Subgroups — #16017191

Order 3328.

770 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	C₂×#11718854	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116888	C₂	1	4	yes	maximal, complemented
4	#13116878	C₂	1	2	yes	maximal, complemented
5	#13116868	C₂	1	2	yes	maximal, complemented
6	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116548	C₂	1	2	yes	maximal, complemented
8	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	—	2	1	no	—
10	C₂×C₁₃×#135	C₂²	1	2	yes	—
11	C₁₃×#532	—	2	1	no	—
12	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₅₂	—	2	1	no	—
14	C₁₃×#495	—	2	1	no	—
15	C₁₃×#495	C₂²	1	2	yes	complemented
16	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
17	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
18	C₁₃×#492	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718854	—	2	8	no	—
21	C₂×#88068	C₂²	1	4	yes	—
22	C₂×#88066	C₂²	1	4	yes	—
23	C₂×#88061	—	2	1	no	—
24	C₂×#88058	C₂²	1	1	yes	characteristic
25	C₂×#88052	C₂²	1	2	yes	—
26	C₄×D₁₀₄	—	2	1	no	—
27	C₄×D₁₀₄	—	2	1	no	—
28	C₄×D₁₀₄	C₂²	1	2	yes	—
29	C₄×D₁₀₄	C₂²	1	2	yes	—
30	#11718156	C₂²	1	4	yes	complemented
31	#11718154	C₂²	1	4	yes	complemented
32	C₄⋊#6090	C₂²	1	2	yes	complemented
33	C₄⋊#6090	C₂²	1	2	yes	complemented
34	C₂×C₁₃×#41	—	2	2	no	—
35	C₂²×D₅₂	—	2	1	no	—
36	C₂²×C₄×D₁₃	—	4	1	no	—
37	C₁₃×#135	—	2	4	no	—
38	C₁₃×#135	—	2	4	no	—
39	C₁₃×#135	—	4	4	no	—
40	C₂×C₁₃×Q₁₆	—	2	1	no	—
41	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×SD₁₆	—	2	2	no	—
43	C₂×C₁₃×SD₁₆	—	2	2	no	—
44	C₂×C₁₃×D₈	—	2	1	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×#127	—	2	1	no	—
50	C₁₃×#124	—	2	2	no	—
51	C₁₃×#121	—	4	1	no	—
52	C₁₃×#121	—	4	1	no	—
53	C₄×D₄×C₁₃	—	4	1	no	—
54	C₄×D₄×C₁₃	—	4	1	no	—
55	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
56	C₂×#6103	—	4	1	no	—
57	C₂×C₄⋊Dic₁₃	—	2	1	no	—
58	C₂×D₁₀₄	—	2	1	no	—
59	C₂×D₁₀₄	—	2	2	no	—
60	C₂×D₁₀₄	D₄	1	4	yes	complemented
61	C₂×D₁₀₄	C₂³	1	2	yes	—
62	C₄×D₅₂	—	2	2	no	—
63	C₄×D₅₂	—	2	2	no	—
64	C₄×D₅₂	—	4	1	no	—
65	C₄×D₅₂	—	4	1	no	—
66	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#88068	—	2	8	no	—
71	#88066	—	2	8	no	—
72	#88061	—	2	2	no	—
73	#88061	—	2	2	no	—
74	#88058	—	2	1	no	—
75	#88058	C₂³	1	2	yes	—
76	#88052	—	2	4	no	—
77	C₂×#6090	C₂³	1	4	yes	—
78	#33580	—	13	1	no	maximal, Sylow
79	C₂³×D₁₃	—	4	1	no	—
80	C₁₃×#41	—	4	4	no	—
81	C₁₃×#41	—	4	4	no	—
82	C₂×Q₈×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	2	2	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	—	4	1	no	—
86	C₂×D₄×C₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₃	—	4	2	no	—
88	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×Dic₁₃	—	4	1	no	—
92	C₂×D₅₂	—	2	4	no	—
93	C₂×D₅₂	—	2	2	no	—
94	C₂×D₅₂	—	4	1	no	—
95	C₂×D₅₂	—	4	2	no	—
