Subgroups — #28375623

Order 13952.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295739	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295677	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295423	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295353	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295350	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295346	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295344	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295315	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295305	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295280	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295279	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295108	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295102	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295067	C₂	1	1	yes	maximal, characteristic, complemented
15	#428×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182029	—	2	1	no	—
17	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
18	C₂×#13182012	C₂²	1	1	yes	characteristic
19	#16812698	C₂²	1	1	yes	characteristic
20	#16812698	C₂²	1	1	yes	characteristic
21	#16812687	—	2	1	no	—
22	#16812685	—	2	1	no	—
23	#16812684	—	2	1	no	—
24	#16812682	—	2	1	no	—
25	#16812677	—	2	1	no	—
26	#16812674	C₂²	1	1	yes	characteristic
27	#16812674	C₂²	1	1	yes	characteristic
28	#16812673	C₂²	1	1	yes	characteristic
29	#16812673	C₂²	1	1	yes	characteristic
30	#16812672	—	2	1	no	—
31	#16812661	C₂²	1	1	yes	characteristic
32	#16812661	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
34	#16812657	C₂²	1	1	yes	characteristic
35	#16812656	—	2	1	no	—
36	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
37	#16812632	C₂²	1	1	yes	characteristic
38	#16812631	C₂²	1	1	yes	characteristic
39	#16812630	C₂²	1	1	yes	characteristic
40	#16812630	C₂²	1	1	yes	characteristic
41	#16812630	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812621	C₂²	1	1	yes	characteristic
44	#16812621	C₂²	1	1	yes	characteristic
45	#16812619	C₂²	1	1	yes	characteristic
46	#16812619	C₂²	1	1	yes	characteristic
47	#16812619	C₂²	1	1	yes	characteristic
48	#16812619	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×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	#117×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
55	#107×C₁₀₉	C₂²	1	1	yes	characteristic
56	#107×C₁₀₉	C₂²	1	1	yes	characteristic
57	#106×C₁₀₉	C₂²	1	1	yes	characteristic
58	#106×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182029	—	4	4	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	—	2	1	no	—
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	—	2	1	no	—
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	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	—	2	1	no	—
72	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
73	D₈₇₂	—	2	2	no	—
74	#13182012	—	2	2	no	—
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₂×#11724784	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
84	#13182006	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	D₄	1	1	yes	characteristic
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₀₉	—	2	1	no	—
95	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
96	M₁₆×C₁₀₉	—	2	2	no	—
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	—	2	1	no	—
100	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
