Subgroups — C₂²×C₁₃×#362 #15719729

Order 3328.

1134 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#362	C₂	1	12	yes	maximal, direct product factor
5	C₂³×Q₈×C₁₃	—	2	1	no	—
6	C₂²×C₁₃×Q₁₆	—	2	3	no	—
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	—	2	3	no	—
9	C₂²×C₁₃×Q₁₆	C₄	1	6	yes	—
10	C₂²×C₁₃×Q₁₆	C₄	1	2	yes	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	6	yes	—
12	C₂²×C₁₃×M₁₆	—	2	1	no	—
13	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₃₂	—	2	3	no	—
15	C₂×C₁₃×M₃₂	C₂²	1	6	yes	—
16	C₂²×C₁₃×C₁₆	—	2	1	no	—
17	C₂×C₁₃×#108	—	2	3	no	—
18	C₂×C₁₃×#108	C₂²	1	6	yes	—
19	C₁₃×#362	C₂²	1	16	yes	direct product factor
20	C₂²×Q₈×C₁₃	—	2	6	no	—
21	C₂²×Q₈×C₁₃	—	2	2	no	—
22	C₂²×Q₈×C₁₃	—	2	6	no	—
23	C₂²×Q₈×C₁₃	—	4	3	no	—
24	C₂²×Q₈×C₁₃	—	4	1	no	—
25	C₂²×Q₈×C₁₃	—	4	3	no	—
26	C₂³×C₄×C₁₃	—	4	1	no	—
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₁₃×Q₁₆	—	2	6	no	—
29	C₂×C₁₃×Q₁₆	—	2	12	no	—
30	C₂×C₁₃×Q₁₆	—	2	6	no	—
31	C₂×C₁₃×Q₁₆	—	2	6	no	—
32	C₂×C₁₃×Q₁₆	—	2	6	no	—
33	C₂×C₁₃×Q₁₆	—	2	6	no	—
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	12	yes	—
35	C₂×C₁₃×Q₁₆	C₂×C₄	1	12	yes	—
36	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
37	C₂×C₁₃×M₁₆	—	2	6	no	—
38	C₂×C₁₃×M₁₆	—	4	3	no	—
39	C₂²×C₈×C₁₃	—	4	1	no	—
40	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
41	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	D₄	1	3	yes	—
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	D₄	1	3	yes	—
45	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
46	C₁₃×M₃₂	—	2	6	no	—
47	C₁₃×M₃₂	C₂³	1	4	yes	—
48	C₂×C₁₃×C₁₆	—	2	6	no	—
49	C₁₃×#108	—	2	6	no	—
50	C₁₃×#108	C₂³	1	4	yes	—
51	C₂²×#362	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
53	C₂×Q₈×C₁₃	—	2	12	no	—
54	C₂×Q₈×C₁₃	—	2	12	no	—
55	C₂×Q₈×C₁₃	—	2	4	no	—
56	C₂×Q₈×C₁₃	—	4	12	no	—
57	C₂×Q₈×C₁₃	—	4	6	no	—
58	C₂×Q₈×C₁₃	—	4	6	no	—
59	C₂×Q₈×C₁₃	—	4	6	no	—
60	C₂×Q₈×C₁₃	—	4	6	no	—
61	C₂×Q₈×C₁₃	—	4	6	no	—
62	C₂²×C₄×C₁₃	—	4	6	no	—
63	C₂²×C₄×C₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₃	—	4	6	no	—
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	3	yes	—
66	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	D₈	1	3	yes	—
68	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	SD₁₆	1	3	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
71	C₁₃×Q₁₆	—	2	12	no	—
72	C₁₃×Q₁₆	—	2	12	no	—
73	C₁₃×Q₁₆	—	2	4	no	—
74	C₁₃×Q₁₆	C₂²×C₄	1	8	yes	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₁₃×M₁₆	—	4	6	no	—
77	C₂×C₈×C₁₃	—	4	6	no	—
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	3	yes	—
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	6	yes	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	3	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₁₃×C₁₆	—	2	4	no	—
88	C₂³×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
89	C₂²×M₃₂	C₂×C₁₃	1	1	yes	characteristic
90	C₂²×#108	C₂×C₁₃	1	1	yes	characteristic
91	C₂×#362	C₂×C₁₃	1	12	yes	direct product factor
92	C₂³×C₁₃	—	2	4	no	—
