Subgroups — #28374939

Order 13952.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295726	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295688	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295381	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295370	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295346	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295261	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295254	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295241	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295170	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×D₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295021	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182030	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×#13182017	C₂²	1	1	yes	characteristic
19	C₂×#13182014	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
22	#16812692	—	2	1	no	—
23	#16812691	C₂²	1	1	yes	characteristic
24	#16812687	—	2	1	no	—
25	#16812687	C₂²	1	1	yes	characteristic
26	#16812682	—	2	1	no	—
27	#16812679	—	2	1	no	—
28	#16812679	C₂²	1	1	yes	characteristic
29	#16812674	—	2	1	no	—
30	#16812674	C₂²	1	1	yes	characteristic
31	#16812672	—	2	1	no	—
32	#16812670	C₂²	1	1	yes	characteristic
33	#16812670	C₂²	1	1	yes	characteristic
34	C₄×D₄₃₆	—	2	1	no	—
35	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
37	#16812657	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
39	#16812632	C₂²	1	1	yes	characteristic
40	#16812630	C₂²	1	1	yes	characteristic
41	C₄⋊#11724784	C₂²	1	1	yes	characteristic
42	#16812624	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812623	C₂²	1	1	yes	characteristic
45	#16812623	C₂²	1	1	yes	characteristic
46	#16812622	C₂²	1	1	yes	characteristic
47	#16812621	C₂²	1	1	yes	characteristic
48	#16812621	C₂²	1	1	yes	characteristic
49	#16812619	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic
51	#16812611	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182030	—	2	2	no	—
60	#13182030	—	4	1	no	—
61	C₂×Dic₂₁₈	—	2	1	no	—
62	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
63	Q₈×D₁₀₉	—	2	2	no	—
64	Q₈×D₁₀₉	—	4	1	no	—
65	C₂×D₄₃₆	—	2	1	no	—
66	C₂×D₄₃₆	—	2	1	no	—
67	C₂×D₄₃₆	D₄	1	1	yes	characteristic
68	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
70	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	—	2	1	no	—
72	C₂×C₄×D₁₀₉	—	2	1	no	—
73	C₂×C₄×D₁₀₉	—	2	1	no	—
74	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
76	#13182017	—	4	1	no	—
77	#13182014	—	4	1	no	—
78	C₂×#11724784	C₂³	1	1	yes	characteristic
79	C₂×#11724784	C₂³	1	1	yes	characteristic
80	C₈×D₁₀₉	—	4	1	no	—
81	C₈×D₁₀₉	—	4	1	no	—
82	C₄⋊Dic₁₀₉	—	2	1	no	—
83	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
84	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
86	#13182008	—	2	1	no	—
87	#13182008	—	2	1	no	—
88	#13182008	—	2	1	no	—
89	C₂²⋊Dic₁₀₉	—	2	1	no	—
90	C₂²⋊Dic₁₀₉	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	—	2	1	no	—
94	#13182006	—	2	1	no	—
95	#13182006	—	2	1	no	—
96	#13182006	D₄	1	1	yes	characteristic
97	#13182006	D₄	1	1	yes	characteristic
98	C₄×Dic₁₀₉	—	2	1	no	—
99	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₀₉	—	2	1	no	—
102	SD₁₆×C₁₀₉	—	2	2	no	—
103	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
109	C₄²×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂³×C₁₀₉	—	2	1	no	—
112	C₂²×D₁₀₉	—	2	1	no	—
113	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	#41	1	2	yes	—
