Subgroups — #28379961

Order 13952.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296078	C₂	1	2	yes	maximal, complemented
2	#20296056	C₂	1	2	yes	maximal, complemented
3	#20296055	C₂	1	2	yes	maximal, complemented
4	#20296053	C₂	1	2	yes	maximal, complemented
5	#20296049	C₂	1	1	yes	maximal, characteristic, complemented
6	#20296046	C₂	1	1	yes	maximal, characteristic, complemented
7	#20296044	C₂	1	1	yes	maximal, characteristic, complemented
8	#20296031	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812766	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₀₉⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295494	C₂	1	1	yes	maximal, characteristic
12	C₂×#13182039	C₂²	1	2	yes	—
13	C₂×D₁₀₉⋊C₈	C₂²	1	2	yes	—
14	C₂²×#11724786	C₂²	1	2	yes	—
15	#16812766	C₂²	1	1	yes	characteristic
16	#16812766	C₂²	1	1	yes	characteristic
17	#16812766	C₂²	1	1	yes	characteristic
18	#16812766	C₂²	1	2	yes	—
19	#16812766	C₂²	1	2	yes	—
20	#16812766	C₂²	1	1	yes	characteristic
21	C₄⋊#11724786	C₂²	1	2	yes	—
22	#16812756	C₂²	1	2	yes	—
23	#16812756	C₂²	1	2	yes	—
24	C₂²⋊#11724786	C₂²	1	2	yes	—
25	#16812754	C₂²	1	2	yes	—
26	#16812754	C₂²	1	2	yes	—
27	C₄×#11724786	C₂²	1	2	yes	—
28	C₂×#13182029	C₄	1	1	yes	characteristic
29	#16812664	C₄	1	1	yes	characteristic
30	#16812664	C₂²	1	1	yes	characteristic
31	#16812661	C₄	1	1	yes	characteristic
32	#16812661	C₂²	1	1	yes	characteristic
33	#16812659	C₄	1	1	yes	characteristic
34	#16812659	C₄	1	1	yes	characteristic
35	#16812659	C₂²	1	1	yes	characteristic
36	#16812659	C₂²	1	1	yes	characteristic
37	#16812656	C₄	1	1	yes	characteristic
38	#16812656	C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
40	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
41	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
42	#117×C₁₀₉	C₄	1	1	yes	characteristic
43	#13182039	—	4	2	no	—
44	D₁₀₉⋊C₈	—	4	2	no	—
45	C₂×#11724786	—	4	2	no	—
46	C₂×#11724786	C₂³	1	2	yes	—
47	C₂×#11724786	C₂³	1	2	yes	—
48	C₂×#11724786	C₂³	1	2	yes	—
49	C₂×#11724786	C₂³	1	2	yes	—
50	#13182029	—	2	4	no	—
51	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
52	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
63	#13182008	C₂×C₄	1	1	yes	characteristic
64	#13182008	C₂×C₄	1	1	yes	characteristic
65	#13182008	C₂×C₄	1	1	yes	characteristic
66	#13182008	C₂×C₄	1	1	yes	characteristic
67	#13182006	C₂×C₄	1	1	yes	characteristic
68	#13182006	C₂×C₄	1	1	yes	characteristic
69	#13182006	C₂×C₄	1	1	yes	characteristic
70	#13182006	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
76	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
78	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
85	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
86	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	—	2	2	no	—
90	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊D₁₀₉	—	2	2	no	—
97	C₂²⋊D₁₀₉	M₁₆	1	4	yes	complemented
98	C₂×Dic₁₀₉	—	2	2	no	—
99	C₂×Dic₁₀₉	—	2	1	no	—
100	C₂×Dic₁₀₉	—	2	1	no	—
101	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