96	C₂×C₄×D₁₃	—	4	2	no	—
97	C₂×C₄×D₁₃	—	4	2	no	—
98	C₂×C₄×D₁₃	—	8	4	no	—
99	C₁₃×Q₁₆	—	2	2	no	—
100	C₁₃×Q₁₆	—	2	2	no	—
101	C₁₃×Q₁₆	—	4	2	no	—
102	C₁₃×SD₁₆	—	4	4	no	—
103	C₁₃×SD₁₆	—	4	4	no	—
104	C₁₃×D₈	—	2	2	no	—
105	C₁₃×D₈	—	2	2	no	—
106	C₁₃×D₈	—	4	2	no	—
107	C₂×C₈×C₁₃	—	2	2	no	—
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₁₃	#41	1	2	yes	—
113	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
114	C₁₃×C₄⋊C₄	—	4	1	no	—
115	C₁₃×C₄⋊C₄	—	4	1	no	—
116	C₁₃×C₂²⋊C₄	—	4	2	no	—
117	C₁₃×C₂²⋊C₄	—	4	2	no	—
118	C₄²×C₁₃	D₈	1	1	yes	characteristic
119	C₄²×C₁₃	D₈	1	1	yes	characteristic
120	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
122	#6103	—	4	2	no	—
123	#6103	—	4	2	no	—
124	C₄⋊Dic₁₃	—	2	2	no	—
125	C₄⋊Dic₁₃	—	4	1	no	—
126	D₁₀₄	—	2	4	no	—
127	D₁₀₄	—	2	2	no	—
128	D₁₀₄	C₂×D₄	1	4	yes	—
129	#6090	—	2	4	no	—
130	C₂×#509	—	13	2	no	—
131	#3147	D₁₃	1	1	yes	characteristic, complemented
132	C₂×C₄×D₈	—	13	1	no	—
133	#2210	—	13	4	no	—
134	#2208	—	13	2	no	—
135	#2206	—	13	2	no	—
136	#2150	—	13	1	no	—
137	#2139	—	13	2	no	—
138	C₂³×C₁₃	—	4	2	no	—
139	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂²×D₁₃	—	4	2	no	—
141	C₂²×D₁₃	—	4	4	no	—
142	C₂²×D₁₃	—	8	2	no	—
143	C₂²×D₁₃	—	8	2	no	—
144	Q₈×C₁₃	—	4	2	no	—
145	Q₈×C₁₃	—	4	2	no	—
146	D₄×C₁₃	—	4	2	no	—
147	D₄×C₁₃	—	4	2	no	—
148	D₄×C₁₃	—	8	2	no	—
149	D₄×C₁₃	—	8	4	no	—
150	D₄×C₁₃	—	8	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	8	4	no	—
155	C₂×C₄×C₁₃	#121	1	2	yes	—
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×Dic₁₃	—	4	2	no	—
164	C₂×Dic₁₃	—	8	2	no	—
165	D₅₂	—	2	4	no	—
166	D₅₂	—	4	2	no	—
167	D₅₂	—	4	4	no	—
168	C₄×D₁₃	—	8	4	no	—
169	C₄×D₁₃	—	8	4	no	—
170	C₈×C₁₃	—	2	1	no	—
171	C₈×C₁₃	—	2	1	no	—
172	C₈×C₁₃	#121	1	2	yes	—
173	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
175	C₂×#135	—	2	1	no	—
176	C₂×#135	C₂×D₁₃	1	2	yes	—
177	C₂²×D₈	—	13	1	no	—
178	#532	—	2	1	no	—
179	C₂×C₄×D₄	—	26	1	no	—
180	#509	—	26	8	no	—
181	C₂×SD₃₂	—	13	4	no	—
182	C₂×D₁₆	—	13	4	no	—
183	C₂×M₃₂	—	13	2	no	—
184	#495	—	2	1	no	—
185	#495	C₂×D₁₃	1	2	yes	complemented
186	#494	C₂×D₁₃	1	1	yes	characteristic, complemented
187	#493	C₂×D₁₃	1	1	yes	characteristic, complemented
188	#492	—	2	1	no	—
189	C₄×D₈	—	13	2	no	—
190	C₄×D₈	—	13	2	no	—
191	C₄×D₈	—	26	1	no	—
192	C₄×D₈	—	26	1	no	—
193	C₂×#107	—	13	1	no	—
194	C₂×#102	—	26	1	no	—
195	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
196	C₄⋊C₁₆	—	13	2	no	—
197	C₄⋊C₁₆	—	13	2	no	—
198	#358	—	13	4	no	—
199	#357	—	13	4	no	—
200	C₂²×C₁₃	—	2	1	no	—
201	C₂²×C₁₃	—	2	1	no	—
202	C₂²×C₁₃	—	8	2	no	—
203	C₂²×C₁₃	—	8	2	no	—
204	C₂²×C₁₃	—	8	2	no	—
205	C₂²×C₁₃	#459	1	2	yes	—
206	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
207	C₂×D₁₃	—	4	4	no	—
208	C₂×D₁₃	—	8	8	no	—
