Subgroups — C₂×C₄×#88103 #16797209

Order 3328.

1086 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88103	C₂	1	2	yes	maximal
2	C₂×C₄×#6118	C₂	1	2	yes	maximal
3	C₄×#88103	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718268	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6118	C₂²	1	4	yes	—
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88110	—	2	2	no	—
11	C₂×C₄×#1057	—	2	2	no	—
12	C₂×#88103	C₄	1	16	yes	direct product factor
13	C₂×#88103	C₂²	1	8	yes	—
14	C₄×#6118	—	2	4	no	—
15	C₄×#6118	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
19	C₂×C₄×#1055	C₄	1	1	yes	characteristic
20	C₄×C₈×D₁₃	C₄	1	4	yes	—
21	C₄×C₈×D₁₃	C₂²	1	4	yes	—
22	#11718268	C₂²	1	8	yes	—
23	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
24	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
25	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
26	C₂²×#1057	—	2	4	no	—
27	C₂×#6118	—	2	4	no	—
28	C₂×#6118	—	2	8	no	—
29	C₂×#6118	C₂×C₄	1	16	yes	—
30	C₂×#6118	C₂³	1	8	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₁₃	—	2	4	no	—
40	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
43	C₂×C₈×D₁₃	C₂³	1	2	yes	—
44	C₂×C₈×D₁₃	C₂³	1	2	yes	—
45	C₄²×D₁₃	C₂×C₄	1	2	yes	—
46	C₄²×D₁₃	D₄	1	2	yes	—
47	C₄²×D₁₃	Q₈	1	2	yes	—
48	C₄²×D₁₃	C₂³	1	2	yes	—
49	#88110	—	2	8	no	—
50	C₄×#1057	—	2	8	no	—
51	#88103	C₂×C₄	1	32	yes	direct product factor
52	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
53	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
54	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
55	C₄×#1055	C₂×C₄	1	4	yes	—
56	C₂×C₄×#108	—	13	1	no	maximal, Sylow
57	C₂³×D₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×D₁₃	—	2	4	no	—
65	C₂×C₄×D₁₃	—	2	4	no	—
66	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
70	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
72	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
73	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₄×D₁₃	#41	1	4	yes	—
75	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
76	C₂×#1057	—	2	16	no	—
77	C₂×#1057	—	2	8	no	—
78	#6118	—	2	16	no	—
79	#6118	—	2	8	no	—
80	#6118	C₂²×C₄	1	16	yes	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	C₄²	1	4	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
87	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
89	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
90	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
91	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
92	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	C₄²	1	4	yes	—
95	C₂×#1055	C₂²×C₄	1	2	yes	—
96	C₂×#1055	C₂²×C₄	1	2	yes	—
97	C₈×D₁₃	—	2	4	no	—
98	C₈×D₁₃	—	2	4	no	—
99	C₈×D₁₃	C₄²	1	8	yes	—
100	C₈×D₁₃	C₂²×C₄	1	8	yes	—
101	C₂²×#108	—	13	2	no	—
102	C₂×C₄×M₁₆	—	13	2	no	—
103	C₂²×C₄×C₈	—	13	1	no	—
104	C₄×#108	—	13	8	no	—
105	C₂×#341	—	13	2	no	—
106	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×D₁₃	—	2	2	no	—
108	C₂²×D₁₃	—	2	2	no	—
109	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×D₁₃	C₄×Q₈	1	4	yes	—
111	C₂²×D₁₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	#108	1	4	yes	complemented
114	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	#117	1	2	yes	—
