Subgroups — C₂×#11727000 #13508524

Order 1792.

558 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×#89968	C₂	1	1	yes	maximal, characteristic
2	C₂×#89627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#360	C₂	1	1	yes	maximal, characteristic
4	C₂×#89361	C₂	1	1	yes	maximal, characteristic
5	C₂×#89308	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#89306	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#89282	C₂	1	1	yes	maximal, characteristic, complemented
8	#11727000	C₂	1	8	yes	maximal, direct product factor
9	C₂×C₇×#135	C₂²	1	1	yes	characteristic
10	C₂×#6511	—	2	1	no	—
11	C₂×#6451	—	2	1	no	—
12	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₇×#106	C₂²	1	1	yes	characteristic
14	C₂×#6411	—	2	1	no	—
15	#89968	—	2	2	no	—
16	#89968	C₂²	1	4	yes	—
17	C₂×#6393	C₂²	1	1	yes	characteristic
18	C₂×#6391	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
20	C₂×#6386	C₂²	1	1	yes	characteristic
21	C₂×#6382	—	2	1	no	—
22	#89627	—	2	2	no	—
23	#89627	C₂²	1	4	yes	complemented
24	C₇×#360	C₂²	1	4	yes	—
25	#89361	C₂²	1	4	yes	—
26	#89308	C₂²	1	4	yes	complemented
27	#89306	C₂²	1	4	yes	complemented
28	#89282	C₂²	1	4	yes	complemented
29	C₂×#2230	—	7	1	no	maximal, Sylow
30	C₂×C₇×#41	—	2	1	no	—
31	C₂²×Dic₁₄	—	4	1	no	—
32	C₇×#135	—	2	1	no	—
33	C₇×#135	—	2	1	no	—
34	C₇×#135	Q₈	1	2	yes	complemented
35	C₇×#135	C₂³	1	2	yes	—
36	C₂×C₇×Q₁₆	Q₈	1	1	yes	characteristic, complemented
37	C₂×C₇×SD₁₆	—	2	1	no	—
38	C₂×C₇×D₈	Q₈	1	1	yes	characteristic, complemented
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₇×C₄⋊C₄	—	2	1	no	—
41	#6511	—	2	4	no	—
42	#6511	—	4	2	no	—
43	C₂×C₄⋊Dic₇	—	2	1	no	—
44	C₂×Dic₇⋊C₄	—	4	1	no	—
45	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
46	C₂×#1136	—	2	1	no	—
47	C₂×#1136	—	4	1	no	—
48	C₂²×C₇⋊C₈	—	4	1	no	—
49	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
50	#6451	—	2	4	no	—
51	#6451	—	4	2	no	—
52	C₇×M₃₂	—	2	1	no	—
53	C₇×M₃₂	C₂³	1	2	yes	—
54	C₂×C₇×C₁₆	—	2	1	no	—
55	C₇×#106	—	2	1	no	—
56	C₇×#106	C₂³	1	2	yes	—
57	#6411	—	2	4	no	—
58	#6393	—	2	1	no	—
59	#6393	C₂³	1	2	yes	—
60	#6391	—	2	1	no	—
61	#6391	C₂³	1	2	yes	—
62	C₈×Dic₇	D₄	1	2	yes	—
63	C₈×Dic₇	C₂³	1	2	yes	—
64	#6386	—	2	1	no	—
65	#6386	C₂³	1	2	yes	—
66	C₂×C₇⋊C₁₆	—	2	1	no	—
67	#6382	—	2	4	no	—
68	C₂×#498	—	7	1	no	—
69	C₂×#443	—	7	1	no	—
70	#2230	—	7	8	no	—
71	C₂×#365	—	7	1	no	—
72	C₂×#365	—	7	1	no	—
73	C₂×#364	—	7	1	no	—
74	C₂×#360	—	7	1	no	—
75	C₂×#360	D₇	1	1	yes	characteristic
76	C₇×#41	—	2	2	no	—
77	C₇×#41	—	2	2	no	—
78	C₇×#41	—	4	1	no	—
79	C₇×#41	—	4	1	no	—
80	C₂×C₇×Q₈	—	2	1	no	—
81	C₂×C₇×D₄	—	2	1	no	—
82	C₂×C₇×D₄	—	4	1	no	—