93	C₂³×C₁₃	#102	1	3	yes	—
94	C₂³×C₁₃	#102	1	1	yes	characteristic
95	C₂³×C₁₃	C₂×SD₁₆	1	3	yes	—
96	Q₈×C₁₃	—	2	8	no	—
97	Q₈×C₁₃	—	4	4	no	—
98	Q₈×C₁₃	—	4	12	no	—
99	Q₈×C₁₃	—	4	12	no	—
100	C₂×C₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	—	4	12	no	—
102	C₂×C₄×C₁₃	—	4	12	no	—
103	C₂×C₄×C₁₃	#102	1	6	yes	—
104	C₂×C₄×C₁₃	#102	1	3	yes	—
105	C₂×C₄×C₁₃	#102	1	3	yes	—
106	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
108	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
109	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
110	C₂×C₄×C₁₃	C₂×SD₁₆	1	3	yes	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₁₃	—	4	4	no	—
113	C₈×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
114	C₈×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
115	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂³×Q₈	—	2	1	no	—
118	C₂²×Q₁₆	—	2	3	no	—
119	C₂²×Q₁₆	—	2	1	no	—
120	C₂²×Q₁₆	—	2	3	no	—
121	C₂²×Q₁₆	C₄×C₁₃	1	6	yes	—
122	C₂²×Q₁₆	C₄×C₁₃	1	2	yes	—
123	C₂²×Q₁₆	C₂²×C₁₃	1	6	yes	—
124	C₂²×M₁₆	—	2	1	no	—
125	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
126	C₂×M₃₂	—	2	3	no	—
127	C₂×M₃₂	C₂²×C₁₃	1	6	yes	—
128	C₂²×C₁₆	—	2	1	no	—
129	C₂×#108	—	2	3	no	—
130	C₂×#108	C₂²×C₁₃	1	6	yes	—
131	#362	C₂²×C₁₃	1	16	yes	direct product factor
132	C₂²×C₁₃	—	2	6	no	—
133	C₂²×C₁₃	—	2	6	no	—
134	C₂²×C₁₃	#362	1	4	yes	direct product factor
135	C₂²×C₁₃	C₂×#102	1	3	yes	—
136	C₂²×C₁₃	C₂×#102	1	3	yes	—
137	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
138	C₄×C₁₃	—	4	8	no	—
139	C₄×C₁₃	C₂×#102	1	3	yes	—
140	C₄×C₁₃	C₂×#102	1	3	yes	—
141	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
143	C₂²×Q₈	—	2	6	no	—
144	C₂²×Q₈	—	2	2	no	—
145	C₂²×Q₈	—	2	6	no	—
146	C₂²×Q₈	—	4	1	no	—
147	C₂²×Q₈	—	4	3	no	—
148	C₂²×Q₈	—	4	3	no	—
149	C₂³×C₄	—	4	1	no	—
150	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
151	C₂×Q₁₆	—	2	6	no	—
152	C₂×Q₁₆	—	2	12	no	—
153	C₂×Q₁₆	—	2	6	no	—
154	C₂×Q₁₆	—	2	6	no	—
155	C₂×Q₁₆	—	2	6	no	—
156	C₂×Q₁₆	—	2	6	no	—
157	C₂×Q₁₆	C₂×C₄×C₁₃	1	12	yes	—
158	C₂×Q₁₆	C₂×C₄×C₁₃	1	12	yes	—
159	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
160	C₂×M₁₆	—	2	6	no	—
161	C₂×M₁₆	—	4	3	no	—
162	C₂²×C₈	—	4	1	no	—
163	C₂²×C₈	C₂×C₄×C₁₃	1	3	yes	—
164	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
165	C₂²×C₈	D₄×C₁₃	1	3	yes	—
166	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₁₃	1	3	yes	—
168	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
169	M₃₂	—	2	6	no	—
170	M₃₂	C₂³×C₁₃	1	4	yes	—
171	C₂×C₁₆	—	2	6	no	—
172	#108	—	2	6	no	—
173	#108	C₂³×C₁₃	1	4	yes	—
174	C₂×C₁₃	—	2	1	no	—
175	C₂×C₁₃	—	2	3	no	—
176	C₂×C₁₃	C₂×#362	1	6	yes	direct product factor
177	C₂×C₁₃	C₂²×#102	1	1	yes	characteristic
178	C₂⁴	C₁₃×SD₁₆	1	1	yes	characteristic
179	C₂×Q₈	—	2	12	no	—
180	C₂×Q₈	—	2	12	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	4	12	no	—
183	C₂×Q₈	—	4	6	no	—
184	C₂×Q₈	—	4	6	no	—
185	C₂×Q₈	—	4	6	no	—
186	C₂×Q₈	—	4	6	no	—
187	C₂×Q₈	—	4	6	no	—
188	C₂²×C₄	—	4	6	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	6	no	—