117	D₄×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
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	—	8	1	no	—
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₁₀₉	—	4	2	no	—
127	C₂²⋊D₁₀₉	—	8	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
133	D₄₃₆	—	2	2	no	—
134	D₄₃₆	—	2	1	no	—
135	D₄₃₆	—	2	1	no	—
136	D₄₃₆	—	4	2	no	—
137	D₄₃₆	—	4	2	no	—
138	C₄×D₁₀₉	—	2	1	no	—
139	C₄×D₁₀₉	—	2	1	no	—
140	C₄×D₁₀₉	—	2	1	no	—
141	C₄×D₁₀₉	—	2	1	no	—
142	C₄×D₁₀₉	—	4	2	no	—
143	C₄×D₁₀₉	—	4	2	no	—
144	C₄×D₁₀₉	—	8	1	no	—
145	Dic₂₁₈	—	2	1	no	—
146	Dic₂₁₈	—	2	1	no	—
147	Dic₂₁₈	—	4	2	no	—
148	C₈×C₁₀₉	—	2	1	no	—
149	C₈×C₁₀₉	—	2	1	no	—
150	#11724784	—	2	1	no	—
151	#11724784	—	2	1	no	—
152	C₂²×C₁₀₉	—	4	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	C₂×D₁₀₉	—	2	1	no	—
160	C₂×D₁₀₉	—	2	1	no	—
161	C₂×D₁₀₉	—	4	1	no	—
162	C₂×D₁₀₉	—	4	1	no	—
163	C₂×D₁₀₉	—	4	1	no	—
164	C₂×D₁₀₉	#135	1	2	yes	—
165	C₄×C₁₀₉	—	2	2	no	—
166	C₄×C₁₀₉	—	2	1	no	—
167	C₄×C₁₀₉	—	2	1	no	—
168	C₄×C₁₀₉	—	4	1	no	—
169	C₄×C₁₀₉	—	4	1	no	—
170	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
171	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₁₀₉	—	2	1	no	—
173	Dic₁₀₉	—	2	1	no	—
174	Dic₁₀₉	—	4	1	no	—
175	Dic₁₀₉	—	4	1	no	—
176	Dic₁₀₉	—	4	1	no	—
177	C₂×C₁₀₉	—	8	1	no	—
178	C₂×C₁₀₉	#546	1	1	yes	characteristic
179	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
180	C₂×C₁₀₉	#577	1	1	yes	characteristic
181	D₁₀₉	—	2	2	no	—
182	D₁₀₉	—	4	1	no	—
183	D₁₀₉	—	8	1	no	—
184	#3301	—	109	1	no	maximal, Sylow
185	C₁₀₉	#3301	1	1	yes	characteristic, complemented, Sylow
186	#553	—	109	1	no	—
187	#548	—	109	1	no	—
188	#488	—	109	1	no	—
189	#485	—	109	1	no	—
190	#481	—	109	1	no	—
191	#479	—	109	1	no	—
192	#466	—	109	1	no	—
193	#465	—	109	1	no	—
194	#463	—	109	1	no	—
195	#449	—	109	1	no	—
196	#448	—	109	1	no	—
197	C₄×SD₁₆	D₁₀₉	1	1	yes	characteristic, complemented
198	C₈×D₄	—	109	1	no	—
199	#416	—	109	1	no	—
200	C₂×#103	—	109	1	no	—
201	C₂×#41	—	109	1	no	—
202	C₂²×Q₈	—	109	1	no	—
203	C₂×SD₁₆	—	109	1	no	—
204	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
205	C₂×D₈	—	109	1	no	—
206	C₂²×C₈	—	109	1	no	—
207	C₂²×C₈	—	109	1	no	—
208	#126	—	218	1	no	—
209	#125	—	109	1	no	—
210	#124	—	109	1	no	—
211	#124	—	218	1	no	—
212	#123	—	218	1	no	—
213	C₂²⋊Q₈	—	109	1	no	—
214	C₂²⋊Q₈	—	218	1	no	—
215	#121	—	109	1	no	—
216	#121	—	109	1	no	—
217	#121	—	109	1	no	—
218	#121	—	218	1	no	—
219	#121	—	218	1	no	—
220	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
221	C₄×D₄	—	109	1	no	—
222	C₄×D₄	—	218	1	no	—
223	C₄×D₄	—	218	1	no	—
224	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
225	#117	—	109	1	no	—
226	C₂×C₄⋊C₄	—	109	1	no	—
227	#107	—	109	1	no	—
228	#106	—	109	1	no	—
229	#106	C₂×D₁₀₉	1	1	yes	characteristic
230	C₄⋊C₈	—	109	1	no	—
231	#103	—	109	1	no	—
232	#103	—	109	1	no	—
233	#103	—	109	1	no	—
234	#103	—	109	1	no	—
235	#103	—	109	1	no	—
236	#103	C₂×D₁₀₉	1	1	yes	characteristic
237	#102	—	109	1	no	—
238	#102	—	109	1	no	—
239	#102	—	109	1	no	—
240	#102	C₂×D₁₀₉	1	1	yes	characteristic
241	C₂²⋊C₈	—	109	1	no	—
242	C₂²⋊C₈	—	109	1	no	—
243	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
244	#41	—	436	1	no	—
245	#41	—	218	2	no	—
246	C₂×Q₈	—	109	1	no	—
247	C₂×Q₈	—	436	1	no	—
248	C₂×Q₈	—	218	1	no	—
249	C₂×Q₈	—	218	2	no	—
250	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
251	C₂×D₄	—	109	1	no	—
252	C₂×D₄	—	109	1	no	—
253	C₂×D₄	—	218	1	no	—