209	C₂×D₁₃	—	8	2	no	—
210	C₂×D₁₃	—	8	2	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	2	1	no	—
213	C₄×C₁₃	—	2	1	no	—
214	C₄×C₁₃	—	8	2	no	—
215	C₄×C₁₃	#459	1	2	yes	—
216	C₄×C₁₃	#509	1	2	yes	complemented
217	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
218	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
219	Dic₁₃	—	8	2	no	—
220	C₂×#41	—	2	2	no	—
221	C₂²×D₄	—	26	1	no	—
222	C₂³×C₄	—	52	1	no	—
223	#135	—	2	4	no	—
224	#135	—	2	4	no	—
225	#135	—	4	4	no	—
226	C₂×Q₁₆	—	2	1	no	—
227	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
228	C₂×SD₁₆	—	2	2	no	—
229	C₂×SD₁₆	—	2	2	no	—
230	C₂×D₈	—	2	1	no	—
231	C₂×D₈	—	13	4	no	—
232	C₂×D₈	—	13	2	no	—
233	C₂×D₈	—	26	2	no	—
234	C₂×D₈	—	26	1	no	—
235	C₂×D₈	C₂²×D₁₃	1	2	yes	—
236	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
238	C₄⋊Q₈	—	2	1	no	—
239	#127	—	2	1	no	—
240	#124	—	2	2	no	—
241	#121	—	4	1	no	—
242	#121	—	4	1	no	—
243	C₄×D₄	—	4	1	no	—
244	C₄×D₄	—	4	1	no	—
245	C₄×D₄	—	26	2	no	—
246	C₄×D₄	—	26	2	no	—
247	C₄×D₄	—	52	1	no	—
248	C₄×D₄	—	52	1	no	—
249	C₂×C₄⋊C₄	—	26	1	no	—
250	C₂×C₂²⋊C₄	—	52	1	no	—
251	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
252	SD₃₂	—	26	8	no	—
253	D₁₆	—	26	8	no	—
254	M₃₂	—	26	4	no	—
255	C₂×C₁₆	—	13	4	no	—
256	#107	—	13	2	no	—
257	#107	—	26	1	no	—
258	#102	—	26	2	no	—
259	#102	—	26	2	no	—
260	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
261	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
264	C₂×C₁₃	—	2	2	no	—
265	C₂×C₁₃	—	16	2	no	—
266	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
267	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
268	C₂×C₁₃	#3414	1	1	yes	characteristic
269	D₁₃	—	8	4	no	—
270	C₂⁴	—	52	1	no	—
271	#41	—	4	4	no	—
272	#41	—	4	4	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	26	4	no	—
279	C₂×D₄	—	26	2	no	—
280	C₂×D₄	—	52	1	no	—
281	C₂×D₄	—	52	2	no	—
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	—	52	1	no	—
284	C₂²×C₄	—	52	2	no	—
285	C₂²×C₄	—	52	2	no	—
286	C₂²×C₄	—	104	4	no	—
287	C₂²×C₄	#6104	1	1	yes	characteristic
288	C₂²×C₄	#6104	1	1	yes	characteristic
289	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
290	Q₁₆	—	2	2	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	4	2	no	—
293	SD₁₆	—	4	4	no	—
294	SD₁₆	—	4	4	no	—
295	D₈	—	2	2	no	—
296	D₈	—	2	2	no	—
297	D₈	—	4	2	no	—
298	D₈	—	13	4	no	—
299	D₈	—	26	4	no	—
300	D₈	—	26	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	D₄×D₁₃	1	2	yes	—
303	C₂×C₈	#6129	1	2	yes	—
304	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	26	2	no	—
311	C₄⋊C₄	—	52	1	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₂²⋊C₄	—	52	2	no	—
315	C₂²⋊C₄	—	52	2	no	—
316	C₄²	#6104	1	1	yes	characteristic
317	C₄²	#6104	1	1	yes	characteristic