191	C₂²×C₄	C₁₃×C₂²⋊C₄	1	3	yes	—
192	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
194	C₂²×C₄	C₁₃×D₈	1	3	yes	—
195	C₂²×C₄	C₁₃×SD₁₆	1	3	yes	—
196	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
197	Q₁₆	—	2	12	no	—
198	Q₁₆	—	2	12	no	—
199	Q₁₆	—	2	4	no	—
200	Q₁₆	C₂²×C₄×C₁₃	1	8	yes	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	6	no	—
203	C₂×C₈	—	4	6	no	—
204	C₂×C₈	C₁₃×C₂²⋊C₄	1	3	yes	—
205	C₂×C₈	C₁₃×C₂²⋊C₄	1	6	yes	—
206	C₂×C₈	C₁₃×C₂²⋊C₄	1	3	yes	—
207	C₂×C₈	C₂²×C₄×C₁₃	1	3	yes	—
208	C₂×C₈	C₂²×C₄×C₁₃	1	3	yes	—
209	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
210	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
211	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
212	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
213	C₁₆	—	2	4	no	—
214	C₁₃	C₂²×#362	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	2	4	no	—
216	C₂³	C₁₃×#102	1	3	yes	—
217	C₂³	C₁₃×#102	1	1	yes	characteristic
218	C₂³	C₂×C₁₃×SD₁₆	1	3	yes	—
219	Q₈	—	2	8	no	—
220	Q₈	—	4	12	no	—
221	Q₈	—	4	12	no	—
222	Q₈	—	4	4	no	—
223	C₂×C₄	—	4	12	no	—
224	C₂×C₄	—	4	12	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	C₁₃×#102	1	6	yes	—
227	C₂×C₄	C₁₃×#102	1	3	yes	—
228	C₂×C₄	C₁₃×#102	1	3	yes	—
229	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
230	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
231	C₂×C₄	C₂×C₁₃×D₈	1	3	yes	—
232	C₂×C₄	C₂×C₁₃×D₈	1	3	yes	—
233	C₂×C₄	C₂×C₁₃×SD₁₆	1	3	yes	—
234	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
235	C₈	—	4	4	no	—
236	C₈	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
237	C₈	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
238	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
239	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
240	C₂²	—	2	6	no	—
241	C₂²	—	2	6	no	—
242	C₂²	C₁₃×#362	1	4	yes	direct product factor
243	C₂²	C₂×C₁₃×#102	1	3	yes	—
244	C₂²	C₂×C₁₃×#102	1	3	yes	—
245	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
246	C₄	—	4	8	no	—
247	C₄	C₂×C₁₃×#102	1	3	yes	—
248	C₄	C₂×C₁₃×#102	1	3	yes	—
249	C₄	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
250	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
251	C₂	—	2	3	no	—
252	C₂	—	2	1	no	—
253	C₂	C₂×C₁₃×#362	1	6	yes	direct product factor
254	C₂	C₂²×C₁₃×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	#102
commutator	C₈	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₁₃×#362	C₂	direct (N × H)
19. C₁₃×#362	C₂²	direct (N × H)
51. C₂²×#362	C₁₃	direct (N × H)
91. C₂×#362	C₂×C₁₃	direct (N × H)
131. #362	C₂²×C₁₃	direct (N × H)
134. C₂²×C₁₃	#362	direct (N × H)
176. C₂×C₁₃	C₂×#362	direct (N × H)
214. C₁₃	C₂²×#362	direct (N × H)
242. C₂²	C₁₃×#362	direct (N × H)
253. C₂	C₂×C₁₃×#362	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×Q₁₆	2
C₂²×C₁₃×M₃₂	2
C₂²×C₁₃×#108	2
C₂×C₁₃×#362	2
C₂²×#362	13

Subgroups — C22×C13×#362 #15719729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#362 #15719729