83	C₂²×C₄×C₇	—	4	1	no	—
84	C₂²×C₄×C₇	—	4	1	no	—
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₇	—	2	1	no	—
87	C₂²×Dic₇	—	4	1	no	—
88	C₂×Dic₁₄	—	4	4	no	—
89	C₂×Dic₁₄	—	8	4	no	—
90	C₇×Q₁₆	—	2	1	no	—
91	C₇×Q₁₆	C₂×Q₈	1	2	yes	complemented
92	C₇×SD₁₆	—	2	2	no	—
93	C₇×SD₁₆	—	2	2	no	—
94	C₇×D₈	—	2	1	no	—
95	C₇×D₈	C₂×Q₈	1	2	yes	complemented
96	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₇×C₈	#41	1	1	yes	characteristic
100	C₂×C₇×C₈	#41	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
102	C₇×C₄⋊C₄	—	2	2	no	—
103	C₇×C₄⋊C₄	—	4	1	no	—
104	C₄⋊Dic₇	—	2	2	no	—
105	C₄⋊Dic₇	—	4	1	no	—
106	Dic₇⋊C₄	—	4	4	no	—
107	C₄×Dic₇	—	2	1	no	—
108	C₄×Dic₇	C₂×D₄	1	2	yes	—
109	#1136	—	2	2	no	—
110	#1136	—	4	1	no	—
111	#1136	—	4	4	no	—
112	C₂×C₇⋊C₈	—	4	1	no	—
113	C₂×C₇⋊C₈	—	4	4	no	—
114	C₂×C₇⋊C₈	SD₁₆	1	1	yes	characteristic
115	C₂×C₇⋊C₈	SD₁₆	1	1	yes	characteristic
116	C₂×C₇⋊C₈	SD₁₆	1	2	yes	—
117	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
118	C₇×C₁₆	—	2	2	no	—
119	C₇⋊C₁₆	—	2	2	no	—
120	C₂×#135	C₂×D₇	1	1	yes	characteristic
121	C₂×#131	—	14	1	no	—
122	C₂×C₄⋊Q₈	—	14	1	no	—
123	C₂×M₃₂	—	7	1	no	—
124	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
125	#498	—	7	4	no	—
126	#498	—	14	2	no	—
127	#443	—	7	4	no	—
128	#443	—	14	2	no	—
129	C₂×#108	—	7	1	no	—
130	C₂×#106	—	7	1	no	—
131	C₂×#106	—	14	1	no	—
132	C₂×#106	C₂×D₇	1	1	yes	characteristic
133	C₂×#104	—	14	1	no	—
134	C₂×C₄×C₈	—	7	1	no	—
135	#365	—	7	4	no	—
136	#365	—	7	4	no	—
137	#364	—	7	4	no	—
138	#360	—	7	4	no	—
139	#360	C₂×D₇	1	4	yes	—
140	C₂³×C₇	—	4	1	no	—
141	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₇×Q₈	—	2	2	no	—
143	C₇×Q₈	—	4	1	no	—
144	C₇×D₄	—	2	2	no	—
145	C₇×D₄	—	4	2	no	—
146	C₇×D₄	—	4	2	no	—
147	C₇×D₄	—	4	1	no	—
148	C₂×C₄×C₇	—	4	1	no	—
149	C₂×C₄×C₇	—	4	2	no	—
150	C₂×C₄×C₇	—	4	2	no	—
151	C₂×C₄×C₇	—	4	2	no	—
152	C₂×C₄×C₇	—	8	2	no	—
153	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
157	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×Dic₇	—	2	4	no	—
160	C₂×Dic₇	—	2	2	no	—
161	C₂×Dic₇	—	4	2	no	—
162	C₂×Dic₇	—	8	2	no	—
163	Dic₁₄	—	8	8	no	—
164	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
165	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
166	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
167	C₇×C₈	C₂×#41	1	1	yes	characteristic
168	C₇⋊C₈	—	4	2	no	—
169	C₇⋊C₈	C₂×SD₁₆	1	2	yes	—
170	C₇⋊C₈	C₂×SD₁₆	1	2	yes	—
171	C₂×#41	—	2	1	no	—
172	C₂²×Q₈	—	28	1	no	—
173	#135	—	2	1	no	—
174	#135	—	2	1	no	—
175	#135	Dic₁₄	1	2	yes	complemented
176	#135	C₂²×D₇	1	2	yes	—
177	C₂×Q₁₆	Dic₁₄	1	1	yes	characteristic, complemented
178	C₂×SD₁₆	—	2	1	no	—