318	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
319	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
320	C₁₆	—	26	4	no	—
321	C₁₃	#33580	1	1	yes	characteristic, complemented, Sylow
322	C₂³	—	4	2	no	—
323	C₂³	—	52	2	no	—
324	C₂³	—	52	4	no	—
325	C₂³	—	104	2	no	—
326	C₂³	—	104	2	no	—
327	C₂³	C₂×#6104	1	1	yes	characteristic
328	Q₈	—	4	2	no	—
329	Q₈	—	4	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	8	2	no	—
333	D₄	—	8	4	no	—
334	D₄	—	8	2	no	—
335	D₄	—	26	4	no	—
336	D₄	—	52	2	no	—
337	D₄	—	52	4	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	8	4	no	—
342	C₂×C₄	—	52	2	no	—
343	C₂×C₄	—	104	4	no	—
344	C₂×C₄	—	104	2	no	—
345	C₂×C₄	—	104	4	no	—
346	C₂×C₄	C₂×#6104	1	1	yes	characteristic
347	C₂×C₄	C₂×#6104	1	1	yes	characteristic
348	C₂×C₄	C₂×#6104	1	1	yes	characteristic
349	C₂×C₄	C₂×#6104	1	1	yes	characteristic
350	C₂×C₄	C₂×#6104	1	1	yes	characteristic
351	C₂×C₄	#88186	1	2	yes	—
352	C₂×C₄	#88192	1	2	yes	—
353	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	#88192	1	2	yes	—
357	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
358	C₈	C₂×#6129	1	1	yes	characteristic
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	8	2	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	52	4	no	—
365	C₂²	—	104	8	no	—
366	C₂²	—	104	2	no	—
367	C₂²	—	104	2	no	—
368	C₂²	#11718733	1	2	yes	—
369	C₂²	C₂²×#6104	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	8	2	no	—
374	C₄	—	104	2	no	—
375	C₄	#11718733	1	2	yes	—
376	C₄	#11718854	1	2	yes	complemented
377	C₄	C₂×#88186	1	1	yes	characteristic
378	C₄	C₂×#88192	1	1	yes	characteristic
379	C₂	—	2	2	no	—
380	C₂	—	16	2	no	—
381	C₂	—	104	4	no	—
382	C₂	C₂×#11718733	1	1	yes	characteristic
383	C₂	C₂×#11718854	1	1	yes	characteristic
384	C₂	#13127970	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3147	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3147	D₁₃
hypocenter	C₁₃	#33580
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#11718854	C₂	semidirect
2. C₂×C₄×D₁₀₄	C₂	semidirect
3. #13116888	C₂	semidirect
4. #13116878	C₂	semidirect
5. #13116868	C₂	semidirect
6. #13116592	C₂	semidirect
7. #13116548	C₂	semidirect
8. C₁₃×#3147	C₂	semidirect
15. C₁₃×#495	C₂²	semidirect
16. C₁₃×#494	C₂²	semidirect
17. C₁₃×#493	C₂²	semidirect
30. #11718156	C₂²	semidirect
31. #11718154	C₂²	semidirect
32. C₄⋊#6090	C₂²	semidirect
33. C₄⋊#6090	C₂²	semidirect
60. C₂×D₁₀₄	D₄	semidirect
131. #3147	D₁₃	semidirect
185. #495	C₂×D₁₃	semidirect
186. #494	C₂×D₁₃	semidirect
187. #493	C₂×D₁₃	semidirect
216. C₄×C₁₃	#509	semidirect
321. C₁₃	#33580	semidirect
376. C₄	#11718854	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718854	2
C₂×C₄×D₁₀₄	2
#13116888	2
#13116878	2
#13116868	2
#13116592	2
#13116548	2
C₁₃×#3147	2
#33580	13