Subgroups — #28378702

Order 13952.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₀₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812778	C₂	1	1	yes	maximal, characteristic
3	#20295893	C₂	1	4	yes	maximal, complemented
4	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182039	—	2	1	no	—
6	D₁₀₉⋊M₁₆	—	2	1	no	—
7	D₁₀₉⋊M₁₆	—	2	1	no	—
8	D₁₀₉⋊M₁₆	C₄	1	2	yes	complemented
9	D₁₀₉⋊M₁₆	C₂²	1	2	yes	—
10	C₂×#13182038	—	2	1	no	—
11	C₂×D₁₀₉⋊C₈	C₄	1	2	yes	complemented
12	#16812778	—	2	1	no	—
13	#16812778	—	2	1	no	—
14	#16812778	C₄	1	2	yes	complemented
15	#16812778	C₂²	1	2	yes	—
16	C₂×C₄⋊#89137	—	2	1	no	—
17	C₂×C₂²⋊#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	C₂²	1	2	yes	—
23	#16812655	—	2	1	no	—
24	C₄⋊C₄×D₁₀₉	—	2	1	no	—
25	C₂²⋊C₄×D₁₀₉	—	2	1	no	—
26	C₄²×D₁₀₉	C₄	1	2	yes	complemented
27	#117×C₁₀₉	C₄	1	1	yes	characteristic, complemented
28	C₂²×#89137	—	2	2	no	—
29	#13182039	—	2	2	no	—
30	#13182039	—	2	2	no	—
31	#13182038	—	2	2	no	—
32	#13182038	—	2	2	no	—
33	D₁₀₉⋊C₈	—	2	2	no	—
34	D₁₀₉⋊C₈	C₂×C₄	1	4	yes	complemented
35	C₂×#11724786	—	2	2	no	—
36	C₄⋊#89137	—	2	2	no	—
37	C₄⋊#89137	—	2	2	no	—
38	C₂²⋊#89137	—	2	2	no	—
39	C₂²⋊#89137	—	2	2	no	—
40	C₄×#89137	—	2	2	no	—
41	C₄×#89137	C₂×C₄	1	4	yes	complemented
42	C₂³×D₁₀₉	D₄	1	1	yes	characteristic
43	C₂²×Dic₁₀₉	D₄	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	C₂×C₄	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	Q₈	1	1	yes	characteristic
55	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
56	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₁₀₉	—	2	1	no	—
58	#13182008	—	2	2	no	—
59	C₂²⋊Dic₁₀₉	—	2	1	no	—
60	#13182006	—	2	2	no	—
61	C₄×Dic₁₀₉	—	2	2	no	—
62	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	complemented
63	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₀₉	—	2	1	no	—
65	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
66	C₄²×C₁₀₉	C₂×C₄	1	2	yes	complemented
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	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
75	C₂×#89137	—	2	4	no	—
76	C₂×#89137	—	4	4	no	—
77	C₂×C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₀₉	—	2	2	no	—
85	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
89	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
90	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₄×D₁₀₉	—	4	4	no	—
92	C₄×D₁₀₉	C₄²	1	1	yes	characteristic
93	C₄×D₁₀₉	C₄²	1	1	yes	characteristic
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	C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
100	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
101	C₄×D₁₀₉	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
105	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
106	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₄×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
108	#11724786	—	2	4	no	—
109	C₂²×C₁₀₉	—	4	1	no	—
110	C₂²×C₁₀₉	#95	1	1	yes	characteristic
111	C₂²×C₁₀₉	#95	1	1	yes	characteristic
112	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×D₁₀₉	—	2	2	no	—
114	C₂×D₁₀₉	—	2	2	no	—
115	C₂×D₁₀₉	—	2	2	no	—
116	C₂×D₁₀₉	—	4	1	no	—
117	C₂×D₁₀₉	—	4	1	no	—
118	C₂×D₁₀₉	—	4	1	no	—
119	C₂×D₁₀₉	#95	1	1	yes	characteristic
120	C₂×D₁₀₉	#95	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	#89137	—	4	4	no	—
124	C₄×C₁₀₉	—	4	2	no	—
125	C₄×C₁₀₉	#95	1	1	yes	characteristic
126	C₄×C₁₀₉	#95	1	1	yes	characteristic
127	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
128	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Dic₁₀₉	—	4	2	no	—
130	Dic₁₀₉	#95	1	1	yes	characteristic
131	Dic₁₀₉	#95	1	1	yes	characteristic
132	Dic₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	Dic₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₁₀₉	—	2	1	no	—
135	C₂×C₁₀₉	—	2	1	no	—
136	C₂×C₁₀₉	—	2	1	no	—
137	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
138	D₁₀₉	—	2	1	no	—
139	D₁₀₉	—	2	1	no	—
140	D₁₀₉	—	2	1	no	—
141	D₁₀₉	H(3,ℤ/4)	1	2	yes	complemented
142	C₂×H(3,ℤ/4)	—	109	1	no	maximal, Sylow
143	C₁₀₉	C₂×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
