Subgroups — C₂×#13128241 #16769763

Order 3328.

7022 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13128241	C₂	1	32	yes	maximal, direct product factor
2	C₂×#11719205	C₂	1	4	yes	maximal, complemented
3	C₂×#11719224	C₂	1	8	yes	maximal, complemented
4	C₂×#11719188	C₂	1	4	yes	maximal
5	C₂×#11719175	C₂	1	4	yes	maximal, complemented
6	C₂×#11719222	C₂	1	8	yes	maximal, complemented
7	C₂²×#88180	C₂	1	2	yes	maximal
8	C₂²×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	C₂²	1	4	yes	—
11	C₂×C₁₃×#117	C₂²	1	16	yes	—
12	C₂×C₁₃×#117	C₂²	1	4	yes	—
13	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
16	C₂²×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
17	C₂×#88180	C₂²	1	4	yes	—
18	C₂×#88180	C₂²	1	4	yes	—
19	C₂×#88180	C₂²	1	32	yes	—
20	C₂×#88180	C₂²	1	4	yes	—
21	C₂×#88180	C₂²	1	4	yes	—
22	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
23	C₂×#88178	C₂²	1	16	yes	—
24	C₂²×C₄×Dic₁₃	C₂²	1	4	yes	—
25	#11719224	C₂²	1	128	yes	complemented
26	#11719222	C₂²	1	128	yes	complemented
27	#11719205	C₂²	1	64	yes	complemented
28	C₂×#88144	C₂²	1	16	yes	complemented
29	C₂×#88142	C₂²	1	16	yes	complemented
30	#11719188	C₂²	1	64	yes	—
31	C₂×#88132	C₂²	1	32	yes	—
32	#11719175	C₂²	1	64	yes	complemented
33	C₂×#88124	C₂²	1	8	yes	complemented
34	C₂×#88123	C₂²	1	8	yes	complemented
35	C₂×C₄×Dic₂₆	C₂²	1	16	yes	complemented
36	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₁₃	Q₈	1	16	yes	complemented
38	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₁₃	C₂³	1	8	yes	—
44	C₂³×Dic₁₃	C₂³	1	8	yes	—
45	C₂²×Dic₂₆	C₂³	1	16	yes	—
46	C₁₃×#117	C₂³	1	32	yes	—
47	C₁₃×#117	C₂³	1	32	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
50	C₂×C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
51	C₂×C₁₃×C₂²⋊C₄	C₂³	1	16	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
54	C₂×C₄²×C₁₃	C₂³	1	16	yes	—
55	C₂×C₂²⋊Dic₁₃	C₂³	1	8	yes	—
56	C₂×C₂²⋊Dic₁₃	C₂³	1	32	yes	—
57	C₂×C₂²⋊Dic₁₃	C₂³	1	8	yes	—
58	#88180	C₂³	1	32	yes	—
59	#88180	C₂³	1	32	yes	—
60	#88180	C₂³	1	32	yes	—
61	#88180	C₂³	1	32	yes	—
62	C₂×C₄⋊Dic₁₃	C₂³	1	16	yes	—
63	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
64	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
65	C₂×C₄⋊Dic₁₃	C₂³	1	32	yes	—
66	#88178	C₂³	1	128	yes	—
67	C₂×#6101	C₂³	1	32	yes	—
68	C₂×C₄×Dic₁₃	C₂³	1	8	yes	—
69	C₂×C₄×Dic₁₃	C₂³	1	8	yes	—
70	C₂×C₄×Dic₁₃	C₂³	1	32	yes	—
71	#88144	C₂³	1	128	yes	complemented
72	#88142	C₂³	1	128	yes	complemented
73	#88132	C₂³	1	256	yes	—
74	#88124	C₂³	1	64	yes	complemented
75	#88123	C₂³	1	64	yes	complemented
76	C₄×Dic₂₆	C₂³	1	128	yes	complemented
77	C₂×#3458	—	13	1	no	maximal, Sylow
78	C₂⁴×C₁₃	—	4	4	no	—
79	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
80	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂⁴×C₁₃	C₂⁴	1	2	yes	—
82	C₂⁴×C₁₃	C₂⁴	1	4	yes	—
83	C₂²×C₄×C₁₃	—	4	8	no	—
84	C₂²×C₄×C₁₃	C₂×Q₈	1	16	yes	—
85	C₂²×C₄×C₁₃	C₂×Q₈	1	16	yes	—
86	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
87	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
