Subgroups — C₂²×#11719435 #16769887

Order 3328.

8454 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	C₂×#11719435	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#88144	C₂	1	2	yes	maximal, complemented
3	C₂²×#88151	C₂	1	2	yes	maximal, complemented
4	C₂²×#88200	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
7	C₂²×#88147	C₂	1	2	yes	maximal
8	C₂²×C₄⋊C₄×D₁₃	C₂	1	2	yes	maximal, complemented
9	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂³×D₅₂	C₂²	1	1	yes	characteristic
12	C₂³×C₄×D₁₃	C₂²	1	2	yes	—
13	C₂³×C₄×D₁₃	C₂²	1	4	yes	—
14	C₂³×Dic₂₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊Q₈	C₂²	1	24	yes	complemented
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88200	C₂²	1	96	yes	complemented
20	C₂²×#6103	C₂²	1	4	yes	—
21	C₂²×#6103	C₂²	1	2	yes	—
22	C₂²×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
25	C₂²×#6101	C₂²	1	4	yes	—
26	C₂²×#6101	C₂²	1	2	yes	—
27	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
28	#11719435	C₂²	1	256	yes	direct product factor
29	C₂×#88151	C₂²	1	48	yes	complemented
30	C₂×#88147	C₂²	1	48	yes	—
31	C₂×C₄⋊C₄×D₁₃	C₂²	1	48	yes	complemented
32	C₂×#88144	C₂²	1	48	yes	complemented
33	C₂×C₄×D₅₂	C₂²	1	24	yes	—
34	C₂×C₄×Dic₂₆	C₂²	1	24	yes	complemented
35	C₂⁴×D₁₃	C₂³	1	2	yes	—
36	C₂²×Q₈×C₁₃	—	4	8	no	—
37	C₂²×Q₈×C₁₃	C₂³	1	24	yes	—
38	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
40	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂³×Dic₁₃	C₂³	1	2	yes	—
43	C₂³×Dic₁₃	C₂³	1	2	yes	—
44	C₂³×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×D₅₂	Q₈	1	16	yes	complemented
46	C₂²×D₅₂	C₂³	1	12	yes	—
47	C₂²×C₄×D₁₃	—	2	16	no	—
48	C₂²×C₄×D₁₃	—	4	16	no	—
49	C₂²×C₄×D₁₃	C₂³	1	24	yes	—
50	C₂²×C₄×D₁₃	C₂³	1	48	yes	—
51	C₂²×Dic₂₆	—	2	8	no	—
52	C₂²×Dic₂₆	C₂³	1	12	yes	—
53	C₁₃×C₄⋊Q₈	C₂³	1	64	yes	complemented
54	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
55	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
56	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
57	#88200	C₂³	1	256	yes	complemented
58	C₂×#6103	C₂³	1	24	yes	—
59	C₂×#6103	C₂³	1	48	yes	—
60	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
61	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
62	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	—
63	C₂×#6101	C₂³	1	24	yes	—
64	C₂×#6101	C₂³	1	48	yes	—
65	C₂×C₄×Dic₁₃	C₂³	1	24	yes	—
66	#88151	C₂³	1	128	yes	complemented
67	#88147	C₂³	1	128	yes	—
68	C₄⋊C₄×D₁₃	C₂³	1	128	yes	complemented
69	#88144	C₂³	1	128	yes	complemented
70	C₄×D₅₂	C₂³	1	64	yes	—
71	C₄×Dic₂₆	C₂³	1	64	yes	complemented
72	C₂²×#554	—	13	1	no	maximal, Sylow
73	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₂³×D₁₃	—	2	8	no	—
75	C₂³×D₁₃	—	2	8	no	—
76	C₂³×D₁₃	C₂×Q₈	1	16	yes	—
77	C₂³×D₁₃	C₂⁴	1	12	yes	—
78	C₂×Q₈×C₁₃	—	4	48	no	—
79	C₂×Q₈×C₁₃	C₂⁴	1	32	yes	—
80	C₂²×C₄×C₁₃	—	2	16	no	—
81	C₂²×C₄×C₁₃	—	2	4	no	—
82	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
83	C₂²×C₄×C₁₃	#41	1	8	yes	—
84	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
