Subgroups — C₂×#11725527 #13268712

Order 1792.

630 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×#89878	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6386	C₂	1	1	yes	maximal, characteristic
4	#11725527	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
7	C₂×#6489	—	2	1	no	—
8	C₂×C₇×#106	—	2	1	no	—
9	C₂×C₇×#106	—	2	2	no	—
10	C₂×C₇×#106	C₄	1	4	yes	complemented
11	C₂×C₇×#106	C₂²	1	2	yes	—
12	#89878	—	2	1	no	—
13	#89878	—	2	1	no	—
14	#89878	C₄	1	2	yes	complemented
15	#89878	C₂²	1	2	yes	—
16	C₂×#6392	C₄	1	2	yes	complemented
17	C₂×#6391	—	2	1	no	—
18	C₂×#6386	—	2	1	no	—
19	C₂×#6386	—	2	2	no	—
20	C₂×#6386	C₄	1	4	yes	complemented
21	C₂×#6386	C₂²	1	2	yes	—
22	C₂²×C₇⋊C₁₆	—	2	1	no	—
23	C₂×#1386	—	7	1	no	maximal, Sylow
24	C₂³×C₄×C₇	—	4	1	no	—
25	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₇×C₈	D₄	1	2	yes	—
34	C₂²×C₇×C₈	Q₈	1	2	yes	—
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₇×C₄⋊C₄	—	2	4	no	—
37	C₂×C₇×C₄⋊C₄	—	2	2	no	—
38	C₂×C₇×C₄⋊C₄	—	4	1	no	—
39	C₂×C₇×C₄⋊C₄	—	4	2	no	—
40	C₂×C₂²⋊Dic₇	—	4	1	no	—
41	#6489	—	2	2	no	—
42	#6489	—	2	2	no	—
43	#6489	—	4	1	no	—
44	#6489	—	4	1	no	—
45	C₂×C₄⋊Dic₇	—	2	2	no	—
46	C₂×C₄×Dic₇	—	4	1	no	—
47	C₇×#106	—	2	4	no	—
48	C₇×#106	—	2	2	no	—
49	C₇×#106	C₂×C₄	1	4	yes	complemented
50	#6392	—	2	2	no	—
51	#6392	C₂×C₄	1	4	yes	complemented
52	#6391	—	2	2	no	—
53	#6391	—	2	2	no	—
54	#6386	—	2	4	no	—
55	#6386	—	2	2	no	—
56	#6386	C₂×C₄	1	4	yes	complemented
57	C₂×C₇⋊C₁₆	—	2	4	no	—
58	C₂×C₇⋊C₁₆	—	2	2	no	—
59	C₂²×M₃₂	—	7	1	no	—
60	C₂×#427	—	7	1	no	—
61	C₂²×#106	D₇	1	1	yes	characteristic
62	#1386	—	7	4	no	—
63	C₂⁴×C₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	—	4	4	no	—
65	C₂²×C₄×C₇	—	4	2	no	—
66	C₂²×C₄×C₇	—	8	4	no	—
67	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₇	Q₁₆	1	2	yes	—
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₇	—	4	2	no	—
78	C₂×C₇×C₈	C₄²	1	2	yes	—
79	C₂×C₇×C₈	C₄²	1	2	yes	—
80	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
88	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
89	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
90	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₇×C₄⋊C₄	—	2	4	no	—
97	C₇×C₄⋊C₄	—	4	4	no	—
98	C₇×C₄⋊C₄	—	4	2	no	—
99	C₂²⋊Dic₇	—	4	2	no	—
100	C₂²⋊Dic₇	—	4	2	no	—
101	C₄⋊Dic₇	—	2	4	no	—
102	C₄⋊Dic₇	—	4	2	no	—
103	C₄×Dic₇	—	4	2	no	—
104	C₄×Dic₇	—	4	2	no	—
105	C₇⋊C₁₆	—	2	4	no	—
106	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
107	C₂×#117	—	14	1	no	—
108	C₂²×C₄⋊C₄	—	2	1	no	—
109	C₂×M₃₂	—	7	4	no	—
110	C₂×M₃₂	—	7	2	no	—
111	C₂×M₃₂	—	14	1	no	—
112	C₂×M₃₂	—	14	2	no	—
113	C₂²×C₁₆	—	14	1	no	—
114	#427	—	7	2	no	—
115	#427	—	7	2	no	—
116	#427	—	14	1	no	—
117	#427	—	14	1	no	—
118	C₂×#107	—	7	2	no	—
119	C₂×#106	—	2	1	no	—
120	C₂×#106	—	2	2	no	—
121	C₂×#106	—	14	1	no	—
122	C₂×#106	Dic₇	1	4	yes	complemented
123	C₂×#106	C₂×D₇	1	2	yes	—
124	C₂³×C₇	—	2	4	no	—
125	C₂³×C₇	#102	1	2	yes	—
126	C₂³×C₇	#102	1	1	yes	characteristic
127	C₂³×C₇	#102	1	1	yes	characteristic
128	C₂³×C₇	#107	1	2	yes	—
129	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₇	—	4	4	no	—
131	C₂×C₄×C₇	—	8	4	no	—
132	C₂×C₄×C₇	—	8	8	no	—
133	C₂×C₄×C₇	#95	1	2	yes	—
134	C₂×C₄×C₇	#95	1	2	yes	—
135	C₂×C₄×C₇	#102	1	2	yes	—