105	C₄²×C₁₀₉	—	2	1	no	—
106	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₀₉	—	2	1	no	—
108	C₂²×D₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	4	2	no	—
117	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₁₀₉	—	2	2	no	—
120	C₂²⋊D₁₀₉	—	4	2	no	—
121	C₂²⋊D₁₀₉	—	8	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	D₄₃₆	—	2	1	no	—
128	D₄₃₆	—	2	1	no	—
129	D₄₃₆	—	2	1	no	—
130	D₄₃₆	—	2	1	no	—
131	D₄₃₆	—	4	2	no	—
132	D₄₃₆	—	8	1	no	—
133	C₄×D₁₀₉	—	2	1	no	—
134	C₄×D₁₀₉	—	2	1	no	—
135	C₄×D₁₀₉	—	4	2	no	—
136	C₄×D₁₀₉	—	4	2	no	—
137	C₄×D₁₀₉	—	4	2	no	—
138	Dic₂₁₈	—	4	2	no	—
139	Dic₂₁₈	—	8	1	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	C₈×C₁₀₉	#41	1	2	yes	complemented
142	#11724784	—	2	1	no	—
143	#11724784	—	2	1	no	—
144	C₂²×C₁₀₉	—	2	1	no	—
145	C₂²×C₁₀₉	—	2	1	no	—
146	C₂²×C₁₀₉	—	2	1	no	—
147	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₀₉	—	2	1	no	—
149	C₂×D₁₀₉	—	2	1	no	—
150	C₂×D₁₀₉	—	2	1	no	—
151	C₂×D₁₀₉	—	4	1	no	—
152	C₂×D₁₀₉	—	4	1	no	—
153	C₂×D₁₀₉	—	4	1	no	—
154	C₂×D₁₀₉	—	4	1	no	—
155	C₂×D₁₀₉	—	4	1	no	—
156	C₂×D₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	—	2	1	no	—
158	C₄×C₁₀₉	—	4	1	no	—
159	C₄×C₁₀₉	—	4	1	no	—
160	C₄×C₁₀₉	—	4	1	no	—
161	C₄×C₁₀₉	—	4	1	no	—
162	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
164	Dic₁₀₉	—	2	1	no	—
165	Dic₁₀₉	—	2	1	no	—
166	Dic₁₀₉	—	4	1	no	—
167	Dic₁₀₉	—	4	1	no	—
168	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₁₀₉	—	4	1	no	—
170	C₂×C₁₀₉	#547	1	1	yes	characteristic
171	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₁₀₉	#577	1	1	yes	characteristic
173	D₁₀₉	—	4	1	no	—
174	D₁₀₉	—	8	1	no	—
175	D₁₀₉	—	8	1	no	—
176	#3354	—	109	1	no	maximal, Sylow
177	C₁₀₉	#3354	1	1	yes	characteristic, complemented, Sylow
178	#555	—	109	1	no	—
179	#547	—	109	1	no	—
180	#498	—	109	1	no	—
181	#482	—	109	1	no	—
182	#482	—	109	1	no	—
183	#481	—	109	1	no	—
184	#481	—	109	1	no	—
185	#476	—	109	1	no	—
186	#474	—	109	1	no	—
187	#469	—	109	1	no	—
188	#469	—	109	1	no	—
189	C₄×D₈	—	109	1	no	—
190	#436	—	109	1	no	—
191	#428	—	109	1	no	—
192	#428	D₁₀₉	1	1	yes	characteristic, complemented
193	C₂×#41	—	218	1	no	—
194	C₂×D₈	—	109	1	no	—
195	C₂×M₁₆	—	109	1	no	—
196	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
197	C₄⋊Q₈	—	109	1	no	—
198	C₄⋊Q₈	—	109	1	no	—
199	#124	—	218	1	no	—
200	#124	—	218	1	no	—
201	#123	—	218	1	no	—
202	#123	—	218	1	no	—
203	C₂²⋊Q₈	—	218	1	no	—
204	#121	—	109	1	no	—
205	#121	—	109	1	no	—
206	#121	—	109	1	no	—
207	#121	—	109	1	no	—
208	#121	—	218	1	no	—
209	C₄×D₄	—	109	1	no	—
210	C₄×D₄	—	109	1	no	—
211	C₄×D₄	—	109	1	no	—
212	#117	—	109	1	no	—
213	#117	—	218	1	no	—
214	#117	C₂×D₁₀₉	1	1	yes	characteristic
215	C₂×C₄⋊C₄	—	109	1	no	—
216	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
217	#107	—	109	1	no	—
218	#107	C₂×D₁₀₉	1	1	yes	characteristic
219	#107	C₂×D₁₀₉	1	1	yes	characteristic
220	#106	—	109	1	no	—
221	#106	—	109	1	no	—
222	#106	—	109	1	no	—
223	#106	—	109	1	no	—
224	#106	C₂×D₁₀₉	1	1	yes	characteristic
225	#106	C₂×D₁₀₉	1	1	yes	characteristic
226	C₄⋊C₈	—	109	1	no	—
227	#102	—	109	1	no	—
228	#102	—	109	1	no	—
229	#102	—	109	1	no	—
230	#102	—	109	1	no	—
231	#102	—	109	1	no	—
232	#102	—	109	1	no	—
233	C₂²⋊C₈	—	109	1	no	—
234	C₂²⋊C₈	—	109	1	no	—
235	C₄×C₈	—	109	1	no	—
236	#41	—	436	4	no	—
237	C₂×Q₈	—	218	1	no	—
238	C₂×Q₈	—	218	1	no	—
239	C₂×D₄	—	109	1	no	—
240	C₂×D₄	—	109	1	no	—
241	C₂×D₄	—	109	1	no	—