85	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
86	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
87	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
88	C₂²×Dic₁₃	—	2	8	no	—
89	C₂²×Dic₁₃	—	2	8	no	—
90	C₂²×Dic₁₃	—	2	8	no	—
91	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
92	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
93	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
94	C₂×D₅₂	C₂×Q₈	1	96	yes	complemented
95	C₂×D₅₂	C₂⁴	1	16	yes	—
96	C₂×C₄×D₁₃	—	2	96	no	—
97	C₂×C₄×D₁₃	—	4	96	no	—
98	C₂×C₄×D₁₃	C₂⁴	1	32	yes	—
99	C₂×C₄×D₁₃	C₂⁴	1	64	yes	—
100	C₂×Dic₂₆	—	2	48	no	—
101	C₂×Dic₂₆	C₂⁴	1	16	yes	—
102	C₁₃×C₄⋊C₄	C₂⁴	1	32	yes	—
103	C₁₃×C₄⋊C₄	C₂⁴	1	32	yes	—
104	C₄²×C₁₃	C₂⁴	1	16	yes	—
105	#6103	C₂⁴	1	32	yes	—
106	#6103	C₂⁴	1	64	yes	—
107	C₄⋊Dic₁₃	C₂⁴	1	16	yes	—
108	C₄⋊Dic₁₃	C₂⁴	1	16	yes	—
109	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
110	#6101	C₂⁴	1	32	yes	—
111	#6101	C₂⁴	1	64	yes	—
112	C₄×Dic₁₃	C₂⁴	1	32	yes	—
113	C₂×#554	—	13	48	no	—
114	C₂²×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
115	C₂²×#125	—	13	2	no	—
116	C₂²×C₂²⋊Q₈	—	13	4	no	—
117	C₂²×C₂²⋊Q₈	—	13	2	no	—
118	C₂²×C₄×Q₈	—	13	1	no	—
119	C₂²×C₄×D₄	—	13	1	no	—
120	C₂²×C₄×D₄	—	13	2	no	—
121	C₂³×C₄⋊C₄	—	13	2	no	—
122	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
123	C₂³×C₁₃	C₂×#41	1	4	yes	—
124	C₂³×C₁₃	H(5,𝔽₂)	1	4	yes	—
125	C₂³×C₁₃	C₂⁵	1	3	yes	—
126	C₂²×D₁₃	—	2	64	no	—
127	C₂²×D₁₃	—	2	24	no	—
128	C₂²×D₁₃	—	2	24	no	—
129	C₂²×D₁₃	C₂²×Q₈	1	48	yes	—
130	C₂²×D₁₃	C₂⁵	1	8	yes	—
131	Q₈×C₁₃	—	4	32	no	—
132	C₂×C₄×C₁₃	—	2	48	no	—
133	C₂×C₄×C₁₃	—	2	12	no	—
134	C₂×C₄×C₁₃	C₂²×Q₈	1	24	yes	—
135	C₂×C₄×C₁₃	C₂×#41	1	24	yes	—
136	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
137	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
138	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
139	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
140	C₂×Dic₁₃	—	2	24	no	—
141	C₂×Dic₁₃	—	2	24	no	—
142	C₂×Dic₁₃	—	2	24	no	—
143	C₂×Dic₁₃	C₂⁵	1	8	yes	—
144	C₂×Dic₁₃	C₂⁵	1	8	yes	—
145	C₂×Dic₁₃	C₂⁵	1	8	yes	—
146	D₅₂	C₂²×Q₈	1	64	yes	complemented
147	C₄×D₁₃	—	2	64	no	—
148	C₄×D₁₃	—	4	64	no	—
149	Dic₂₆	—	2	32	no	—
150	C₂³×Q₈	—	13	1	no	—
151	C₂³×Q₈	C₂×D₁₃	1	2	yes	—
152	C₂³×D₄	—	13	1	no	—
153	C₂⁴×C₄	—	13	2	no	—
154	C₂⁴×C₄	—	13	4	no	—
155	#554	—	13	256	no	—
156	C₂×C₄⋊Q₈	C₂×D₁₃	1	24	yes	complemented
157	C₂×#125	—	13	48	no	—
158	C₂×C₂²⋊Q₈	—	13	96	no	—
159	C₂×C₂²⋊Q₈	—	13	48	no	—
160	C₂×C₄×Q₈	—	13	24	no	—
161	C₂×C₄×D₄	—	13	48	no	—
162	C₂×C₄×D₄	—	13	24	no	—
163	C₂²×C₄⋊C₄	—	13	4	no	—
164	C₂²×C₄⋊C₄	—	13	2	no	—
165	C₂²×C₄⋊C₄	—	13	1	no	—
166	C₂²×C₄⋊C₄	—	13	1	no	—
167	C₂²×C₄⋊C₄	—	13	4	no	—
168	C₂²×C₄⋊C₄	—	13	48	no	—
169	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
170	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
171	C₂²×C₂²⋊C₄	—	13	4	no	—
172	C₂²×C₂²⋊C₄	—	13	2	no	—
173	C₂²×C₄²	—	13	2	no	—
174	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