136	C₂×C₄×C₇	#102	1	1	yes	characteristic
137	C₂×C₄×C₇	#102	1	1	yes	characteristic
138	C₂×C₄×C₇	#103	1	2	yes	—
139	C₂×C₄×C₇	#103	1	2	yes	—
140	C₂×C₄×C₇	#103	1	2	yes	—
141	C₂×C₄×C₇	#103	1	2	yes	—
142	C₂×C₄×C₇	#106	1	2	yes	—
143	C₂×C₄×C₇	#106	1	2	yes	—
144	C₂×C₄×C₇	#107	1	2	yes	—
145	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
151	C₂×Dic₇	—	4	4	no	—
152	C₂×Dic₇	—	8	4	no	—
153	C₇×C₈	#95	1	2	yes	—
154	C₇×C₈	#95	1	2	yes	—
155	C₇×C₈	C₂×C₄²	1	1	yes	characteristic
156	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂³×C₄	—	4	1	no	—
160	C₂³×C₄	C₂²⋊D₇	1	1	yes	characteristic
161	C₂²×C₈	Dic₁₄	1	2	yes	—
162	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
163	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₇	1	2	yes	—
165	C₂²×C₈	D₂₈	1	1	yes	characteristic
166	C₂²×C₈	D₂₈	1	1	yes	characteristic
167	C₂²×C₈	C₂×Dic₇	1	2	yes	—
168	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
170	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
171	#117	—	14	2	no	—
172	#117	—	14	2	no	—
173	#117	—	28	1	no	—
174	#117	—	28	1	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₄⋊C₄	—	4	2	no	—
179	C₂×C₄⋊C₄	—	14	2	no	—
180	C₂×C₂²⋊C₄	—	28	1	no	—
181	C₂×C₄²	—	28	1	no	—
182	M₃₂	—	7	4	no	—
183	M₃₂	—	14	4	no	—
184	M₃₂	—	14	2	no	—
185	C₂×C₁₆	—	14	4	no	—
186	C₂×C₁₆	—	14	2	no	—
187	#107	—	7	4	no	—
188	#107	—	14	2	no	—
189	#106	—	2	4	no	—
190	#106	—	2	2	no	—
191	#106	—	14	2	no	—
192	#106	—	14	2	no	—
193	#106	C₂×Dic₇	1	4	yes	complemented
194	C₂²×C₇	—	2	2	no	—
195	C₂²×C₇	—	2	2	no	—
196	C₂²×C₇	—	2	4	no	—
197	C₂²×C₇	—	2	4	no	—
198	C₂²×C₇	#340	1	2	yes	—
199	C₂²×C₇	#340	1	2	yes	—
200	C₂²×C₇	#360	1	4	yes	—
201	C₂²×C₇	C₂×#102	1	1	yes	characteristic
202	C₂²×C₇	C₂×#102	1	1	yes	characteristic
203	C₂²×C₇	C₂×#107	1	1	yes	characteristic
204	C₄×C₇	—	8	4	no	—
205	C₄×C₇	#340	1	2	yes	—
206	C₄×C₇	#340	1	2	yes	—
207	C₄×C₇	C₂×#95	1	1	yes	characteristic
208	C₄×C₇	C₂×#103	1	1	yes	characteristic
209	C₄×C₇	C₂×#103	1	1	yes	characteristic
210	C₄×C₇	C₂×#106	1	1	yes	characteristic
211	Dic₇	—	8	4	no	—
212	C₂⁴	#1141	1	1	yes	characteristic
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	8	4	no	—
216	C₂²×C₄	—	28	2	no	—
217	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
218	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	#1141	1	1	yes	characteristic
221	C₂²×C₄	#1142	1	2	yes	—
222	C₂²×C₄	#1143	1	1	yes	characteristic
223	C₂²×C₄	#1143	1	1	yes	characteristic
224	C₂²×C₄	#1144	1	2	yes	—
225	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
227	C₂×C₈	C₄×Dic₇	1	2	yes	—
228	C₂×C₈	C₄×Dic₇	1	2	yes	—
229	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
230	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
231	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
232	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
233	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
234	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
235	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
236	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
237	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
238	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
239	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