107	D₄₃₆	—	2	2	no	—
108	C₄×D₁₀₉	—	2	2	no	—
109	C₄×D₁₀₉	—	2	2	no	—
110	C₄×D₁₀₉	—	2	2	no	—
111	C₄×D₁₀₉	—	2	2	no	—
112	C₄×D₁₀₉	—	2	1	no	—
113	C₄×D₁₀₉	—	2	1	no	—
114	C₄×D₁₀₉	—	2	1	no	—
115	C₄×D₁₀₉	—	2	1	no	—
116	Dic₂₁₈	—	2	2	no	—
117	#11724786	—	2	2	no	—
118	#11724786	—	2	2	no	—
119	#11724786	—	2	2	no	—
120	#11724786	—	2	2	no	—
121	C₂²×C₁₀₉	—	2	1	no	—
122	C₂²×C₁₀₉	—	2	1	no	—
123	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
124	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₁₀₉	—	2	1	no	—
126	C₂×D₁₀₉	—	2	1	no	—
127	C₂×D₁₀₉	—	2	1	no	—
128	C₂×D₁₀₉	—	2	1	no	—
129	C₂×D₁₀₉	—	2	1	no	—
130	C₂×D₁₀₉	C₂×M₁₆	1	2	yes	—
131	C₄×C₁₀₉	—	2	1	no	—
132	C₄×C₁₀₉	—	2	1	no	—
133	C₄×C₁₀₉	—	2	1	no	—
134	C₄×C₁₀₉	—	2	1	no	—
135	C₄×C₁₀₉	—	2	1	no	—
136	C₄×C₁₀₉	—	2	1	no	—
137	Dic₁₀₉	—	2	1	no	—
138	Dic₁₀₉	—	2	1	no	—
139	Dic₁₀₉	—	2	1	no	—
140	Dic₁₀₉	C₂×M₁₆	1	2	yes	—
141	Dic₁₀₉	#131	1	2	yes	—
142	Dic₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
143	Dic₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂×C₁₀₉	—	2	2	no	—
145	C₂×C₁₀₉	#520	1	1	yes	characteristic
146	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
147	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
148	D₁₀₉	—	2	2	no	—
149	D₁₀₉	—	4	1	no	—
150	#2990	—	109	1	no	maximal, Sylow
151	C₁₀₉	#2990	1	1	yes	characteristic, complemented, Sylow
152	C₄×#41	—	109	1	no	—
153	#446	—	109	2	no	—
154	#435	—	109	2	no	—
155	#435	—	109	2	no	—
156	C₈×D₄	—	109	2	no	—
157	#432	—	109	1	no	—
158	#432	—	109	1	no	—
159	#431	—	109	1	no	—
160	#423	—	109	1	no	—
161	C₂×C₄⋊C₈	—	109	1	no	—
162	C₂×C₄⋊C₈	—	109	1	no	—
163	C₂×#41	—	109	1	no	—
164	C₂×M₁₆	—	109	2	no	—
165	C₂²×C₈	—	109	2	no	—
166	C₂²×C₈	—	109	2	no	—
167	C₄×Q₈	—	109	1	no	—
168	C₄×Q₈	—	109	1	no	—
169	C₄×D₄	—	109	1	no	—
170	C₄×D₄	—	109	1	no	—
171	C₄×D₄	—	109	1	no	—
172	C₄×D₄	—	109	1	no	—
173	C₄×D₄	—	109	1	no	—
174	C₄×D₄	—	109	1	no	—
175	#117	—	109	1	no	—
176	#117	—	109	1	no	—
177	#117	#89137	1	1	yes	characteristic
178	C₂×C₄²	—	109	1	no	—
179	C₂×C₄²	—	109	1	no	—
180	C₂×C₄²	—	109	1	no	—
181	C₄⋊C₈	—	109	1	no	—
182	C₄⋊C₈	—	109	1	no	—
183	C₄⋊C₈	—	109	1	no	—
184	C₄⋊C₈	—	109	2	no	—
185	C₄⋊C₈	—	109	2	no	—
186	C₄⋊C₈	—	109	2	no	—
187	C₄⋊C₈	—	109	1	no	—
188	C₂²⋊C₈	—	109	2	no	—
189	C₂²⋊C₈	—	109	2	no	—
190	C₂²⋊C₈	—	109	2	no	—
191	#97	—	109	2	no	—
192	#97	—	109	2	no	—
193	C₄×C₈	—	109	2	no	—
194	#41	—	218	4	no	—
195	C₂×Q₈	—	109	1	no	—
196	C₂×D₄	—	109	1	no	—
197	C₂×D₄	—	109	1	no	—
198	C₂×D₄	—	109	1	no	—
199	C₂²×C₄	—	109	1	no	—
200	C₂²×C₄	—	109	1	no	—
201	C₂²×C₄	—	109	1	no	—
202	C₂²×C₄	—	109	1	no	—
203	C₂²×C₄	—	109	1	no	—
204	C₂²×C₄	—	109	1	no	—
205	C₂²×C₄	—	109	1	no	—
206	C₂²×C₄	—	109	1	no	—
207	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
208	M₁₆	—	436	2	no	—
209	C₂×C₈	—	109	2	no	—
210	C₂×C₈	—	109	2	no	—
211	C₂×C₈	—	109	2	no	—
212	C₂×C₈	—	109	2	no	—
213	C₂×C₈	—	436	2	no	—
214	C₂×C₈	—	436	2	no	—
215	C₄⋊C₄	—	109	1	no	—
216	C₄⋊C₄	—	109	1	no	—
217	C₄⋊C₄	—	109	1	no	—
218	C₄⋊C₄	—	109	1	no	—