175	C₂²×C₁₃	#554	1	16	yes	complemented
176	C₂²×C₁₃	C₂³×Q₈	1	6	yes	—
177	C₂²×C₁₃	C₂²×#41	1	6	yes	—
178	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	6	yes	—
179	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
180	C₂×D₁₃	—	2	8	no	—
181	C₂×D₁₃	—	2	8	no	—
182	C₂×D₁₃	—	2	96	no	—
183	C₂×D₁₃	C₂³×Q₈	1	16	yes	—
184	C₄×C₁₃	—	2	16	no	—
185	C₄×C₁₃	—	2	4	no	—
186	C₄×C₁₃	C₂³×Q₈	1	8	yes	—
187	C₄×C₁₃	C₂²×#41	1	8	yes	—
188	Dic₁₃	—	2	8	no	—
189	Dic₁₃	—	2	8	no	—
190	Dic₁₃	—	2	8	no	—
191	C₂⁵	—	13	2	no	—
192	C₂²×Q₈	—	4	8	no	—
193	C₂²×Q₈	—	13	12	no	—
194	C₂²×Q₈	—	26	8	no	—
195	C₂²×Q₈	C₂²×D₁₃	1	24	yes	—
196	C₂²×D₄	—	13	16	no	—
197	C₂²×D₄	—	13	12	no	—
198	C₂³×C₄	—	13	2	no	—
199	C₂³×C₄	—	13	2	no	—
200	C₂³×C₄	—	13	2	no	—
201	C₂³×C₄	—	13	24	no	—
202	C₂³×C₄	—	13	48	no	—
203	C₂³×C₄	—	26	16	no	—
204	C₂³×C₄	—	52	16	no	—
205	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
206	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
207	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
208	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊Q₈	C₂²×D₁₃	1	64	yes	complemented
210	#125	—	13	128	no	—
211	C₂²⋊Q₈	—	13	256	no	—
212	C₂²⋊Q₈	—	13	128	no	—
213	C₄×Q₈	—	13	64	no	—
214	C₄×D₄	—	13	128	no	—
215	C₄×D₄	—	13	64	no	—
216	C₂×C₄⋊C₄	—	13	12	no	—
217	C₂×C₄⋊C₄	—	13	48	no	—
218	C₂×C₄⋊C₄	—	13	12	no	—
219	C₂×C₄⋊C₄	—	13	24	no	—
220	C₂×C₄⋊C₄	—	13	48	no	—
221	C₂×C₄⋊C₄	—	13	128	no	—
222	C₂×C₄⋊C₄	C₂²×D₁₃	1	24	yes	—
223	C₂×C₄⋊C₄	C₂²×D₁₃	1	24	yes	—
224	C₂×C₂²⋊C₄	—	13	48	no	—
225	C₂×C₂²⋊C₄	—	13	24	no	—
226	C₂×C₄²	—	13	24	no	—
227	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
228	C₂×C₁₃	C₂×#554	1	12	yes	complemented
229	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
230	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
231	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
232	D₁₃	—	2	16	no	—
233	C₂⁴	—	13	16	no	—
234	C₂⁴	—	13	12	no	—
235	C₂⁴	—	26	8	no	—
236	C₂⁴	—	26	8	no	—
237	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
238	C₂×Q₈	—	4	48	no	—
239	C₂×Q₈	—	13	16	no	—
240	C₂×Q₈	—	26	48	no	—
241	C₂×Q₈	C₂³×D₁₃	1	32	yes	—
242	C₂×D₄	—	13	96	no	—
243	C₂×D₄	—	13	16	no	—
244	C₂²×C₄	—	2	16	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	13	12	no	—
247	C₂²×C₄	—	13	12	no	—
248	C₂²×C₄	—	13	12	no	—
249	C₂²×C₄	—	13	32	no	—
250	C₂²×C₄	—	13	64	no	—
251	C₂²×C₄	—	26	8	no	—
252	C₂²×C₄	—	26	8	no	—
253	C₂²×C₄	—	26	8	no	—
254	C₂²×C₄	—	26	96	no	—
255	C₂²×C₄	—	52	96	no	—
256	C₂²×C₄	Q₈×D₁₃	1	8	yes	—
257	C₂²×C₄	#6131	1	8	yes	—
258	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
259	C₂²×C₄	C₂³×D₁₃	1	24	yes	—
260	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
261	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
262	C₄⋊C₄	—	13	16	no	—
263	C₄⋊C₄	—	13	64	no	—
264	C₄⋊C₄	—	13	16	no	—
265	C₄⋊C₄	—	13	32	no	—
266	C₄⋊C₄	—	13	64	no	—
267	C₄⋊C₄	C₂³×D₁₃	1	32	yes	—
268	C₄⋊C₄	C₂³×D₁₃	1	32	yes	—
269	C₂²⋊C₄	—	13	64	no	—
270	C₂²⋊C₄	—	13	32	no	—
271	C₄²	—	13	32	no	—
272	C₄²	C₂³×D₁₃	1	16	yes	—
