Subgroups — #28376069

Order 13952.

418 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295758	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295697	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295416	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295412	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295401	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295244	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295224	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295222	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295220	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295219	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295155	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295136	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295135	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295111	C₂	1	1	yes	maximal, characteristic, complemented
15	#463×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182029	—	2	1	no	—
17	C₂×#13182016	C₂²	1	1	yes	characteristic
18	C₂×#13182016	C₂²	1	1	yes	characteristic
19	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
20	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
21	C₂×#13182014	C₂²	1	1	yes	characteristic
22	C₂×#13182014	C₂²	1	1	yes	characteristic
23	C₂×#13182014	C₂²	1	1	yes	characteristic
24	#16812696	C₂²	1	1	yes	characteristic
25	#16812695	C₂²	1	1	yes	characteristic
26	#16812695	C₂²	1	1	yes	characteristic
27	#16812695	C₂²	1	1	yes	characteristic
28	#16812688	C₂²	1	1	yes	characteristic
29	#16812685	—	2	1	no	—
30	#16812683	—	2	1	no	—
31	#16812681	—	2	1	no	—
32	#16812673	—	2	1	no	—
33	#16812672	—	2	1	no	—
34	#16812672	—	2	1	no	—
35	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
37	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
38	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
39	#16812661	C₂²	1	1	yes	characteristic
40	#16812661	C₂²	1	1	yes	characteristic
41	#16812659	—	2	1	no	—
42	#16812630	C₂²	1	1	yes	characteristic
43	#16812628	C₂²	1	1	yes	characteristic
44	#16812628	C₂²	1	1	yes	characteristic
45	#16812621	C₂²	1	1	yes	characteristic
46	#16812621	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	#16812610	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	#124×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182029	—	4	4	no	—
60	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
61	C₂×D₄₃₆	—	2	1	no	—
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
64	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
65	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
67	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
68	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
69	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
70	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
71	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
72	C₂²×Dic₁₀₉	—	2	1	no	—
73	C₂×C₄×D₁₀₉	—	2	1	no	—
74	C₂×C₄×D₁₀₉	—	2	1	no	—
75	C₂×C₄×D₁₀₉	—	2	1	no	—
76	#13182016	—	2	2	no	—
77	#13182016	—	4	1	no	—
78	D₈₇₂	—	4	1	no	—
79	D₈₇₂	—	4	1	no	—
80	#13182014	—	2	2	no	—
81	#13182014	—	2	2	no	—
82	#13182014	—	4	1	no	—
83	C₂×#11724784	C₂³	1	1	yes	characteristic
84	C₂×#11724784	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₁₀₉	—	2	1	no	—
86	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
87	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
88	#13182008	D₄	1	1	yes	characteristic
89	#13182008	D₄	1	1	yes	characteristic
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	C₄×Dic₁₀₉	—	2	1	no	—
97	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
98	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₁₀₉	—	2	1	no	—
103	SD₁₆×C₁₀₉	—	4	1	no	—
104	D₈×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	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
109	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