144	C₂²×M₁₆	—	109	1	no	—
145	C₂×#117	—	109	1	no	—
146	C₂×#117	—	109	1	no	—
147	H(3,ℤ/4)	—	109	4	no	—
148	C₂³×C₄	—	109	1	no	—
149	C₂×M₁₆	—	109	2	no	—
150	C₂×M₁₆	—	109	2	no	—
151	C₂×M₁₆	—	218	1	no	—
152	C₂×M₁₆	—	218	1	no	—
153	C₂×M₁₆	—	218	1	no	—
154	C₂×M₁₆	—	218	1	no	—
155	C₂²×C₈	—	109	2	no	—
156	#117	—	109	2	no	—
157	#117	—	109	2	no	—
158	#117	—	109	1	no	—
159	#117	—	109	2	no	—
160	#117	—	218	1	no	—
161	#117	—	218	1	no	—
162	#117	—	218	1	no	—
163	#117	—	218	1	no	—
164	#117	#89137	1	1	yes	characteristic, complemented
165	C₂×C₄⋊C₄	—	218	1	no	—
166	C₂×C₄⋊C₄	—	218	1	no	—
167	C₂×C₂²⋊C₄	—	218	1	no	—
168	C₂×C₂²⋊C₄	—	218	1	no	—
169	C₂×C₄²	—	109	2	no	—
170	C₂×C₄²	—	109	2	no	—
171	C₂⁴	—	109	1	no	—
172	C₂²×C₄	—	109	1	no	—
173	C₂²×C₄	—	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	—	218	2	no	—
186	C₂²×C₄	—	218	2	no	—
187	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
188	M₁₆	—	218	2	no	—
189	M₁₆	—	218	2	no	—
190	M₁₆	—	218	2	no	—
191	M₁₆	—	218	2	no	—
192	C₂×C₈	—	109	4	no	—
193	C₂×C₈	—	218	2	no	—
194	C₂×C₈	—	218	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	218	2	no	—
197	C₄⋊C₄	—	218	2	no	—
198	C₄⋊C₄	—	218	2	no	—
199	C₄⋊C₄	—	218	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	218	2	no	—
202	C₂²⋊C₄	—	218	2	no	—
203	C₂²⋊C₄	—	218	2	no	—
204	C₂²⋊C₄	—	218	1	no	—
205	C₄²	—	109	4	no	—
206	C₄²	—	109	2	no	—
207	C₄²	—	218	2	no	—
208	C₄²	—	218	2	no	—
209	C₄²	C₂×#89137	1	2	yes	complemented
210	C₂³	—	109	1	no	—
211	C₂³	—	109	1	no	—
212	C₂³	—	109	1	no	—
213	C₂³	—	109	1	no	—
214	C₂³	—	109	1	no	—
215	C₂³	—	109	1	no	—
216	C₂³	—	436	2	no	—
217	C₂³	C₂²⋊#89137	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	109	1	no	—
220	C₂×C₄	—	109	1	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	—	436	4	no	—
242	C₂×C₄	—	436	4	no	—
243	C₂×C₄	—	218	4	no	—
244	C₂×C₄	—	218	2	no	—
245	C₂×C₄	C₄×#89137	1	1	yes	characteristic
246	C₂×C₄	C₄×#89137	1	1	yes	characteristic
247	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
248	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
249	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
250	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
251	C₈	—	218	4	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	109	1	no	—
254	C₂²	—	109	1	no	—
255	C₂²	—	109	1	no	—
256	C₂²	—	109	1	no	—
257	C₂²	—	436	1	no	—
258	C₂²	—	436	1	no	—
259	C₂²	—	436	1	no	—
260	C₂²	—	218	2	no	—
261	C₂²	—	218	2	no	—
262	C₂²	—	218	2	no	—
263	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
264	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
265	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
266	C₄	—	4	2	no	—
267	C₄	—	109	1	no	—
268	C₄	—	109	1	no	—
269	C₄	—	109	1	no	—
270	C₄	—	109	1	no	—
271	C₄	—	436	2	no	—
272	C₄	—	436	4	no	—
273	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
274	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
275	C₄	C₂×C₄×#89137	1	1	yes	characteristic
276	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	109	2	no	—
281	C₂	—	218	1	no	—
282	C₂	—	218	1	no	—
283	C₂	—	218	1	no	—
284	C₂	C₂×(C₂×#89137)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄×#89137
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×#89137
Fitting	#117×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×#95
prime radical	C₂³×D₁₀₉	D₄
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#117	#89137
hypocenter	C₁₀₉	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #20295893	C₂	semidirect
8. D₁₀₉⋊M₁₆	C₄	semidirect
11. C₂×D₁₀₉⋊C₈	C₄	semidirect
14. #16812778	C₄	semidirect
18. C₂×C₄×#89137	C₄	semidirect
20. #16812658	C₄	semidirect
26. C₄²×D₁₀₉	C₄	semidirect
27. #117×C₁₀₉	C₄	semidirect
34. D₁₀₉⋊C₈	C₂×C₄	semidirect
41. C₄×#89137	C₂×C₄	semidirect
62. C₄×Dic₁₀₉	C₂×C₄	semidirect
66. C₄²×C₁₀₉	C₂×C₄	semidirect
141. D₁₀₉	H(3,ℤ/4)	semidirect
143. C₁₀₉	C₂×H(3,ℤ/4)	semidirect
164. #117	#89137	semidirect
209. C₄²	C₂×#89137	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₀₉⋊M₁₆	2
C₂×#16812778	2
#20295893	2
#117×D₁₀₉	2
C₂×H(3,ℤ/4)	109