88	C₂²×C₄×C₁₃	C₂×Q₈	1	64	yes	complemented
89	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
90	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
91	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
93	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
95	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
96	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
97	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₂²×Dic₁₃	—	4	16	no	—
99	C₂²×Dic₁₃	C₂⁴	1	16	yes	—
100	C₂²×Dic₁₃	C₂⁴	1	32	yes	—
101	C₂×Dic₂₆	—	4	32	no	—
102	C₂×Dic₂₆	C₂⁴	1	64	yes	—
103	C₁₃×C₄⋊C₄	C₂⁴	1	32	yes	—
104	C₁₃×C₂²⋊C₄	C₂⁴	1	32	yes	—
105	C₄²×C₁₃	C₂⁴	1	16	yes	—
106	C₄²×C₁₃	C₂⁴	1	16	yes	—
107	C₂²⋊Dic₁₃	C₂⁴	1	32	yes	—
108	C₂²⋊Dic₁₃	C₂⁴	1	32	yes	—
109	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
110	C₄⋊Dic₁₃	C₂⁴	1	32	yes	—
111	C₄⋊Dic₁₃	C₂⁴	1	32	yes	—
112	#6101	C₂⁴	1	128	yes	—
113	C₄×Dic₁₃	C₂⁴	1	32	yes	—
114	C₄×Dic₁₃	C₂⁴	1	32	yes	—
115	#3458	—	13	32	no	—
116	C₂×#544	—	13	8	no	—
117	C₂×#544	—	13	4	no	—
118	C₂×#543	—	13	4	no	—
119	C₂×#533	—	13	4	no	—
120	C₂×#533	—	13	8	no	—
121	C₂²×#117	—	13	2	no	—
122	C₂²×#117	D₁₃	1	1	yes	characteristic
123	C₂³×C₁₃	—	2	8	no	—
124	C₂³×C₁₃	—	2	4	no	—
125	C₂³×C₁₃	—	4	16	no	—
126	C₂³×C₁₃	—	4	8	no	—
127	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
128	C₂³×C₁₃	C₂²×Q₈	1	8	yes	—
129	C₂³×C₁₃	C₂²×Q₈	1	16	yes	—
130	C₂³×C₁₃	C₂⁵	1	2	yes	—
131	C₂³×C₁₃	C₂⁵	1	4	yes	—
132	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
133	C₂×C₄×C₁₃	—	2	16	no	—
134	C₂×C₄×C₁₃	—	4	16	no	—
135	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
136	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
137	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
138	C₂×C₄×C₁₃	C₂²×Q₈	1	32	yes	—
139	C₂×C₄×C₁₃	C₂²×Q₈	1	32	yes	—
140	C₂×C₄×C₁₃	C₂²×Q₈	1	16	yes	—
141	C₂×C₄×C₁₃	C₂²×Q₈	1	16	yes	—
142	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
143	C₂×C₄×C₁₃	C₂⁵	1	2	yes	—
144	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
145	C₂×C₄×C₁₃	C₂⁵	1	2	yes	—
146	C₂×Dic₁₃	—	2	32	no	—
147	C₂×Dic₁₃	—	4	32	no	—
148	C₂×Dic₁₃	C₂⁵	1	32	yes	—
149	Dic₂₆	—	4	64	no	—
150	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
151	#544	—	13	128	no	—
152	#544	—	13	64	no	—
153	#543	—	13	64	no	—
154	#533	—	13	128	no	—
155	#533	—	13	64	no	—
156	C₂×C₄⋊Q₈	—	13	16	no	—
157	C₂×C₄⋊Q₈	—	13	8	no	—
158	C₂×#125	—	13	16	no	—
159	C₂×#125	—	13	8	no	—
160	C₂×C₂²⋊Q₈	—	13	16	no	—
161	C₂×C₂²⋊Q₈	—	13	32	no	—
162	C₂×C₄×Q₈	—	13	16	no	—
163	C₂×#117	—	13	4	no	—
164	C₂×#117	—	13	4	no	—
165	C₂×#117	—	13	4	no	—
166	C₂×#117	—	13	4	no	—
167	C₂×#117	—	13	32	no	—
168	C₂×#117	C₂×D₁₃	1	4	yes	—
169	C₂×#117	C₂×D₁₃	1	4	yes	—
170	C₂×#117	C₂×D₁₃	1	16	yes	—
171	C₂²×C₄⋊C₄	—	13	4	no	—
172	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
173	C₂²×C₂²⋊C₄	—	13	4	no	—
174	C₂²×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
175	C₂²×C₄²	—	13	4	no	—
176	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
177	C₂²×C₁₃	—	2	8	no	—
178	C₂²×C₁₃	—	2	4	no	—