110	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
111	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
112	C₂³×C₁₀₉	—	2	1	no	—
113	C₂³×C₁₀₉	—	2	1	no	—
114	C₂²×D₁₀₉	—	2	1	no	—
115	C₂²×D₁₀₉	—	2	1	no	—
116	C₂²×D₁₀₉	—	2	1	no	—
117	Q₈×C₁₀₉	—	2	1	no	—
118	Q₈×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	—	2	1	no	—
120	D₄×C₁₀₉	—	2	1	no	—
121	D₄×C₁₀₉	—	2	1	no	—
122	D₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	—	2	1	no	—
124	C₂×C₄×C₁₀₉	—	2	1	no	—
125	C₂×C₄×C₁₀₉	—	4	2	no	—
126	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₁₀₉	—	4	2	no	—
130	C₂²⋊D₁₀₉	—	4	2	no	—
131	C₂²⋊D₁₀₉	—	4	2	no	—
132	C₂²⋊D₁₀₉	—	8	1	no	—
133	C₂²⋊D₁₀₉	—	8	1	no	—
134	C₂²⋊D₁₀₉	—	8	1	no	—
135	C₂×Dic₁₀₉	—	2	1	no	—
136	C₂×Dic₁₀₉	—	8	1	no	—
137	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
141	D₄₃₆	—	2	1	no	—
142	D₄₃₆	—	2	1	no	—
143	D₄₃₆	—	2	1	no	—
144	D₄₃₆	—	2	1	no	—
145	D₄₃₆	—	2	1	no	—
146	D₄₃₆	—	2	1	no	—
147	D₄₃₆	—	4	2	no	—
148	D₄₃₆	—	8	1	no	—
149	C₄×D₁₀₉	—	4	2	no	—
150	C₄×D₁₀₉	—	4	2	no	—
151	C₄×D₁₀₉	—	8	1	no	—
152	Dic₂₁₈	—	2	2	no	—
153	C₈×C₁₀₉	—	2	1	no	—
154	C₈×C₁₀₉	—	2	1	no	—
155	#11724784	—	2	1	no	—
156	#11724784	#41	1	2	yes	complemented
157	C₂²×C₁₀₉	—	4	1	no	—
158	C₂²×C₁₀₉	—	4	1	no	—
159	C₂²×C₁₀₉	—	4	1	no	—
160	C₂²×C₁₀₉	—	4	1	no	—
161	C₂²×C₁₀₉	—	4	1	no	—
162	C₂²×C₁₀₉	—	4	1	no	—
163	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₁₀₉	—	4	1	no	—
165	C₂×D₁₀₉	—	4	1	no	—
166	C₂×D₁₀₉	—	4	1	no	—
167	C₂×D₁₀₉	—	4	1	no	—
168	C₂×D₁₀₉	—	4	1	no	—
169	C₂×D₁₀₉	—	4	1	no	—
170	C₂×D₁₀₉	—	4	1	no	—
171	C₂×D₁₀₉	—	4	1	no	—
172	C₂×D₁₀₉	—	4	1	no	—
173	C₄×C₁₀₉	—	2	1	no	—
174	C₄×C₁₀₉	—	2	1	no	—
175	C₄×C₁₀₉	—	4	1	no	—
176	C₄×C₁₀₉	—	4	1	no	—
177	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
178	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
179	Dic₁₀₉	—	2	1	no	—
180	Dic₁₀₉	—	2	1	no	—
181	Dic₁₀₉	—	2	1	no	—
182	Dic₁₀₉	—	4	1	no	—
183	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
184	C₂×C₁₀₉	—	8	1	no	—
185	C₂×C₁₀₉	—	8	1	no	—
186	C₂×C₁₀₉	#550	1	1	yes	characteristic
187	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₂×C₁₀₉	#576	1	1	yes	characteristic
189	D₁₀₉	—	8	1	no	—
190	D₁₀₉	—	8	1	no	—
191	D₁₀₉	—	8	1	no	—
192	#3399	—	109	1	no	maximal, Sylow
193	C₁₀₉	#3399	1	1	yes	characteristic, complemented, Sylow
194	#559	—	109	1	no	—
195	#550	—	109	1	no	—
196	#496	—	109	1	no	—
197	#496	—	109	1	no	—
198	#493	—	109	1	no	—
199	#463	—	109	1	no	—
200	#463	D₁₀₉	1	1	yes	characteristic, complemented
201	#460	—	109	1	no	—
202	#459	—	109	1	no	—
203	#459	—	109	1	no	—
204	#459	—	109	1	no	—
205	#446	—	109	1	no	—
206	#442	—	109	1	no	—
207	#442	—	109	1	no	—
208	C₄×SD₁₆	—	109	1	no	—
209	C₂×#41	—	218	1	no	—
210	C₂×SD₁₆	—	109	1	no	—
211	C₂×SD₁₆	—	109	1	no	—
212	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
213	C₂×D₈	—	109	1	no	—
214	C₂×D₈	—	109	1	no	—
215	C₂×D₈	—	109	1	no	—
216	C₂×D₈	—	109	1	no	—
217	C₂×D₈	—	109	1	no	—
218	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
219	#127	—	109	1	no	—
220	#127	—	109	1	no	—
221	#127	—	109	1	no	—
222	#127	—	109	1	no	—
223	#124	—	109	1	no	—
224	#124	—	218	1	no	—
225	#124	C₂×D₁₀₉	1	1	yes	characteristic
226	#123	—	218	1	no	—
227	#123	—	218	1	no	—
228	#121	—	218	1	no	—
229	#121	—	218	1	no	—
230	#121	—	218	1	no	—
231	C₄×Q₈	—	109	1	no	—
232	C₄×Q₈	—	109	1	no	—
233	C₄×D₄	—	109	1	no	—
234	C₄×D₄	—	109	1	no	—
235	C₄×D₄	—	109	1	no	—
236	C₄×D₄	—	109	1	no	—
237	C₄×D₄	—	218	1	no	—
238	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
239	#106	—	109	1	no	—
240	C₄⋊C₈	—	109	1	no	—
241	C₄⋊C₈	—	109	1	no	—
242	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
243	#103	C₂×D₁₀₉	1	1	yes	characteristic
244	#102	—	109	1	no	—
245	#102	—	109	1	no	—
246	#102	—	109	1	no	—
247	#102	—	109	1	no	—
248	#102	C₂×D₁₀₉	1	1	yes	characteristic
249	#97	—	109	1	no	—
250	#97	—	109	1	no	—
251	C₄×C₈	—	109	1	no	—
252	#41	—	436	4	no	—
253	C₂×Q₈	—	109	1	no	—
