×D₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×D₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×D₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	#89137	—	4	4	no	—
125	#89137	—	4	4	no	—
126	C₄×C₁₀₉	—	4	2	no	—
127	C₄×C₁₀₉	#95	1	1	yes	characteristic
128	C₄×C₁₀₉	#95	1	1	yes	characteristic
129	C₄×C₁₀₉	#95	1	1	yes	characteristic
130	C₄×C₁₀₉	#95	1	1	yes	characteristic
131	Dic₁₀₉	—	4	2	no	—
132	Dic₁₀₉	#95	1	1	yes	characteristic
133	Dic₁₀₉	#95	1	1	yes	characteristic
134	Dic₁₀₉	#95	1	1	yes	characteristic
135	Dic₁₀₉	#95	1	1	yes	characteristic
136	C₂×C₁₀₉	—	2	1	no	—
137	C₂×C₁₀₉	—	2	1	no	—
138	C₂×C₁₀₉	—	2	1	no	—
139	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
140	D₁₀₉	—	2	1	no	—
141	D₁₀₉	—	2	1	no	—
142	D₁₀₉	—	2	1	no	—
143	D₁₀₉	#410	1	2	yes	—
144	#1740	—	109	1	no	maximal, Sylow
145	C₁₀₉	#1740	1	1	yes	characteristic, complemented, Sylow
146	C₂×#117	—	109	1	no	—
147	C₂×#117	—	109	1	no	—
148	C₂×#117	—	109	1	no	—
149	#342	—	109	4	no	—
150	C₂³×C₄	—	109	1	no	—
151	#117	—	109	2	no	—
152	#117	—	109	2	no	—
153	#117	—	109	1	no	—
154	#117	—	109	2	no	—
155	#117	—	218	1	no	—
156	#117	—	218	1	no	—
157	#117	—	218	1	no	—
158	#117	—	218	1	no	—
159	#117	—	218	1	no	—
160	#117	—	218	1	no	—
161	#117	—	218	1	no	—
162	#117	—	218	1	no	—
163	#117	#89137	1	1	yes	characteristic, complemented
164	C₂×C₄⋊C₄	—	109	2	no	—
165	C₂×C₄⋊C₄	—	218	1	no	—
166	C₂×C₄⋊C₄	—	218	1	no	—
167	C₂×C₂²⋊C₄	—	109	2	no	—
168	C₂×C₂²⋊C₄	—	109	2	no	—
169	C₂×C₂²⋊C₄	—	109	2	no	—
170	C₂×C₄²	—	109	2	no	—
171	C₂×C₄²	—	218	1	no	—
172	C₂×C₄²	—	218	1	no	—
173	C₂⁴	—	109	1	no	—
174	C₂²×C₄	—	109	1	no	—
175	C₂²×C₄	—	109	1	no	—
176	C₂²×C₄	—	109	1	no	—
177	C₂²×C₄	—	109	1	no	—
178	C₂²×C₄	—	109	1	no	—
179	C₂²×C₄	—	109	1	no	—
180	C₂²×C₄	—	109	1	no	—
181	C₂²×C₄	—	109	1	no	—
182	C₂²×C₄	—	109	1	no	—
183	C₂²×C₄	—	109	1	no	—
184	C₂²×C₄	—	109	1	no	—
185	C₂²×C₄	—	109	1	no	—
186	C₂²×C₄	—	109	1	no	—
187	C₂²×C₄	—	109	4	no	—
188	C₂²×C₄	—	218	2	no	—
189	C₂²×C₄	—	218	2	no	—
190	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	218	2	no	—
193	C₄⋊C₄	—	218	2	no	—
194	C₄⋊C₄	—	218	2	no	—
195	C₄⋊C₄	—	218	2	no	—
196	C₄⋊C₄	—	218	2	no	—
197	C₄⋊C₄	—	218	1	no	—
198	C₂²⋊C₄	—	109	4	no	—
199	C₂²⋊C₄	—	109	2	no	—
200	C₂²⋊C₄	—	218	2	no	—
201	C₂²⋊C₄	—	218	2	no	—
202	C₂²⋊C₄	—	218	2	no	—
203	C₂²⋊C₄	—	218	2	no	—
204	C₂²⋊C₄	C₂×#89137	1	2	yes	—
205	C₄²	—	2	1	no	—
206	C₄²	—	218	2	no	—
207	C₄²	—	218	2	no	—
208	C₄²	—	218	2	no	—
209	C₄²	—	218	2	no	—
210	C₄²	—	218	2	no	—
211	C₄²	—	218	1	no	—
212	C₂³	—	109	1	no	—
213	C₂³	—	109	1	no	—
214	C₂³	—	109	1	no	—
215	C₂³	—	109	1	no	—