179	C₂²×C₁₃	—	2	16	no	—
180	C₂²×C₁₃	—	2	8	no	—
181	C₂²×C₁₃	—	4	16	no	—
182	C₂²×C₁₃	C₂³×Q₈	1	4	yes	—
183	C₂²×C₁₃	C₂³×Q₈	1	8	yes	—
184	C₂²×C₁₃	C₂³×Q₈	1	2	yes	—
185	C₂²×C₁₃	#585	1	4	yes	—
186	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
187	C₄×C₁₃	—	2	16	no	—
188	C₄×C₁₃	C₂³×Q₈	1	4	yes	—
189	C₄×C₁₃	C₂³×Q₈	1	8	yes	—
190	C₄×C₁₃	C₂³×Q₈	1	4	yes	—
191	Dic₁₃	—	2	32	no	—
192	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
193	C₂²×Q₈	—	13	16	no	—
194	C₂³×C₄	—	13	8	no	—
195	C₂³×C₄	Dic₂₆	1	16	yes	complemented
196	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
197	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
198	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
199	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
200	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
201	C₂³×C₄	C₂²×D₁₃	1	8	yes	—
202	C₄⋊Q₈	—	13	128	no	—
203	C₄⋊Q₈	—	13	64	no	—
204	#125	—	13	128	no	—
205	#125	—	13	64	no	—
206	C₂²⋊Q₈	—	13	128	no	—
207	C₂²⋊Q₈	—	13	256	no	—
208	C₄×Q₈	—	13	128	no	—
209	#117	—	13	32	no	—
210	#117	—	13	32	no	—
211	#117	—	13	32	no	—
212	#117	—	13	32	no	—
213	#117	C₂²×D₁₃	1	32	yes	—
214	#117	C₂²×D₁₃	1	32	yes	—
215	C₂×C₄⋊C₄	—	13	8	no	—
216	C₂×C₄⋊C₄	—	13	8	no	—
217	C₂×C₄⋊C₄	—	13	32	no	—
218	C₂×C₄⋊C₄	—	13	32	no	—
219	C₂×C₄⋊C₄	—	13	16	no	—
220	C₂×C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
221	C₂×C₄⋊C₄	C₂²×D₁₃	1	16	yes	—
222	C₂×C₂²⋊C₄	—	13	8	no	—
223	C₂×C₂²⋊C₄	—	13	8	no	—
224	C₂×C₂²⋊C₄	—	13	32	no	—
225	C₂×C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
226	C₂×C₂²⋊C₄	C₂²×D₁₃	1	16	yes	—
227	C₂×C₄²	—	13	8	no	—
228	C₂×C₄²	—	13	32	no	—
229	C₂×C₄²	—	13	8	no	—
230	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
231	C₂×C₄²	C₂²×D₁₃	1	16	yes	—
232	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
233	C₂×C₁₃	—	2	8	no	—
234	C₂×C₁₃	—	2	4	no	—
235	C₂×C₁₃	#3458	1	4	yes	complemented
236	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
237	C₂×C₁₃	C₂×#585	1	2	yes	—
238	C₂⁴	—	4	4	no	—
239	C₂⁴	C₂×Dic₂₆	1	8	yes	—
240	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
241	C₂⁴	C₂³×D₁₃	1	2	yes	—
242	C₂⁴	C₂³×D₁₃	1	4	yes	—
243	C₂×Q₈	—	13	64	no	—
244	C₂×Q₈	—	52	32	no	—
245	C₂²×C₄	—	4	8	no	—
246	C₂²×C₄	—	13	16	no	—
247	C₂²×C₄	—	13	32	no	—
248	C₂²×C₄	—	52	16	no	—
249	C₂²×C₄	C₂×Dic₂₆	1	16	yes	—
250	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
251	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
252	C₂²×C₄	C₂×Dic₂₆	1	16	yes	—
253	C₂²×C₄	C₂×Dic₂₆	1	64	yes	complemented
254	C₂²×C₄	C₂³×D₁₃	1	8	yes	—
255	C₂²×C₄	C₂³×D₁₃	1	16	yes	—
256	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
257	C₂²×C₄	C₂³×D₁₃	1	8	yes	—
258	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
260	C₂²×C₄	C₂³×D₁₃	1	8	yes	—
261	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
262	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
263	C₄⋊C₄	—	13	32	no	—
264	C₄⋊C₄	—	13	32	no	—
265	C₄⋊C₄	—	13	128	no	—
266	C₄⋊C₄	—	13	64	no	—
267	C₄⋊C₄	C₂³×D₁₃	1	32	yes	—
268	C₂²⋊C₄	—	13	32	no	—