273	C₁₃	C₂²×#554	1	1	yes	characteristic, complemented, Sylow
274	C₂³	—	13	48	no	—
275	C₂³	—	13	8	no	—
276	C₂³	—	26	64	no	—
277	C₂³	—	26	24	no	—
278	C₂³	—	26	24	no	—
279	C₂³	#88251	1	4	yes	—
280	C₂³	C₂×Q₈×D₁₃	1	4	yes	—
281	C₂³	C₂×#6131	1	4	yes	—
282	C₂³	C₂⁴×D₁₃	1	3	yes	—
283	Q₈	—	4	32	no	—
284	Q₈	—	26	32	no	—
285	D₄	—	13	64	no	—
286	C₂×C₄	—	2	48	no	—
287	C₂×C₄	—	2	12	no	—
288	C₂×C₄	—	13	8	no	—
289	C₂×C₄	—	13	8	no	—
290	C₂×C₄	—	13	8	no	—
291	C₂×C₄	—	26	24	no	—
292	C₂×C₄	—	26	24	no	—
293	C₂×C₄	—	26	24	no	—
294	C₂×C₄	—	26	64	no	—
295	C₂×C₄	—	52	64	no	—
296	C₂×C₄	C₂×Q₈×D₁₃	1	24	yes	—
297	C₂×C₄	C₂×#6131	1	24	yes	—
298	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
299	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
300	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
301	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
302	C₂²	—	13	16	no	—
303	C₂²	—	26	8	no	—
304	C₂²	—	26	8	no	—
305	C₂²	—	26	96	no	—
306	C₂²	#11719435	1	16	yes	direct product factor
307	C₂²	C₂×#88251	1	6	yes	—
308	C₂²	C₂²×Q₈×D₁₃	1	6	yes	—
309	C₂²	C₂²×#6131	1	6	yes	—
310	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
311	C₄	—	2	16	no	—
312	C₄	—	2	4	no	—
313	C₄	—	26	8	no	—
314	C₄	—	26	8	no	—
315	C₄	—	26	8	no	—
316	C₄	C₂²×Q₈×D₁₃	1	8	yes	—
317	C₄	C₂²×#6131	1	8	yes	—
318	C₂	—	26	16	no	—
319	C₂	C₂×#11719435	1	12	yes	direct product factor
320	C₂	C₂³×Q₈×D₁₃	1	1	yes	characteristic
321	C₂	C₂³×#6131	1	1	yes	characteristic
322	C₂	C₂²×#88251	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×C₄⋊Q₈	C₂
socle	C₂⁴×C₁₃	C₂⁴
prime radical	C₂³×D₅₂	C₂²
hypercenter	C₂²×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	C₂²×#554
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#11719435	C₂	direct (N × H)
2. C₂²×#88144	C₂	semidirect
3. C₂²×#88151	C₂	semidirect
4. C₂²×#88200	C₂	semidirect
5. C₂²×C₄×Dic₂₆	C₂	semidirect
8. C₂²×C₄⋊C₄×D₁₃	C₂	semidirect
9. C₂²×C₁₃×C₄⋊Q₈	C₂	semidirect
15. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
19. C₂×#88200	C₂²	semidirect
28. #11719435	C₂²	direct (N × H)
29. C₂×#88151	C₂²	semidirect
31. C₂×C₄⋊C₄×D₁₃	C₂²	semidirect
32. C₂×#88144	C₂²	semidirect
34. C₂×C₄×Dic₂₆	C₂²	semidirect
45. C₂²×D₅₂	Q₈	semidirect
53. C₁₃×C₄⋊Q₈	C₂³	semidirect
57. #88200	C₂³	semidirect
66. #88151	C₂³	semidirect
68. C₄⋊C₄×D₁₃	C₂³	semidirect
69. #88144	C₂³	semidirect
71. C₄×Dic₂₆	C₂³	semidirect
94. C₂×D₅₂	C₂×Q₈	semidirect
114. C₂²×C₄⋊Q₈	D₁₃	semidirect
146. D₅₂	C₂²×Q₈	semidirect
156. C₂×C₄⋊Q₈	C₂×D₁₃	semidirect
175. C₂²×C₁₃	#554	semidirect
209. C₄⋊Q₈	C₂²×D₁₃	semidirect
228. C₂×C₁₃	C₂×#554	semidirect
273. C₁₃	C₂²×#554	semidirect
306. C₂²	#11719435	direct (N × H)
319. C₂	C₂×#11719435	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11719435	2
C₂²×#88144	2
C₂²×#88151	2
C₂²×#88200	2
C₂²×C₄×Dic₂₆	2
C₂²×C₄×D₅₂	2
C₂²×#88147	2
C₂²×C₄⋊C₄×D₁₃	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×#554	13

Subgroups — C22×#11719435 #16769887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719435 #16769887