254	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
255	C₂×D₄	—	109	1	no	—
256	C₂×D₄	—	109	1	no	—
257	C₂×D₄	—	109	1	no	—
258	C₂×D₄	—	218	1	no	—
259	C₂×D₄	—	218	1	no	—
260	C₂×D₄	—	218	1	no	—
261	C₂×D₄	—	218	1	no	—
262	C₂×D₄	—	218	1	no	—
263	C₂×D₄	—	218	1	no	—
264	C₂×D₄	—	218	1	no	—
265	C₂×D₄	—	218	1	no	—
266	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
267	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	218	1	no	—
270	C₂²×C₄	—	218	1	no	—
271	C₂²×C₄	—	218	1	no	—
272	C₂²×C₄	—	218	1	no	—
273	SD₁₆	—	4	1	no	—
274	SD₁₆	—	436	1	no	—
275	SD₁₆	—	218	2	no	—
276	D₈	—	4	1	no	—
277	D₈	—	436	1	no	—
278	D₈	—	436	1	no	—
279	D₈	—	436	1	no	—
280	D₈	—	218	2	no	—
281	D₈	—	218	2	no	—
282	C₂×C₈	—	109	1	no	—
283	C₂×C₈	—	109	1	no	—
284	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
285	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
286	C₄⋊C₄	—	109	1	no	—
287	C₄⋊C₄	—	109	1	no	—
288	C₄⋊C₄	—	109	1	no	—
289	C₄⋊C₄	—	109	1	no	—
290	C₄⋊C₄	—	218	1	no	—
291	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	2	1	no	—
295	C₂²⋊C₄	—	218	1	no	—
296	C₂²⋊C₄	—	218	1	no	—
297	C₂²⋊C₄	—	218	1	no	—
298	C₂²⋊C₄	—	218	1	no	—
299	C₂²⋊C₄	—	218	1	no	—
300	C₂²⋊C₄	—	218	1	no	—
301	C₄²	—	109	1	no	—
302	C₄²	—	109	1	no	—
303	C₄²	—	218	1	no	—
304	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
305	C₂³	—	2	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	218	1	no	—
308	C₂³	—	218	1	no	—
309	C₂³	—	218	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	218	2	no	—
313	D₄	—	2	1	no	—
314	D₄	—	2	1	no	—
315	D₄	—	2	1	no	—
316	D₄	—	2	1	no	—
317	D₄	—	872	1	no	—
318	D₄	—	872	1	no	—
319	D₄	—	872	1	no	—
320	D₄	—	872	1	no	—
321	D₄	—	436	2	no	—
322	D₄	—	436	2	no	—
323	D₄	—	436	2	no	—
324	D₄	—	436	2	no	—
325	D₄	—	218	1	no	—
326	D₄	—	218	1	no	—
327	D₄	—	218	1	no	—
328	D₄	—	218	1	no	—
329	D₄	—	218	1	no	—
330	D₄	—	218	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	872	1	no	—
335	C₂×C₄	—	872	1	no	—
336	C₂×C₄	—	109	1	no	—
337	C₂×C₄	—	109	1	no	—
338	C₂×C₄	—	109	1	no	—
339	C₂×C₄	—	109	1	no	—
340	C₂×C₄	—	436	2	no	—
341	C₂×C₄	—	436	2	no	—
342	C₂×C₄	—	218	1	no	—
343	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
344	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
345	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	109	2	no	—
349	C₈	—	218	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	436	1	no	—
357	C₂²	—	436	1	no	—
358	C₂²	—	436	1	no	—
359	C₂²	—	436	1	no	—
360	C₂²	—	436	1	no	—
361	C₂²	—	436	1	no	—
362	C₂²	—	436	1	no	—
363	C₂²	—	436	1	no	—
364	C₂²	—	436	1	no	—
365	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	109	2	no	—
371	C₄	—	436	1	no	—
372	C₄	—	218	1	no	—
373	C₄	—	218	1	no	—
374	C₄	—	218	1	no	—
375	C₄	#41×D₁₀₉	1	1	yes	characteristic
376	C₄	#16812744	1	1	yes	characteristic
377	C₂	—	8	1	no	—
378	C₂	—	8	1	no	—
379	C₂	—	872	1	no	—
380	C₂	—	872	1	no	—
381	C₂	—	872	1	no	—
382	C₂	#20295694	1	1	yes	characteristic
383	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
384	C₂	#20295828	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#463×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#463	D₁₀₉
hypocenter	C₁₀₉	#3399
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #20295758	C₂	semidirect
2. #20295697	C₂	semidirect
3. #20295416	C₂	semidirect
4. #20295412	C₂	semidirect
5. #20295401	C₂	semidirect
6. #20295244	C₂	semidirect
7. #20295224	C₂	semidirect
8. #20295222	C₂	semidirect
9. #20295220	C₂	semidirect
10. #20295219	C₂	semidirect
11. #20295155	C₂	semidirect
12. #20295136	C₂	semidirect
13. #20295135	C₂	semidirect
14. #20295111	C₂	semidirect
15. #463×C₁₀₉	C₂	semidirect
156. #11724784	#41	semidirect
193. C₁₀₉	#3399	semidirect
200. #463	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295758	2
#20295697	2
#20295416	2
#20295412	2
#20295401	2
#20295244	2
#20295224	2
#20295222	2
#20295220	2
#20295219	2
#20295155	2
#20295136	2
#20295135	2
#20295111	2
#463×C₁₀₉	2
#3399	109