254	C₂×D₄	—	218	1	no	—
255	C₂×D₄	—	218	1	no	—
256	C₂×D₄	—	218	1	no	—
257	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	109	1	no	—
260	C₂²×C₄	—	109	1	no	—
261	C₂²×C₄	—	218	1	no	—
262	C₂²×C₄	—	218	1	no	—
263	C₂²×C₄	—	218	1	no	—
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	436	1	no	—
266	D₈	—	436	1	no	—
267	C₂×C₈	—	109	1	no	—
268	C₂×C₈	—	109	1	no	—
269	C₂×C₈	—	436	1	no	—
270	C₂×C₈	—	436	1	no	—
271	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	109	1	no	—
275	C₄⋊C₄	—	109	1	no	—
276	C₄⋊C₄	—	109	1	no	—
277	C₄⋊C₄	—	218	1	no	—
278	C₄⋊C₄	—	218	1	no	—
279	C₄⋊C₄	—	218	1	no	—
280	C₄⋊C₄	—	218	1	no	—
281	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	109	1	no	—
285	C₂²⋊C₄	—	109	1	no	—
286	C₂²⋊C₄	—	218	1	no	—
287	C₂²⋊C₄	—	218	1	no	—
288	C₂²⋊C₄	—	218	1	no	—
289	C₂²⋊C₄	—	218	1	no	—
290	C₂²⋊C₄	—	218	1	no	—
291	C₂²⋊C₄	—	218	1	no	—
292	C₂²⋊C₄	—	218	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	218	1	no	—
295	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	109	1	no	—
298	C₂³	—	109	1	no	—
299	C₂³	—	218	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	436	2	no	—
302	Q₈	—	218	1	no	—
303	Q₈	—	218	1	no	—
304	Q₈	#13182029	1	2	yes	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	872	1	no	—
308	D₄	—	436	2	no	—
309	D₄	—	436	2	no	—
310	D₄	—	436	2	no	—
311	D₄	—	218	2	no	—
312	D₄	—	218	1	no	—
313	D₄	—	218	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	8	1	no	—
318	C₂×C₄	—	872	1	no	—
319	C₂×C₄	—	109	1	no	—
320	C₂×C₄	—	109	1	no	—
321	C₂×C₄	—	436	2	no	—
322	C₂×C₄	—	436	2	no	—
323	C₂×C₄	—	218	1	no	—
324	C₂×C₄	—	218	1	no	—
325	C₂×C₄	—	218	1	no	—
326	C₂×C₄	—	218	1	no	—
327	C₂×C₄	—	218	1	no	—
328	C₂×C₄	—	218	1	no	—
329	C₂×C₄	—	218	1	no	—
330	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
331	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	218	1	no	—
336	C₈	—	218	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	109	2	no	—
341	C₂²	—	436	1	no	—
342	C₂²	—	436	1	no	—
343	C₂²	—	436	1	no	—
344	C₂²	—	218	1	no	—
345	C₂²	—	218	1	no	—
346	C₂²	—	218	1	no	—
347	C₂²	—	218	1	no	—
348	C₂²	—	218	1	no	—
349	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	436	1	no	—
356	C₄	—	436	1	no	—
357	C₄	—	436	1	no	—
358	C₄	—	218	1	no	—
359	C₄	—	218	1	no	—
360	C₄	C₂×#13182029	1	1	yes	characteristic
361	C₄	#16812745	1	1	yes	characteristic
362	C₂	—	8	1	no	—
363	C₂	—	872	1	no	—
364	C₂	—	436	1	no	—
365	C₂	—	218	2	no	—
366	C₂	#20295672	1	1	yes	characteristic
367	C₂	#135×D₁₀₉	1	1	yes	characteristic
368	C₂	#20295832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×SD₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	C₄×SD₁₆	D₁₀₉
hypocenter	C₁₀₉	#3301
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #20295726	C₂	semidirect
2. #20295688	C₂	semidirect
3. #20295381	C₂	semidirect
4. #20295370	C₂	semidirect
5. #20295346	C₂	semidirect
6. #20295333	C₂	semidirect
7. #20295261	C₂	semidirect
8. #20295254	C₂	semidirect
9. #20295241	C₂	semidirect
10. #20295170	C₂	semidirect
11. #20295163	C₂	semidirect
12. C₈×D₄₃₆	C₂	semidirect
13. #20295021	C₂	semidirect
14. #103×D₁₀₉	C₂	semidirect
15. C₄×SD₁₆×C₁₀₉	C₂	semidirect
185. C₁₀₉	#3301	semidirect
197. C₄×SD₁₆	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295726	2
#20295688	2
#20295381	2
#20295370	2
#20295346	2
#20295333	2
#20295261	2
#20295254	2
#20295241	2
#20295170	2
#20295163	2
C₈×D₄₃₆	2
#20295021	2
#103×D₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#3301	109