242	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
243	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
244	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	14	4	no	—
249	C₄⋊C₄	—	28	2	no	—
250	C₂²⋊C₄	—	28	2	no	—
251	C₂²⋊C₄	—	28	2	no	—
252	C₄²	—	28	2	no	—
253	C₄²	—	28	2	no	—
254	C₁₆	—	14	4	no	—
255	C₂×C₇	—	2	1	no	—
256	C₂×C₇	—	2	2	no	—
257	C₂×C₇	—	2	1	no	—
258	C₂×C₇	#1386	1	4	yes	complemented
259	C₂×C₇	C₂×#340	1	1	yes	characteristic
260	C₂×C₇	C₂×#360	1	2	yes	—
261	C₂³	—	2	4	no	—
262	C₂³	#6381	1	2	yes	—
263	C₂³	#6383	1	1	yes	characteristic
264	C₂³	#6383	1	1	yes	characteristic
265	C₂³	#6406	1	2	yes	—
266	C₂³	C₂×#1141	1	1	yes	characteristic
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	8	8	no	—
270	C₂×C₄	—	28	4	no	—
271	C₂×C₄	—	56	4	no	—
272	C₂×C₄	#6381	1	2	yes	—
273	C₂×C₄	#6382	1	2	yes	—
274	C₂×C₄	#6382	1	2	yes	—
275	C₂×C₄	#6383	1	1	yes	characteristic
276	C₂×C₄	#6383	1	1	yes	characteristic
277	C₂×C₄	#6384	1	2	yes	—
278	C₂×C₄	#6384	1	2	yes	—
279	C₂×C₄	C₇⋊#95	1	2	yes	—
280	C₂×C₄	C₇⋊#95	1	2	yes	—
281	C₂×C₄	#6406	1	2	yes	—
282	C₂×C₄	#6409	1	2	yes	—
283	C₂×C₄	#6409	1	2	yes	—
284	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
285	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₂×#1142	1	1	yes	characteristic
287	C₂×C₄	C₂×#1143	1	1	yes	characteristic
288	C₂×C₄	C₂×#1144	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
290	C₈	C₇⋊#95	1	2	yes	—
291	C₈	C₇⋊#95	1	2	yes	—
292	C₈	C₂×C₄×Dic₇	1	1	yes	characteristic
293	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
294	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
295	C₈	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
296	C₇	C₂×#1386	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	#89283	1	4	yes	—
302	C₂²	#89324	1	2	yes	—
303	C₂²	#89324	1	2	yes	—
304	C₂²	C₂×#6381	1	1	yes	characteristic
305	C₂²	C₂×#6383	1	1	yes	characteristic
306	C₂²	C₂×#6406	1	1	yes	characteristic
307	C₄	—	8	4	no	—
308	C₄	—	56	4	no	—
309	C₄	#89324	1	2	yes	—
310	C₄	#89324	1	2	yes	—
311	C₄	C₂×#6382	1	1	yes	characteristic
312	C₄	C₂×#6384	1	1	yes	characteristic
313	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
314	C₄	C₂×#6409	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	2	no	—
318	C₂	#11725527	1	4	yes	direct product factor
319	C₂	C₂×#89283	1	2	yes	—
320	C₂	C₂×#89324	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6383
commutator	C₇×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂²×C₇×#106	C₂
socle	C₂³×C₇	#102
prime radical	C₂⁴×C₇	D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂²×#106	D₇
hypocenter	C₇	C₂×#1386
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
4. #11725527	C₂	direct (N × H)
10. C₂×C₇×#106	C₄	semidirect
14. #89878	C₄	semidirect
16. C₂×#6392	C₄	semidirect
20. C₂×#6386	C₄	semidirect
49. C₇×#106	C₂×C₄	semidirect
51. #6392	C₂×C₄	semidirect
56. #6386	C₂×C₄	semidirect
122. C₂×#106	Dic₇	semidirect
193. #106	C₂×Dic₇	semidirect
258. C₂×C₇	#1386	semidirect
296. C₇	C₂×#1386	semidirect
318. C₂	#11725527	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×C₇×#106	2
C₂×#89878	2
C₂²×#6386	2
#11725527	2
C₂×#1386	7

Subgroups — C2×#11725527 #13268712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11725527 #13268712