119	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×Dic₁₃	—	2	2	no	—
121	C₂×Dic₁₃	#108	1	4	yes	complemented
122	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×Dic₁₃	C₄×D₄	1	4	yes	—
126	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₄×D₁₃	—	2	4	no	—
128	C₄×D₁₃	—	2	4	no	—
129	C₄×D₁₃	—	2	4	no	—
130	C₄×D₁₃	—	2	4	no	—
131	C₄×D₁₃	C₂×C₄²	1	2	yes	—
132	C₄×D₁₃	#117	1	2	yes	—
133	C₄×D₁₃	C₄×D₄	1	4	yes	—
134	C₄×D₁₃	C₄×Q₈	1	4	yes	—
135	C₄×D₁₃	C₂³×C₄	1	2	yes	—
136	C₄×D₁₃	C₂×#41	1	2	yes	—
137	#1057	—	2	16	no	—
138	C₈×C₁₃	—	2	2	no	—
139	C₈×C₁₃	C₂×C₄²	1	4	yes	—
140	#1055	—	2	2	no	—
141	#1055	C₂×C₄²	1	4	yes	—
142	C₂²×M₁₆	—	13	4	no	—
143	C₂³×C₈	—	13	1	no	—
144	C₂³×C₈	—	13	1	no	—
145	C₂²×C₄²	—	13	1	no	—
146	C₂×#108	—	13	16	no	—
147	C₂×#108	—	13	8	no	—
148	C₄×M₁₆	—	13	8	no	—
149	C₄×M₁₆	—	26	4	no	—
150	C₂×#97	—	26	2	no	—
151	C₂×C₄×C₈	—	13	1	no	—
152	C₂×C₄×C₈	—	13	1	no	—
153	C₂×C₄×C₈	—	13	1	no	—
154	C₂×C₄×C₈	—	13	4	no	—
155	C₂×C₄×C₈	—	13	4	no	—
156	C₂×C₄×C₈	—	26	2	no	—
157	C₂×C₄×C₈	#261	1	1	yes	characteristic
158	#341	—	13	8	no	—
159	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
160	C₂²×C₁₃	C₂×#108	1	2	yes	—
161	C₂²×C₁₃	C₂×#108	1	2	yes	—
162	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₂×D₁₃	—	2	1	no	—
164	C₂×D₁₃	—	2	1	no	—
165	C₂×D₁₃	—	2	1	no	—
166	C₂×D₁₃	—	2	1	no	—
167	C₂×D₁₃	—	2	8	no	—
168	C₂×D₁₃	C₄×C₄⋊C₄	1	2	yes	—
169	C₂×D₁₃	C₂×C₄×Q₈	1	2	yes	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
172	C₄×C₁₃	C₂×#108	1	4	yes	complemented
173	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
174	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
175	Dic₁₃	—	2	2	no	—
176	Dic₁₃	C₄×C₄⋊C₄	1	2	yes	—
177	Dic₁₃	C₂×#108	1	4	yes	complemented
178	Dic₁₃	C₂×C₄×D₄	1	2	yes	—
179	C₂³×C₄	—	13	1	no	—
180	C₂³×C₄	—	13	1	no	—
181	C₂³×C₄	—	13	1	no	—
182	C₂×M₁₆	—	13	8	no	—
183	C₂×M₁₆	—	13	16	no	—
184	C₂×M₁₆	—	26	4	no	—
185	C₂×M₁₆	—	26	8	no	—
186	C₂²×C₈	—	13	1	no	—
187	C₂²×C₈	—	13	8	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	13	1	no	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	—	13	2	no	—
193	C₂²×C₈	—	26	4	no	—
194	C₂²×C₈	—	26	4	no	—
195	C₂²×C₈	C₂×#261	1	1	yes	characteristic
196	C₂²×C₈	C₂×#261	1	1	yes	characteristic
197	C₂×C₄²	—	13	1	no	—
198	C₂×C₄²	—	13	1	no	—
199	C₂×C₄²	—	13	1	no	—
200	C₂×C₄²	—	13	2	no	—
201	C₂×C₄²	—	13	2	no	—
202	C₂×C₄²	—	13	2	no	—
203	C₂×C₄²	—	13	2	no	—
204	C₂×C₄²	C₂×#261	1	1	yes	characteristic
205	#108	—	13	32	no	—
206	#97	—	26	8	no	—
207	C₄×C₈	—	13	4	no	—
208	C₄×C₈	—	13	4	no	—
209	C₄×C₈	—	13	4	no	—
210	C₄×C₈	—	26	8	no	—
211	C₄×C₈	C₂×#261	1	4	yes	—
212	C₂×C₁₃	C₄×#108	1	4	yes	complemented
213	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
214	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
215	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
216	D₁₃	—	2	2	no	—
217	D₁₃	—	2	2	no	—
218	C₂⁴	—	13	1	no	—
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	1	no	—
221	C₂²×C₄	—	13	4	no	—
222	C₂²×C₄	—	13	1	no	—
223	C₂²×C₄	—	13	1	no	—
224	C₂²×C₄	—	13	4	no	—
225	C₂²×C₄	—	13	1	no	—
226	C₂²×C₄	—	13	1	no	—
227	C₂²×C₄	—	13	1	no	—
228	C₂²×C₄	—	13	2	no	—
229	C₂²×C₄	—	13	2	no	—
230	C₂²×C₄	—	13	2	no	—
231	C₂²×C₄	—	13	2	no	—
232	C₂²×C₄	—	26	4	no	—
