Subgroups — #28374945

Order 13952.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295744	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295675	C₂	1	1	yes	maximal, characteristic
3	#20295375	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295367	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295366	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295304	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295267	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295266	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295189	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295188	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295019	C₂	1	1	yes	maximal, characteristic, complemented
15	#416×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182028	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
18	C₂×#13182019	C₂²	1	1	yes	characteristic
19	C₂×#13182017	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
21	#16812698	C₂²	1	1	yes	characteristic
22	#16812697	C₂²	1	1	yes	characteristic
23	#16812693	—	2	1	no	—
24	#16812692	—	2	1	no	—
25	#16812685	—	2	1	no	—
26	#16812682	—	2	1	no	—
27	#16812680	—	2	1	no	—
28	#16812679	C₂²	1	1	yes	characteristic
29	#16812678	C₂²	1	1	yes	characteristic
30	#16812678	C₂²	1	1	yes	characteristic
31	#16812675	C₂²	1	1	yes	characteristic
32	#16812671	—	2	1	no	—
33	C₄×Dic₂₁₈	—	2	1	no	—
34	#16812664	C₂²	1	1	yes	characteristic
35	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
37	#16812657	C₂²	1	1	yes	characteristic
38	C₂×#13182008	—	2	1	no	—
39	#16812633	C₂²	1	1	yes	characteristic
40	#16812632	C₂²	1	1	yes	characteristic
41	#16812630	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812623	C₂²	1	1	yes	characteristic
46	#16812622	C₂²	1	1	yes	characteristic
47	#16812622	C₂²	1	1	yes	characteristic
48	#16812620	C₂²	1	1	yes	characteristic
49	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	#117×C₁₀₉	C₂²	1	1	yes	characteristic
55	#103×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182028	—	2	2	no	—
60	#13182028	—	4	1	no	—
61	C₂×Dic₂₁₈	—	2	1	no	—
62	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
63	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₀₉	D₄	1	1	yes	characteristic
66	C₂²×Dic₁₀₉	—	2	1	no	—
67	C₂²×Dic₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
69	Dic₄₃₆	—	4	1	no	—
70	#13182019	—	4	1	no	—
71	#13182017	—	2	2	no	—
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₈×D₁₀₉	—	4	1	no	—
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	C₂²⋊Dic₁₀₉	—	2	1	no	—
88	C₂²⋊Dic₁₀₉	—	2	1	no	—
89	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	#41×C₁₀₉	—	2	2	no	—
98	#41×C₁₀₉	—	4	1	no	—
99	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₀₉	—	2	1	no	—
101	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₁₀₉	—	2	1	no	—
103	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	—	4	1	no	—
105	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	—	2	1	no	—
111	C₂³×C₁₀₉	—	2	1	no	—
112	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	D₄×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	—	4	2	no	—
118	D₄×C₁₀₉	#41	1	2	yes	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	2	1	no	—
