Subgroups — C₂×#11727625 #13354061

Order 1792.

730 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×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇⋊#338	C₂	1	1	yes	maximal, characteristic
3	C₂×#89351	C₂	1	2	yes	maximal
4	C₂×#89352	C₂	1	1	yes	maximal, characteristic
5	C₂×#89318	C₂	1	2	yes	maximal
6	#11727625	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×#6489	C₂²	1	2	yes	—
11	C₂²×#1136	C₂²	1	2	yes	—
12	C₄×C₇×M₁₆	C₄	1	4	yes	complemented
13	C₄×C₇×M₁₆	C₂²	1	4	yes	—
14	C₂×C₇×#97	C₄	1	2	yes	complemented
15	C₂×C₄×C₇×C₈	C₄	1	2	yes	complemented
16	C₇⋊#338	C₂²	1	4	yes	—
17	#89352	C₂²	1	4	yes	—
18	#89351	C₂²	1	8	yes	—
19	#89318	C₂²	1	8	yes	—
20	C₂×#1841	—	7	1	no	maximal, Sylow
21	C₂³×C₄×C₇	—	2	1	no	—
22	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×C₇×M₁₆	—	2	2	no	—
24	C₂×C₇×M₁₆	—	2	2	no	—
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×M₁₆	—	2	2	no	—
27	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₇×M₁₆	C₂³	1	2	yes	—
30	C₂×C₇×M₁₆	C₂³	1	2	yes	—
31	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇	—	2	1	no	—
34	C₂×C₄²×C₇	—	2	1	no	—
35	C₂×C₄²×C₇	—	2	1	no	—
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇	C₂³	1	2	yes	—
40	C₂×C₂²⋊Dic₇	—	2	2	no	—
41	#6489	—	2	2	no	—
42	#6489	—	2	2	no	—
43	#6489	—	2	2	no	—
44	#6489	C₂³	1	4	yes	—
45	C₂×C₄⋊Dic₇	—	2	2	no	—
46	C₂×C₄×Dic₇	—	2	2	no	—
47	C₂×#1136	—	2	2	no	—
48	C₂×#1136	—	2	2	no	—
49	C₂×#1136	—	2	2	no	—
50	C₂×#1136	—	2	2	no	—
51	C₂×#1136	—	2	2	no	—
52	C₂×#1136	C₂³	1	4	yes	—
53	C₂²×C₇⋊C₈	—	2	2	no	—
54	C₇×#97	C₂×C₄	1	8	yes	complemented
55	C₄×C₇×C₈	C₂×C₄	1	8	yes	complemented
56	C₂×C₄×M₁₆	D₇	1	1	yes	characteristic
57	#1841	—	7	8	no	—
58	C₂×#344	—	7	2	no	—
59	C₂×#339	—	7	2	no	—
60	C₂×#338	—	7	1	no	—
61	C₂×H(3,ℤ/4)	—	7	1	no	—
62	C₂⁴×C₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₇	—	2	1	no	—
64	C₂²×C₄×C₇	—	2	2	no	—
65	C₂²×C₄×C₇	—	2	1	no	—
66	C₂²×C₄×C₇	—	2	2	no	—
67	C₂²×C₄×C₇	—	4	4	no	—
68	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₇	#41	1	2	yes	—
72	C₂²×C₄×C₇	#41	1	2	yes	—
73	C₂²×C₄×C₇	#41	1	2	yes	—
74	C₂²×C₄×C₇	#41	1	2	yes	—
75	C₂²×C₄×C₇	#41	1	1	yes	characteristic
76	C₂²×C₄×C₇	#41	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₇	—	4	2	no	—
79	C₇×M₁₆	—	2	4	no	—
80	C₇×M₁₆	—	2	4	no	—
81	C₇×M₁₆	—	2	4	no	—
82	C₇×M₁₆	—	2	4	no	—
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
89	C₄²×C₇	—	2	1	no	—
90	C₄²×C₇	—	2	2	no	—
91	C₄²×C₇	—	2	1	no	—
92	C₄²×C₇	—	2	2	no	—
93	C₄²×C₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₇	C₂²×C₄	1	2	yes	—
95	C₂²⋊Dic₇	—	2	4	no	—
96	C₂²⋊Dic₇	—	4	2	no	—
97	C₄⋊Dic₇	—	2	4	no	—
98	C₄⋊Dic₇	—	4	2	no	—
99	C₄×Dic₇	—	2	4	no	—
100	C₄×Dic₇	—	4	2	no	—
101	#1136	—	2	4	no	—
102	#1136	—	2	4	no	—
103	#1136	—	4	2	no	—
104	#1136	—	4	2	no	—
105	C₂×C₇⋊C₈	—	2	4	no	—
106	C₂×C₇⋊C₈	—	4	2	no	—
107	C₂×C₇⋊C₈	—	4	2	no	—
108	C₂²×M₁₆	—	7	2	no	—
109	C₂²×M₁₆	C₂×D₇	1	1	yes	characteristic
110	C₂²×M₁₆	C₂×D₇	1	1	yes	characteristic
111	C₂×#117	—	7	2	no	—
112	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
113	C₄×M₁₆	Dic₇	1	4	yes	complemented
114	C₄×M₁₆	C₂×D₇	1	4	yes	—
115	C₂×#97	Dic₇	1	2	yes	complemented
116	C₂×C₄×C₈	Dic₇	1	2	yes	complemented