233	C₂²×C₄	—	26	4	no	—
234	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
235	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
236	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
237	M₁₆	—	13	16	no	—
238	M₁₆	—	26	16	no	—
239	M₁₆	—	26	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	13	4	no	—
242	C₂×C₈	—	13	2	no	—
243	C₂×C₈	—	13	8	no	—
244	C₂×C₈	—	13	8	no	—
245	C₂×C₈	—	13	2	no	—
246	C₂×C₈	—	26	4	no	—
247	C₂×C₈	—	26	4	no	—
248	C₂×C₈	—	26	8	no	—
249	C₂×C₈	—	26	16	no	—
250	C₂×C₈	—	26	2	no	—
251	C₂×C₈	C₄×#261	1	4	yes	—
252	C₂×C₈	C₂²×#261	1	2	yes	—
253	C₂×C₈	C₂²×#261	1	2	yes	—
254	C₄²	—	13	2	no	—
255	C₄²	—	13	2	no	—
256	C₄²	—	13	2	no	—
257	C₄²	—	13	2	no	—
258	C₄²	—	13	2	no	—
259	C₄²	—	13	2	no	—
260	C₄²	C₄⋊#261	1	2	yes	—
261	C₄²	C₂²×#261	1	2	yes	—
262	C₁₃	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
263	C₂³	—	13	4	no	—
264	C₂³	—	13	1	no	—
265	C₂³	—	13	1	no	—
266	C₂³	—	26	2	no	—
267	C₂³	—	26	2	no	—
268	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	13	4	no	—
271	C₂×C₄	—	13	4	no	—
272	C₂×C₄	—	13	1	no	—
273	C₂×C₄	—	13	1	no	—
274	C₂×C₄	—	13	2	no	—
275	C₂×C₄	—	13	4	no	—
276	C₂×C₄	—	13	4	no	—
277	C₂×C₄	—	13	2	no	—
278	C₂×C₄	—	13	2	no	—
279	C₂×C₄	—	13	2	no	—
280	C₂×C₄	—	13	2	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	26	4	no	—
283	C₂×C₄	—	26	4	no	—
284	C₂×C₄	—	26	2	no	—
285	C₂×C₄	—	26	4	no	—
286	C₂×C₄	—	26	4	no	—
287	C₂×C₄	#88103	1	4	yes	direct product factor
288	C₂×C₄	C₂×C₄×#261	1	2	yes	—
289	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
292	C₂×C₄	#88237	1	2	yes	—
293	C₂×C₄	C₂³×#261	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	13	4	no	—
296	C₈	—	26	16	no	—
297	C₈	—	26	2	no	—
298	C₈	C₂×C₄×#261	1	4	yes	—
299	C₂²	—	13	2	no	—
300	C₂²	—	13	2	no	—
301	C₂²	—	26	8	no	—
302	C₂²	—	26	1	no	—
303	C₂²	—	26	1	no	—
304	C₂²	—	26	1	no	—
305	C₂²	—	26	1	no	—
306	C₂²	C₄×C₄⋊#261	1	2	yes	—
307	C₂²	C₂×#88103	1	2	yes	—
308	C₂²	D₁₃⋊#108	1	2	yes	—
309	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	13	2	no	—
312	C₄	—	13	2	no	—
313	C₄	—	13	4	no	—
314	C₄	—	26	2	no	—
315	C₄	C₄×C₄⋊#261	1	2	yes	—
316	C₄	C₂×#88103	1	4	yes	direct product factor
317	C₄	C₂²×C₄×#261	1	1	yes	characteristic
318	C₄	C₂×#88237	1	1	yes	characteristic
319	C₂	—	26	2	no	—
320	C₂	—	26	2	no	—
321	C₂	C₄×#88103	1	4	yes	direct product factor
322	C₂	C₂×C₄×C₄⋊#261	1	1	yes	characteristic
323	C₂	C₂²×#88103	1	1	yes	characteristic
324	C₂	C₂×D₁₃⋊#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×Q₈
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×C₄×#108
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#88103	C₂	direct (N × H)
12. C₂×#88103	C₄	direct (N × H)
51. #88103	C₂×C₄	direct (N × H)
113. C₂×C₄×C₁₃	#108	semidirect
121. C₂×Dic₁₃	#108	semidirect
172. C₄×C₁₃	C₂×#108	semidirect
177. Dic₁₃	C₂×#108	semidirect
212. C₂×C₁₃	C₄×#108	semidirect
262. C₁₃	C₂×C₄×#108	semidirect
287. C₂×C₄	#88103	direct (N × H)
316. C₄	C₂×#88103	direct (N × H)
321. C₂	C₄×#88103	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88103	2
C₂×C₄×#6118	2
C₄×#88103	2
C₂×#11718268	2
C₂×C₄×C₈×D₁₃	2
C₂×C₄×#108	13

Subgroups — C2×C4×#88103 #16797209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88103 #16797209