Subgroups — #16789987

Order 3328.

440 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718233	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718233	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131078	C₂	1	2	yes	maximal
4	#13129886	C₂	1	2	yes	maximal, complemented
5	C₄×#88202	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
7	C₂×#88202	C₄	1	1	yes	characteristic
8	C₂×#88202	C₂²	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	2	yes	—
10	C₂×#88106	C₂²	1	2	yes	—
11	#11718658	—	2	1	no	—
12	#11718658	—	2	1	no	—
13	#11718658	C₄	1	1	yes	characteristic
14	#11718658	C₂²	1	1	yes	characteristic
15	C₄×#6099	—	2	1	no	—
16	C₄×#6099	C₄	1	1	yes	characteristic
17	C₂×C₄×#1055	—	2	1	no	—
18	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
19	#11718304	—	2	4	no	—
20	#11718233	—	2	1	no	—
21	#11718233	—	2	1	no	—
22	#11718233	—	2	1	no	—
23	#11718233	—	2	1	no	—
24	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₃	—	2	1	no	—
26	C₄×D₄×C₁₃	—	2	1	no	—
27	C₄×D₄×C₁₃	—	2	2	no	—
28	C₁₃×#117	—	2	1	no	—
29	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	—	2	1	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	#88202	—	2	2	no	—
33	#88202	—	2	1	no	—
34	#88202	—	2	1	no	—
35	#88202	C₈	1	4	yes	—
36	#88202	D₄	1	2	yes	—
37	#88202	Q₈	1	2	yes	—
38	C₂×#6099	—	2	1	no	—
39	C₂×#6099	—	2	1	no	—
40	C₂×#6099	C₂×C₄	1	1	yes	characteristic
41	C₂×#6099	C₂×C₄	1	1	yes	characteristic
42	C₂²×#1055	—	2	1	no	—
43	C₂²×#1055	—	2	1	no	—
44	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
45	C₂²×#1055	C₂³	1	1	yes	characteristic
46	#88106	—	2	2	no	—
47	#88106	—	2	2	no	—
48	#88106	—	2	2	no	—
49	#88106	—	2	2	no	—
50	C₂×#6092	—	2	2	no	—
51	C₂×#6092	—	2	2	no	—
52	#88043	—	2	1	no	—
53	#88043	—	2	1	no	—
54	#88043	—	2	1	no	—
55	C₄×#1055	—	2	1	no	—
56	C₄×#1055	—	2	1	no	—
57	C₄×#1055	—	2	1	no	—
58	C₄×#1055	D₄	1	1	yes	characteristic
59	C₄×#1055	D₄	1	1	yes	characteristic
60	C₄×#1055	D₄	1	1	yes	characteristic
61	C₄×#1055	D₄	1	1	yes	characteristic
62	#12792	—	13	1	no	maximal, Sylow
63	C₁₃×#41	—	2	1	no	—
64	C₁₃×#41	—	2	1	no	—
65	C₁₃×#41	C₄⋊C₄	1	2	yes	—
66	C₁₃×#41	C₂×C₈	1	2	yes	—
67	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
68	C₂×D₄×C₁₃	—	2	1	no	—
69	C₂×D₄×C₁₃	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	2	1	no	—
76	C₁₃×C₂²⋊C₄	—	2	2	no	—
77	C₁₃×C₂²⋊C₄	M₁₆	1	2	yes	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	#6099	—	2	1	no	—
84	#6099	—	2	1	no	—
85	#6099	—	2	4	no	—
86	#6099	—	2	2	no	—
87	#6099	—	2	2	no	—
88	#6099	C₄⋊C₄	1	2	yes	—
89	#6099	C₂×C₈	1	2	yes	—
90	C₂×#1055	—	2	1	no	—
91	C₂×#1055	—	2	1	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×#1055	C₂×C₈	1	2	yes	—
98	C₂×#1055	M₁₆	1	2	yes	—
99	C₂×#1055	C₂×D₄	1	1	yes	characteristic
100	C₂×#1055	C₂×D₄	1	1	yes	characteristic
101	C₂×#1055	C₂×D₄	1	1	yes	characteristic
102	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
103	C₂×#1055	#41	1	1	yes	characteristic
104	C₂×#1055	#41	1	1	yes	characteristic
105	#6092	—	4	2	no	—
106	#6092	—	4	2	no	—
107	C₄×#131	—	13	1	no	—
108	C₂×#364	—	13	1	no	—
109	C₂×#364	—	13	1	no	—
110	#2117	—	13	2	no	—
111	#1378	—	13	2	no	—
112	C₂³×C₁₃	—	2	1	no	—
113	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
114	Q₈×C₁₃	—	2	1	no	—
115	Q₈×C₁₃	C₄⋊C₈	1	2	yes	—
116	D₄×C₁₃	—	2	2	no	—
117	D₄×C₁₃	—	2	2	no	—
118	D₄×C₁₃	—	2	1	no	—
119	D₄×C₁₃	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
134	#1055	—	2	2	no	—
135	#1055	—	2	2	no	—
136	#1055	C₂²⋊C₈	1	2	yes	—
137	#1055	C₂²⋊C₈	1	2	yes	—
138	#1055	#121	1	1	yes	characteristic
139	#1055	#121	1	1	yes	characteristic
140	#1055	C₂²⋊Q₈	1	1	yes	characteristic
141	#1055	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×#131	—	13	1	no	—
143	C₂×#131	—	13	1	no	—
144	C₄×#41	#261	1	1	yes	characteristic
145	C₂×M₃₂	—	13	2	no	—
146	C₂×M₃₂	—	13	2	no	—
147	#405	—	13	1	no	—
148	#405	—	13	1	no	—
149	#405	—	26	1	no	—
150	#405	—	26	1	no	—
151	C₄×M₁₆	—	13	1	no	—
152	C₄×M₁₆	—	26	1	no	—
153	C₂×C₄×C₈	—	13	1	no	—
154	C₂×C₄×C₈	—	26	1	no	—
155	#364	—	26	1	no	—
156	#364	—	26	1	no	—
157	#364	—	26	1	no	—
158	#364	—	26	1	no	—