117	#344	—	7	8	no	—
118	#339	—	7	8	no	—
119	#338	—	7	4	no	—
120	H(3,ℤ/4)	—	7	4	no	—
121	C₂³×C₇	—	4	2	no	—
122	C₂³×C₇	#117	1	1	yes	characteristic
123	C₂³×C₇	#117	1	1	yes	characteristic
124	C₂³×C₇	#126	1	2	yes	—
125	C₂³×C₇	#126	1	2	yes	—
126	C₂³×C₇	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₇	—	2	4	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	4	4	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	—	4	2	no	—
133	C₂×C₄×C₇	#117	1	1	yes	characteristic
134	C₂×C₄×C₇	#117	1	1	yes	characteristic
135	C₂×C₄×C₇	#117	1	1	yes	characteristic
136	C₂×C₄×C₇	#117	1	2	yes	—
137	C₂×C₄×C₇	#117	1	2	yes	—
138	C₂×C₄×C₇	#117	1	1	yes	characteristic
139	C₂×C₄×C₇	#117	1	1	yes	characteristic
140	C₂×C₄×C₇	#117	1	1	yes	characteristic
141	C₂×C₄×C₇	#126	1	2	yes	—
142	C₂×C₄×C₇	#126	1	2	yes	—
143	C₂×C₄×C₇	#126	1	2	yes	—
144	C₂×C₄×C₇	#126	1	2	yes	—
145	C₂×C₄×C₇	#126	1	2	yes	—
146	C₂×C₄×C₇	#126	1	2	yes	—
147	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂×#41	1	2	yes	—
149	C₂×C₄×C₇	C₂×#41	1	2	yes	—
150	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
151	C₂×Dic₇	—	4	4	no	—
152	C₂×Dic₇	—	8	4	no	—
153	C₇×C₈	—	2	4	no	—
154	C₇×C₈	—	2	4	no	—
155	C₇⋊C₈	—	4	4	no	—
156	C₂³×C₄	—	2	1	no	—
157	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	7	4	no	—
163	C₂×M₁₆	—	14	2	no	—
164	C₂×M₁₆	—	14	2	no	—
165	C₂×M₁₆	—	14	2	no	—
166	C₂×M₁₆	—	14	2	no	—
167	C₂×M₁₆	—	14	2	no	—
168	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
169	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
170	C₂×M₁₆	C₂²×D₇	1	2	yes	—
171	C₂×M₁₆	C₂²×D₇	1	2	yes	—
172	C₂²×C₈	—	14	2	no	—
173	C₂²×C₈	C₂×Dic₇	1	2	yes	—
174	C₂²×C₈	C₂×Dic₇	1	2	yes	—
175	#117	—	7	4	no	—
176	#117	—	14	2	no	—
177	#117	—	14	2	no	—
178	#117	—	14	2	no	—
179	C₂×C₄⋊C₄	—	14	2	no	—
180	C₂×C₂²⋊C₄	—	14	2	no	—
181	C₂×C₄²	—	2	1	no	—
182	C₂×C₄²	—	2	1	no	—
183	C₂×C₄²	—	2	1	no	—
184	C₂×C₄²	—	14	2	no	—
185	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₇	1	2	yes	—
188	C₂×C₄²	C₂²×D₇	1	2	yes	—
189	#97	C₂×Dic₇	1	8	yes	complemented
190	C₄×C₈	C₂×Dic₇	1	8	yes	complemented
191	C₂²×C₇	—	2	4	no	—
192	C₂²×C₇	—	2	2	no	—
193	C₂²×C₇	—	4	1	no	—
194	C₂²×C₇	—	4	1	no	—
195	C₂²×C₇	—	4	2	no	—
196	C₂²×C₇	#383	1	1	yes	characteristic
197	C₂²×C₇	#383	1	2	yes	—
198	C₂²×C₇	#383	1	1	yes	characteristic
199	C₂²×C₇	C₂×#117	1	1	yes	characteristic
200	C₂²×C₇	C₂×#126	1	2	yes	—
201	C₄×C₇	—	2	4	no	—
202	C₄×C₇	—	4	2	no	—
203	C₄×C₇	#383	1	1	yes	characteristic
204	C₄×C₇	#383	1	2	yes	—
205	C₄×C₇	#383	1	1	yes	characteristic
206	C₄×C₇	C₂×#117	1	1	yes	characteristic
207	C₄×C₇	C₂×#117	1	1	yes	characteristic
208	C₄×C₇	C₂×#126	1	2	yes	—
209	Dic₇	—	8	4	no	—
210	C₂⁴	#1157	1	1	yes	characteristic
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	28	2	no	—
217	C₂²×C₄	#1157	1	1	yes	characteristic
218	C₂²×C₄	#1157	1	1	yes	characteristic
219	C₂²×C₄	#1157	1	1	yes	characteristic
220	C₂²×C₄	#1157	1	2	yes	—
221	C₂²×C₄	#1157	1	2	yes	—
222	C₂²×C₄	#1157	1	2	yes	—
223	C₂²×C₄	#1157	1	2	yes	—
224	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
226	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
227	M₁₆	—	2	4	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	14	4	no	—