216	C₂³	—	109	1	no	—
217	C₂³	—	109	1	no	—
218	C₂³	—	436	2	no	—
219	C₂³	C₂²×#89137	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	109	1	no	—
222	C₂×C₄	—	109	1	no	—
223	C₂×C₄	—	109	1	no	—
224	C₂×C₄	—	109	1	no	—
225	C₂×C₄	—	109	1	no	—
226	C₂×C₄	—	109	1	no	—
227	C₂×C₄	—	109	1	no	—
228	C₂×C₄	—	109	1	no	—
229	C₂×C₄	—	109	1	no	—
230	C₂×C₄	—	109	1	no	—
231	C₂×C₄	—	109	1	no	—
232	C₂×C₄	—	109	1	no	—
233	C₂×C₄	—	109	1	no	—
234	C₂×C₄	—	109	1	no	—
235	C₂×C₄	—	109	1	no	—
236	C₂×C₄	—	109	1	no	—
237	C₂×C₄	—	109	1	no	—
238	C₂×C₄	—	109	1	no	—
239	C₂×C₄	—	109	1	no	—
240	C₂×C₄	—	109	1	no	—
241	C₂×C₄	—	109	1	no	—
242	C₂×C₄	—	109	1	no	—
243	C₂×C₄	—	436	4	no	—
244	C₂×C₄	—	436	4	no	—
245	C₂×C₄	—	436	4	no	—
246	C₂×C₄	—	218	4	no	—
247	C₂×C₄	—	218	4	no	—
248	C₂×C₄	—	218	2	no	—
249	C₂×C₄	C₄×#89137	1	1	yes	characteristic
250	C₂×C₄	C₄×#89137	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
253	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
254	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
255	C₂²	—	4	1	no	—
256	C₂²	—	109	1	no	—
257	C₂²	—	109	1	no	—
258	C₂²	—	109	1	no	—
259	C₂²	—	109	1	no	—
260	C₂²	—	436	1	no	—
261	C₂²	—	436	1	no	—
262	C₂²	—	436	1	no	—
263	C₂²	—	218	2	no	—
264	C₂²	—	218	2	no	—
265	C₂²	—	218	2	no	—
266	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
267	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
268	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	109	1	no	—
271	C₄	—	109	1	no	—
272	C₄	—	109	1	no	—
273	C₄	—	109	1	no	—
274	C₄	—	436	2	no	—
275	C₄	—	436	4	no	—
276	C₄	—	436	4	no	—
277	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
278	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
279	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
280	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	—	109	2	no	—
285	C₂	—	218	1	no	—
286	C₂	—	218	1	no	—
287	C₂	—	218	1	no	—
288	C₂	C₂×(C₂×#89137)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×#89137)⋊C₄
commutator	C₂²×C₁₀₉	C₂×C₄²
Frattini	C₂³	C₂²×#89137
Fitting	#117×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×#95
prime radical, cosocle	C₂³×D₁₀₉	C₂³
hypercenter	#117	#89137
hypocenter	C₁₀₉	#1740
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
11. #16812778	C₄	semidirect
12. #16812778	C₄	semidirect
14. C₂×C₄⋊#89137	C₄	semidirect
18. C₂×C₄×#89137	C₄	semidirect
20. #16812658	C₄	semidirect
23. #16812655	C₄	semidirect
27. #117×C₁₀₉	C₄	semidirect
145. C₁₀₉	#1740	semidirect
163. #117	#89137	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812778	2
#20295903	2
#117×D₁₀₉	2
#1740	109