219	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic
220	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic
221	C₂²⋊C₄	—	109	1	no	—
222	C₂²⋊C₄	—	109	1	no	—
223	C₂²⋊C₄	—	109	1	no	—
224	C₂²⋊C₄	—	109	1	no	—
225	C₂²⋊C₄	C₂×#89137	1	1	yes	characteristic
226	C₂²⋊C₄	C₂×#89137	1	1	yes	characteristic
227	C₄²	—	109	1	no	—
228	C₄²	—	109	1	no	—
229	C₄²	—	109	1	no	—
230	C₄²	—	109	1	no	—
231	C₄²	—	109	1	no	—
232	C₄²	—	109	1	no	—
233	C₄²	—	109	1	no	—
234	C₄²	—	109	1	no	—
235	C₄²	C₂×#89137	1	1	yes	characteristic
236	C₄²	C₂×#89137	1	1	yes	characteristic
237	C₂³	—	109	1	no	—
238	C₂³	—	109	1	no	—
239	C₂³	C₂²×#89137	1	1	yes	characteristic
240	Q₈	—	218	2	no	—
241	D₄	—	109	4	no	—
242	D₄	—	218	2	no	—
243	D₄	—	218	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	109	1	no	—
246	C₂×C₄	—	109	1	no	—
247	C₂×C₄	—	109	1	no	—
248	C₂×C₄	—	109	1	no	—
249	C₂×C₄	—	109	1	no	—
250	C₂×C₄	—	109	1	no	—
251	C₂×C₄	—	218	1	no	—
252	C₂×C₄	—	218	1	no	—
253	C₂×C₄	—	218	2	no	—
254	C₂×C₄	—	218	1	no	—
255	C₂×C₄	—	218	1	no	—
256	C₂×C₄	—	218	2	no	—
257	C₂×C₄	—	218	2	no	—
258	C₂×C₄	—	218	2	no	—
259	C₂×C₄	—	218	1	no	—
260	C₂×C₄	—	218	1	no	—
261	C₂×C₄	—	218	2	no	—
262	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
263	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
264	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
265	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
266	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
267	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
268	C₈	—	218	2	no	—
269	C₈	—	218	2	no	—
270	C₈	—	218	2	no	—
271	C₈	—	218	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	109	2	no	—
275	C₂²	—	218	1	no	—
276	C₂²	—	218	1	no	—
277	C₂²	—	218	1	no	—
278	C₂²	—	218	1	no	—
279	C₂²	—	218	1	no	—
280	C₂²	D₁₀₉⋊M₁₆	1	2	yes	—
281	C₂²	C₂³×#89137	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	109	1	no	—
289	C₄	—	109	1	no	—
290	C₄	—	109	2	no	—
291	C₄	—	109	2	no	—
292	C₄	—	218	1	no	—
293	C₄	—	218	1	no	—
294	C₄	—	218	1	no	—
295	C₂	—	2	2	no	—
296	C₂	—	436	1	no	—
297	C₂	—	218	2	no	—
298	C₂	#20296095	1	1	yes	characteristic
299	C₂	C₂×D₁₀₉⋊M₁₆	1	1	yes	characteristic
300	C₂	D₁₀₉⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#117×C₁₀₉	C₄
prime radical	C₂×#13182029	C₄
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	#117	#89137
hypocenter	C₁₀₉	#2990
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #20296078	C₂	semidirect
2. #20296056	C₂	semidirect
3. #20296055	C₂	semidirect
4. #20296053	C₂	semidirect
5. #20296049	C₂	semidirect
6. #20296046	C₂	semidirect
7. #20296044	C₂	semidirect
8. #20296031	C₂	semidirect
9. C₂×#16812766	C₂	semidirect
10. D₁₀₉⋊(C₄⋊C₈)	C₂	semidirect
97. C₂²⋊D₁₀₉	M₁₆	semidirect
151. C₁₀₉	#2990	semidirect

Maximal subgroups

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

subgroup	index
#20296078	2
#20296056	2
#20296055	2
#20296053	2
#20296049	2
#20296046	2
#20296044	2
#20296031	2
C₂×#16812766	2
D₁₀₉⋊(C₄⋊C₈)	2
#20295494	2
#2990	109