122	C₂×C₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	—	2	1	no	—
124	C₂×C₄×C₁₀₉	—	4	2	no	—
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₁₀₉	—	2	2	no	—
128	C₂²⋊D₁₀₉	—	4	2	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	4	2	no	—
134	C₂×Dic₁₀₉	—	4	2	no	—
135	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
138	C₄×D₁₀₉	—	2	1	no	—
139	C₄×D₁₀₉	—	2	1	no	—
140	Dic₂₁₈	—	2	1	no	—
141	Dic₂₁₈	—	2	1	no	—
142	Dic₂₁₈	—	2	1	no	—
143	Dic₂₁₈	—	2	1	no	—
144	Dic₂₁₈	—	4	2	no	—
145	C₈×C₁₀₉	—	2	1	no	—
146	C₈×C₁₀₉	—	2	1	no	—
147	#11724784	—	2	1	no	—
148	#11724784	—	2	1	no	—
149	C₂²×C₁₀₉	—	2	2	no	—
150	C₂²×C₁₀₉	—	2	1	no	—
151	C₂²×C₁₀₉	—	2	1	no	—
152	C₂²×C₁₀₉	—	2	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	—	4	1	no	—
155	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₀₉	—	2	1	no	—
157	C₂×D₁₀₉	—	2	1	no	—
158	C₂×D₁₀₉	—	2	1	no	—
159	C₄×C₁₀₉	—	2	1	no	—
160	C₄×C₁₀₉	—	4	1	no	—
161	C₄×C₁₀₉	—	4	1	no	—
162	C₄×C₁₀₉	—	4	1	no	—
163	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
164	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₁₀₉	—	2	1	no	—
166	Dic₁₀₉	—	2	1	no	—
167	Dic₁₀₉	—	4	1	no	—
168	Dic₁₀₉	—	4	1	no	—
169	Dic₁₀₉	—	4	1	no	—
170	Dic₁₀₉	—	4	1	no	—
171	Dic₁₀₉	#135	1	2	yes	—
172	C₂×C₁₀₉	—	4	1	no	—
173	C₂×C₁₀₉	—	4	2	no	—
174	C₂×C₁₀₉	#546	1	1	yes	characteristic
175	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
176	C₂×C₁₀₉	#578	1	1	yes	characteristic
177	D₁₀₉	—	4	1	no	—
178	#3302	—	109	1	no	maximal, Sylow
179	C₁₀₉	#3302	1	1	yes	characteristic, complemented, Sylow
180	#556	—	109	1	no	—
181	#547	—	109	1	no	—
182	#487	—	109	1	no	—
183	#485	—	109	1	no	—
184	#485	—	109	1	no	—
185	#474	—	109	1	no	—
186	#467	—	109	1	no	—
187	#467	—	109	1	no	—
188	#462	—	109	1	no	—
189	#452	—	109	1	no	—
190	#452	—	109	1	no	—
191	C₄×SD₁₆	—	109	1	no	—
192	C₈×D₄	—	109	1	no	—
193	#416	—	109	1	no	—
194	#416	D₁₀₉	1	1	yes	characteristic, complemented
195	C₂×#41	—	109	1	no	—
196	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
197	C₂×Q₁₆	—	109	1	no	—
198	C₂×Q₁₆	—	109	1	no	—
199	C₂×SD₁₆	—	109	1	no	—
200	C₂²×C₈	—	109	1	no	—
201	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
202	C₄⋊Q₈	—	109	1	no	—
203	C₄⋊Q₈	—	109	1	no	—
204	#126	—	218	1	no	—
205	#126	—	218	1	no	—
206	#124	—	218	1	no	—
207	#123	—	218	1	no	—
208	#123	—	218	1	no	—
209	C₂²⋊Q₈	—	109	1	no	—
210	C₂²⋊Q₈	—	109	1	no	—
211	C₂²⋊Q₈	—	109	1	no	—
212	C₂²⋊Q₈	—	109	1	no	—
213	#121	—	218	1	no	—
214	C₄×Q₈	—	109	1	no	—
215	C₄×Q₈	—	218	1	no	—
216	C₄×D₄	—	109	1	no	—
217	C₄×D₄	—	109	1	no	—
218	#117	—	109	1	no	—
219	#117	C₂×D₁₀₉	1	1	yes	characteristic
220	C₂×C₄⋊C₄	—	218	1	no	—
221	#107	—	109	1	no	—
222	#107	—	109	1	no	—
223	#106	—	109	1	no	—
224	C₄⋊C₈	—	109	1	no	—
225	#103	—	109	1	no	—
226	#103	—	109	1	no	—
227	#103	—	109	1	no	—
228	#103	—	109	1	no	—
229	#103	—	109	1	no	—
230	#103	C₂×D₁₀₉	1	1	yes	characteristic
231	#103	C₂×D₁₀₉	1	1	yes	characteristic
232	#102	—	109	1	no	—
233	#102	C₂×D₁₀₉	1	1	yes	characteristic
234	#102	C₂×D₁₀₉	1	1	yes	characteristic
235	C₂²⋊C₈	—	109	1	no	—
236	C₂²⋊C₈	—	109	1	no	—
237	C₄×C₈	—	109	1	no	—
238	#41	—	2	2	no	—
239	#41	—	4	1	no	—
240	#41	—	436	1	no	—
241	#41	—	218	2	no	—
242	C₂×Q₈	—	109	1	no	—
243	C₂×Q₈	—	109	1	no	—
244	C₂×Q₈	—	218	1	no	—