159	#350	—	26	4	no	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	4	1	no	—
162	C₂²×C₁₃	—	4	2	no	—
163	C₂²×C₁₃	#407	1	1	yes	characteristic
164	C₂²×C₁₃	#423	1	1	yes	characteristic
165	C₂²×C₁₃	#431	1	1	yes	characteristic
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	4	2	no	—
169	C₄×C₁₃	—	4	1	no	—
170	C₄×C₁₃	#380	1	1	yes	characteristic
171	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
172	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
173	C₄×C₁₃	#432	1	1	yes	characteristic
174	C₂×#41	C₂×#261	1	1	yes	characteristic
175	#131	—	13	4	no	—
176	#131	—	13	2	no	—
177	#131	—	13	2	no	—
178	#131	—	26	2	no	—
179	#131	—	26	1	no	—
180	#131	—	26	1	no	—
181	C₂×M₁₆	—	13	1	no	—
182	C₂×M₁₆	—	13	1	no	—
183	C₂×M₁₆	—	26	1	no	—
184	C₂×M₁₆	—	26	1	no	—
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	13	1	no	—
187	C₂²×C₈	—	26	1	no	—
188	C₂²×C₈	—	26	1	no	—
189	C₄×Q₈	—	2	1	no	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	2	no	—
192	#117	—	2	1	no	—
193	#117	C₂×#261	1	1	yes	characteristic
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	C₂×#261	1	1	yes	characteristic
196	M₃₂	—	26	2	no	—
197	M₃₂	—	26	2	no	—
198	M₃₂	—	26	2	no	—
199	M₃₂	—	26	2	no	—
200	C₂×C₁₆	—	26	2	no	—
201	C₂×C₁₆	—	26	2	no	—
202	#97	—	26	1	no	—
203	#97	—	26	1	no	—
204	#97	—	26	1	no	—
205	C₄×C₈	—	13	1	no	—
206	C₄×C₈	—	13	1	no	—
207	C₄×C₈	—	13	1	no	—
208	C₄×C₈	—	13	1	no	—
209	C₄×C₈	—	26	1	no	—
210	C₄×C₈	—	26	1	no	—
211	C₄×C₈	—	26	1	no	—
212	C₂×C₁₃	—	2	1	no	—
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	4	2	no	—
216	C₂×C₁₃	#1797	1	1	yes	characteristic
217	#41	—	2	1	no	—
218	#41	—	2	1	no	—
219	#41	D₁₃⋊C₈	1	2	yes	—
220	#41	C₄⋊#261	1	2	yes	—
221	C₂×Q₈	#6122	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	#6122	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
228	M₁₆	—	13	2	no	—
229	M₁₆	—	13	2	no	—
230	M₁₆	—	26	4	no	—
231	M₁₆	—	26	2	no	—
232	M₁₆	—	26	2	no	—
233	M₁₆	—	26	1	no	—
234	M₁₆	—	26	1	no	—
235	C₂×C₈	—	13	2	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	1	no	—
238	C₂×C₈	—	13	1	no	—
239	C₂×C₈	—	13	1	no	—
240	C₂×C₈	—	13	1	no	—
241	C₂×C₈	—	13	1	no	—
242	C₂×C₈	—	13	1	no	—
243	C₂×C₈	—	13	1	no	—
244	C₂×C₈	—	13	1	no	—
245	C₂×C₈	—	26	4	no	—
246	C₂×C₈	—	26	2	no	—
247	C₂×C₈	—	26	2	no	—
248	C₂×C₈	—	26	1	no	—
249	C₂×C₈	—	26	1	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	#6122	1	2	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	C₂²⋊#261	1	1	yes	characteristic
258	C₄²	C₂²⋊#261	1	1	yes	characteristic
259	C₁₆	—	52	2	no	—
260	C₁₆	—	52	2	no	—
261	C₁₃	#12792	1	1	yes	characteristic, complemented, Sylow
262	C₂³	—	2	1	no	—
263	C₂³	C₂×#6122	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	#88112	1	2	yes	—
266	D₄	—	2	2	no	—
267	D₄	—	2	2	no	—
268	D₄	—	2	1	no	—
269	D₄	#88112	1	2	yes	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
279	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
281	C₂×C₄	#88237	1	1	yes	characteristic
282	C₂×C₄	C₂×#6122	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
284	C₈	—	13	2	no	—
285	C₈	—	13	2	no	—
286	C₈	—	13	1	no	—
287	C₈	—	13	1	no	—
288	C₈	—	13	1	no	—
289	C₈	—	13	1	no	—
290	C₈	—	26	2	no	—
291	C₈	—	26	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#11719125	1	1	yes	characteristic
296	C₂²	#11719129	1	1	yes	characteristic
297	C₂²	#11719169	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	1	no	—
302	C₄	C₂×#88112	1	1	yes	characteristic
303	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
304	C₄	#11719128	1	1	yes	characteristic
305	C₄	#11719136	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	#13130692	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11719136
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₄×C₁₃×#41	C₄
socle	C₂×C₁₃	#1797
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂²×#1055	C₂³
hypercenter	C₄×#41	#261
hypocenter	C₁₃	#12792
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#11718233	C₂	semidirect
2. C₂×#11718233	C₂	semidirect
4. #13129886	C₂	semidirect
261. C₁₃	#12792	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718233	2
#13131078	2
#13129886	2
C₄×#88202	2
#12792	13