232	M₁₆	—	14	4	no	—
233	M₁₆	—	28	2	no	—
234	M₁₆	—	28	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	14	4	no	—
240	C₂×C₈	—	28	2	no	—
241	C₂×C₈	—	28	2	no	—
242	C₂×C₈	C₂²×Dic₇	1	4	yes	—
243	C₂×C₈	C₂²×Dic₇	1	4	yes	—
244	C₄⋊C₄	—	14	4	no	—
245	C₄⋊C₄	—	28	2	no	—
246	C₂²⋊C₄	—	14	4	no	—
247	C₂²⋊C₄	—	28	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	14	4	no	—
253	C₄²	—	28	2	no	—
254	C₄²	C₂²×Dic₇	1	2	yes	—
255	C₄²	C₂²×Dic₇	1	2	yes	—
256	C₂×C₇	—	2	2	no	—
257	C₂×C₇	—	2	1	no	—
258	C₂×C₇	—	2	2	no	—
259	C₂×C₇	—	2	1	no	—
260	C₂×C₇	#1841	1	2	yes	complemented
261	C₂×C₇	C₂×#383	1	1	yes	characteristic
262	C₂³	—	4	2	no	—
263	C₂³	#6439	1	2	yes	—
264	C₂³	#6439	1	2	yes	—
265	C₂³	#6489	1	1	yes	characteristic
266	C₂³	#6489	1	1	yes	characteristic
267	C₂³	C₂×#1157	1	1	yes	characteristic
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	28	4	no	—
275	C₂×C₄	—	56	4	no	—
276	C₂×C₄	#6439	1	2	yes	—
277	C₂×C₄	#6439	1	2	yes	—
278	C₂×C₄	#6439	1	2	yes	—
279	C₂×C₄	#6439	1	2	yes	—
280	C₂×C₄	#6439	1	2	yes	—
281	C₂×C₄	#6439	1	2	yes	—
282	C₂×C₄	#6489	1	1	yes	characteristic
283	C₂×C₄	#6489	1	1	yes	characteristic
284	C₂×C₄	#6489	1	1	yes	characteristic
285	C₂×C₄	#6489	1	1	yes	characteristic
286	C₂×C₄	#6489	1	1	yes	characteristic
287	C₂×C₄	#6489	1	2	yes	—
288	C₂×C₄	#6489	1	2	yes	—
289	C₂×C₄	#6489	1	1	yes	characteristic
290	C₂×C₄	C₂×#1157	1	1	yes	characteristic
291	C₂×C₄	C₂×#1157	1	2	yes	—
292	C₂×C₄	C₂×#1157	1	2	yes	—
293	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
294	C₈	—	2	4	no	—
295	C₈	—	2	4	no	—
296	C₈	—	28	4	no	—
297	C₇	C₂×#1841	1	1	yes	characteristic, complemented, Sylow
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#89709	1	2	yes	—
304	C₂²	#89709	1	1	yes	characteristic
305	C₂²	#89709	1	1	yes	characteristic
306	C₂²	C₂×#6439	1	2	yes	—
307	C₂²	C₂×#6489	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	56	4	no	—
311	C₄	#89709	1	2	yes	—
312	C₄	#89709	1	1	yes	characteristic
313	C₄	#89709	1	1	yes	characteristic
314	C₄	C₂×#6439	1	2	yes	—
315	C₄	C₂×#6489	1	1	yes	characteristic
316	C₄	C₂×#6489	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	#11727625	1	2	yes	direct product factor
322	C₂	C₂×#89709	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6489
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₂×C₄×C₇×M₁₆	C₂
socle	C₂²×C₇	#383
prime radical	C₂⁴×C₇	#41
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₇
hypocenter	C₇	C₂×#1841
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
6. #11727625	C₂	direct (N × H)
12. C₄×C₇×M₁₆	C₄	semidirect
14. C₂×C₇×#97	C₄	semidirect
15. C₂×C₄×C₇×C₈	C₄	semidirect
54. C₇×#97	C₂×C₄	semidirect
55. C₄×C₇×C₈	C₂×C₄	semidirect
113. C₄×M₁₆	Dic₇	semidirect
115. C₂×#97	Dic₇	semidirect
116. C₂×C₄×C₈	Dic₇	semidirect
189. #97	C₂×Dic₇	semidirect
190. C₄×C₈	C₂×Dic₇	semidirect
260. C₂×C₇	#1841	semidirect
297. C₇	C₂×#1841	semidirect
321. C₂	#11727625	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×C₄×C₇×M₁₆	2
C₂×C₇⋊#338	2
C₂×#89351	2
C₂×#89352	2
C₂×#89318	2
#11727625	2
C₂×#1841	7

Subgroups — C2×#11727625 #13354061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11727625 #13354061