242	C₂×D₄	—	218	1	no	—
243	C₂×D₄	—	218	1	no	—
244	C₂×D₄	—	218	1	no	—
245	C₂×D₄	—	218	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	109	1	no	—
248	C₂²×C₄	—	218	1	no	—
249	C₂²×C₄	—	218	1	no	—
250	C₂²×C₄	—	218	1	no	—
251	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
252	D₈	—	218	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	218	2	no	—
255	C₂×C₈	—	109	1	no	—
256	C₂×C₈	—	109	1	no	—
257	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	109	1	no	—
261	C₄⋊C₄	—	109	1	no	—
262	C₄⋊C₄	—	109	1	no	—
263	C₄⋊C₄	—	218	1	no	—
264	C₄⋊C₄	—	218	1	no	—
265	C₄⋊C₄	—	218	1	no	—
266	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
267	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	109	1	no	—
272	C₂²⋊C₄	—	109	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₂²⋊C₄	—	218	1	no	—
276	C₂²⋊C₄	—	218	1	no	—
277	C₂²⋊C₄	—	218	1	no	—
278	C₂²⋊C₄	—	218	1	no	—
279	C₂²⋊C₄	—	218	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	109	1	no	—
282	C₄²	—	218	1	no	—
283	C₂³	—	109	1	no	—
284	C₂³	—	218	1	no	—
285	C₂³	—	218	1	no	—
286	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
287	Q₈	—	872	1	no	—
288	Q₈	—	436	2	no	—
289	D₄	—	872	1	no	—
290	D₄	—	872	1	no	—
291	D₄	—	436	2	no	—
292	D₄	—	436	2	no	—
293	D₄	—	218	2	no	—
294	D₄	—	218	1	no	—
295	D₄	—	218	1	no	—
296	D₄	—	218	1	no	—
297	D₄	—	218	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	109	1	no	—
306	C₂×C₄	—	109	1	no	—
307	C₂×C₄	—	109	1	no	—
308	C₂×C₄	—	436	2	no	—
309	C₂×C₄	—	436	2	no	—
310	C₂×C₄	—	436	2	no	—
311	C₂×C₄	—	218	1	no	—
312	C₂×C₄	—	218	1	no	—
313	C₂×C₄	—	218	1	no	—
314	C₂×C₄	—	218	1	no	—
315	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	218	1	no	—
319	C₈	—	218	1	no	—
320	C₈	#13182029	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	436	1	no	—
325	C₂²	—	436	1	no	—
326	C₂²	—	436	1	no	—
327	C₂²	—	436	1	no	—
328	C₂²	—	436	1	no	—
329	C₂²	—	436	1	no	—
330	C₂²	—	218	1	no	—
331	C₂²	—	218	1	no	—
332	C₂²	—	218	1	no	—
333	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	109	2	no	—
340	C₄	—	436	1	no	—
341	C₄	—	436	1	no	—
342	C₄	—	218	1	no	—
343	C₄	—	218	1	no	—
344	C₄	C₂×#13182029	1	1	yes	characteristic
345	C₄	#16812747	1	1	yes	characteristic
346	C₂	—	4	1	no	—
347	C₂	—	872	1	no	—
348	C₂	—	872	1	no	—
349	C₂	—	436	1	no	—
350	C₂	#20295677	1	1	yes	characteristic
351	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
352	C₂	#20295835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#428×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#428	D₁₀₉
hypocenter	C₁₀₉	#3354
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
1. #20295739	C₂	semidirect
2. #20295677	C₂	semidirect
3. #20295423	C₂	semidirect
4. #20295353	C₂	semidirect
5. #20295350	C₂	semidirect
6. #20295346	C₂	semidirect
7. #20295344	C₂	semidirect
8. #20295315	C₂	semidirect
9. #20295305	C₂	semidirect
10. #20295280	C₂	semidirect
11. #20295279	C₂	semidirect
12. #20295108	C₂	semidirect
13. #20295102	C₂	semidirect
14. #20295067	C₂	semidirect
15. #428×C₁₀₉	C₂	semidirect
141. C₈×C₁₀₉	#41	semidirect
177. C₁₀₉	#3354	semidirect
192. #428	D₁₀₉	semidirect
320. C₈	#13182029	semidirect

Maximal subgroups

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

subgroup	index
#20295739	2
#20295677	2
#20295423	2
#20295353	2
#20295350	2
#20295346	2
#20295344	2
#20295315	2
#20295305	2
#20295280	2
#20295279	2
#20295108	2
#20295102	2
#20295067	2
#428×C₁₀₉	2
#3354	109