245	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	109	1	no	—
248	C₂×D₄	—	218	1	no	—
249	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	109	1	no	—
252	C₂²×C₄	—	218	1	no	—
253	C₂²×C₄	—	218	1	no	—
254	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
255	Q₁₆	—	436	1	no	—
256	Q₁₆	—	436	1	no	—
257	SD₁₆	—	218	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	109	1	no	—
260	C₂×C₈	—	109	1	no	—
261	C₂×C₈	—	436	1	no	—
262	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
264	C₄⋊C₄	—	109	1	no	—
265	C₄⋊C₄	—	109	1	no	—
266	C₄⋊C₄	—	109	1	no	—
267	C₄⋊C₄	—	218	1	no	—
268	C₄⋊C₄	—	218	1	no	—
269	C₄⋊C₄	—	218	1	no	—
270	C₄⋊C₄	—	218	1	no	—
271	C₄⋊C₄	—	218	1	no	—
272	C₄⋊C₄	—	218	1	no	—
273	C₄⋊C₄	—	218	1	no	—
274	C₄⋊C₄	—	218	1	no	—
275	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	109	1	no	—
279	C₂²⋊C₄	—	109	1	no	—
280	C₂²⋊C₄	—	218	1	no	—
281	C₂²⋊C₄	—	218	1	no	—
282	C₂²⋊C₄	—	218	1	no	—
283	C₂²⋊C₄	—	218	1	no	—
284	C₂²⋊C₄	—	218	1	no	—
285	C₂²⋊C₄	—	218	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	109	1	no	—
288	C₄²	—	218	1	no	—
289	C₄²	—	218	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	109	1	no	—
292	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	436	2	no	—
296	Q₈	—	218	1	no	—
297	Q₈	—	218	1	no	—
298	Q₈	—	218	1	no	—
299	Q₈	—	218	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	4	2	no	—
302	D₄	—	436	2	no	—
303	D₄	—	218	2	no	—
304	D₄	#13182029	1	2	yes	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	109	1	no	—
312	C₂×C₄	—	109	1	no	—
313	C₂×C₄	—	109	1	no	—
314	C₂×C₄	—	436	2	no	—
315	C₂×C₄	—	436	2	no	—
316	C₂×C₄	—	218	1	no	—
317	C₂×C₄	—	218	1	no	—
318	C₂×C₄	—	218	1	no	—
319	C₂×C₄	—	218	1	no	—
320	C₂×C₄	—	218	1	no	—
321	C₂×C₄	—	218	1	no	—
322	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	218	1	no	—
327	C₈	—	218	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	218	1	no	—
335	C₂²	—	218	1	no	—
336	C₂²	—	218	1	no	—
337	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	109	2	no	—
343	C₄	—	436	1	no	—
344	C₄	—	436	1	no	—
345	C₄	—	436	1	no	—
346	C₄	—	436	1	no	—
347	C₄	—	218	1	no	—
348	C₄	—	218	1	no	—
349	C₄	C₂×#13182029	1	1	yes	characteristic
350	C₄	#16812744	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	4	2	no	—
353	C₂	—	436	1	no	—
354	C₂	#20295666	1	1	yes	characteristic
355	C₂	#135×D₁₀₉	1	1	yes	characteristic
356	C₂	#20295840	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#416×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295675	C₂
hypercenter	#416	D₁₀₉
hypocenter	C₁₀₉	#3302
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
1. #20295744	C₂	semidirect
3. #20295375	C₂	semidirect
4. #20295367	C₂	semidirect
5. #20295366	C₂	semidirect
6. #20295304	C₂	semidirect
7. #20295267	C₂	semidirect
8. #20295266	C₂	semidirect
9. #20295235	C₂	semidirect
10. #20295189	C₂	semidirect
11. #20295188	C₂	semidirect
12. #20295117	C₂	semidirect
13. C₈×C₂²⋊D₁₀₉	C₂	semidirect
14. #20295019	C₂	semidirect
15. #416×C₁₀₉	C₂	semidirect
179. C₁₀₉	#3302	semidirect
194. #416	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295744	2
#20295675	2
#20295375	2
#20295367	2
#20295366	2
#20295304	2
#20295267	2
#20295266	2
#20295235	2
#20295189	2
#20295188	2
#20295117	2
C₈×C₂²⋊D₁₀₉	2
#20295019	2
#416×C₁₀₉	2
#3302	109