269	C₂²⋊C₄	—	13	32	no	—
270	C₂²⋊C₄	C₂³×D₁₃	1	32	yes	—
271	C₄²	—	13	32	no	—
272	C₄²	—	13	32	no	—
273	C₄²	C₂³×D₁₃	1	16	yes	—
274	C₄²	C₂³×D₁₃	1	16	yes	—
275	C₁₃	C₂×#3458	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	2	4	no	—
277	C₂³	—	2	8	no	—
278	C₂³	—	4	16	no	—
279	C₂³	—	4	8	no	—
280	C₂³	C₂²×Dic₂₆	1	4	yes	—
281	C₂³	C₂²×Dic₂₆	1	8	yes	—
282	C₂³	C₂²×Dic₂₆	1	16	yes	—
283	C₂³	C₂⁴×D₁₃	1	2	yes	—
284	C₂³	C₂⁴×D₁₃	1	4	yes	—
285	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
286	Q₈	—	52	64	no	—
287	C₂×C₄	—	2	16	no	—
288	C₂×C₄	—	4	16	no	—
289	C₂×C₄	—	13	32	no	—
290	C₂×C₄	—	26	32	no	—
291	C₂×C₄	—	52	32	no	—
292	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
293	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
294	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
295	C₂×C₄	C₂²×Dic₂₆	1	32	yes	—
296	C₂×C₄	C₂²×Dic₂₆	1	16	yes	—
297	C₂×C₄	C₂²×Dic₂₆	1	32	yes	—
298	C₂×C₄	C₂²×Dic₂₆	1	16	yes	—
299	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
300	C₂×C₄	C₂⁴×D₁₃	1	2	yes	—
301	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
302	C₂×C₄	C₂⁴×D₁₃	1	2	yes	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	16	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	4	16	no	—
308	C₂²	C₂³×Dic₂₆	1	4	yes	—
309	C₂²	C₂³×Dic₂₆	1	8	yes	—
310	C₂²	C₂³×Dic₂₆	1	2	yes	—
311	C₂²	#11719648	1	4	yes	—
312	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
313	C₄	—	2	16	no	—
314	C₄	—	26	32	no	—
315	C₄	C₂³×Dic₂₆	1	8	yes	—
316	C₄	C₂³×Dic₂₆	1	4	yes	—
317	C₄	C₂³×Dic₂₆	1	4	yes	—
318	C₂	—	2	4	no	—
319	C₂	—	2	8	no	—
320	C₂	#13128241	1	4	yes	direct product factor
321	C₂	C₂⁴×Dic₂₆	1	1	yes	characteristic
322	C₂	C₂×#11719648	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×#117	C₂
socle	C₂³×C₁₃	C₂⁵
prime radical	C₂⁵×C₁₃	C₂³
hypercenter	C₂²×#117	D₁₃
hypocenter	C₁₃	C₂×#3458
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. #13128241	C₂	direct (N × H)
2. C₂×#11719205	C₂	semidirect
3. C₂×#11719224	C₂	semidirect
5. C₂×#11719175	C₂	semidirect
6. C₂×#11719222	C₂	semidirect
25. #11719224	C₂²	semidirect
26. #11719222	C₂²	semidirect
27. #11719205	C₂²	semidirect
28. C₂×#88144	C₂²	semidirect
29. C₂×#88142	C₂²	semidirect
32. #11719175	C₂²	semidirect
33. C₂×#88124	C₂²	semidirect
34. C₂×#88123	C₂²	semidirect
35. C₂×C₄×Dic₂₆	C₂²	semidirect
37. C₂³×C₄×C₁₃	Q₈	semidirect
71. #88144	C₂³	semidirect
72. #88142	C₂³	semidirect
74. #88124	C₂³	semidirect
75. #88123	C₂³	semidirect
76. C₄×Dic₂₆	C₂³	semidirect
88. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
195. C₂³×C₄	Dic₂₆	semidirect
235. C₂×C₁₃	#3458	semidirect
253. C₂²×C₄	C₂×Dic₂₆	semidirect
275. C₁₃	C₂×#3458	semidirect
320. C₂	#13128241	direct (N × H)

Maximal subgroups

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

subgroup	index
#13128241	2
C₂×#11719205	2
C₂×#11719224	2
C₂×#11719188	2
C₂×#11719175	2
C₂×#11719222	2
C₂²×#88180	2
C₂²×C₁₃×#117	2
C₂×#3458	13

Subgroups — C2×#13128241 #16769763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13128241 #16769763