179	C₂×D₈	Dic₁₄	1	1	yes	characteristic, complemented
180	#131	—	14	4	no	—
181	#131	—	28	2	no	—
182	C₂×M₁₆	—	14	1	no	—
183	C₂×M₁₆	—	28	1	no	—
184	C₂²×C₈	—	7	1	no	—
185	C₂²×C₈	—	28	1	no	—
186	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
187	C₄⋊Q₈	—	14	4	no	—
188	C₄⋊Q₈	—	28	2	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	14	1	no	—
191	C₂×C₄⋊C₄	—	28	1	no	—
192	C₂×C₄²	—	7	1	no	—
193	M₃₂	—	2	1	no	—
194	M₃₂	—	7	2	no	—
195	M₃₂	—	14	1	no	—
196	M₃₂	C₂²×D₇	1	2	yes	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	14	1	no	—
199	#108	—	7	2	no	—
200	#108	—	14	1	no	—
201	#106	—	2	1	no	—
202	#106	—	7	2	no	—
203	#106	—	14	4	no	—
204	#106	—	14	1	no	—
205	#106	C₂²×D₇	1	2	yes	—
206	#104	—	14	4	no	—
207	C₄×C₈	—	7	2	no	—
208	C₄×C₈	—	7	2	no	—
209	C₂²×C₇	—	2	2	no	—
210	C₂²×C₇	—	4	2	no	—
211	C₂²×C₇	—	8	2	no	—
212	C₂²×C₇	#476	1	1	yes	characteristic
213	C₂²×C₇	#476	1	1	yes	characteristic
214	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₄×C₇	—	4	2	no	—
216	C₄×C₇	—	8	2	no	—
217	C₄×C₇	#476	1	1	yes	characteristic
218	C₄×C₇	#476	1	1	yes	characteristic
219	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
220	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
221	Dic₇	—	2	4	no	—
222	Dic₇	—	8	2	no	—
223	#41	—	2	2	no	—
224	#41	—	2	2	no	—
225	#41	—	4	1	no	—
226	#41	—	4	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	28	4	no	—
229	C₂×Q₈	—	56	4	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	14	1	no	—
235	C₂²×C₄	—	28	1	no	—
236	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
237	Q₁₆	—	2	1	no	—
238	Q₁₆	C₂×Dic₁₄	1	2	yes	complemented
239	SD₁₆	—	2	2	no	—
240	SD₁₆	—	2	2	no	—
241	D₈	—	2	1	no	—
242	D₈	C₂×Dic₁₄	1	2	yes	complemented
243	M₁₆	—	14	2	no	—
244	M₁₆	—	28	4	no	—
245	M₁₆	—	28	1	no	—
246	C₂×C₈	—	7	1	no	—
247	C₂×C₈	—	7	1	no	—
248	C₂×C₈	—	7	2	no	—
249	C₂×C₈	—	7	2	no	—
250	C₂×C₈	—	28	4	no	—
251	C₂×C₈	—	28	1	no	—
252	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
253	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
254	C₂×C₈	D₄×D₇	1	1	yes	characteristic
255	C₂×C₈	#1159	1	1	yes	characteristic
256	C₂×C₈	#1159	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	14	2	no	—
261	C₄⋊C₄	—	28	4	no	—
262	C₄⋊C₄	—	28	1	no	—
263	C₄²	—	7	2	no	—
264	C₄²	—	14	1	no	—
265	C₁₆	—	2	2	no	—
266	C₁₆	—	14	2	no	—
267	C₂×C₇	—	2	1	no	—
268	C₂×C₇	—	2	1	no	—
269	C₂×C₇	—	8	2	no	—
270	C₂×C₇	#2230	1	2	yes	complemented
271	C₂×C₇	C₂×#476	1	1	yes	characteristic
272	C₂³	—	4	1	no	—
273	C₂³	#6474	1	1	yes	characteristic
274	Q₈	—	2	2	no	—
275	Q₈	—	4	1	no	—
276	Q₈	—	56	8	no	—
277	D₄	—	2	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	14	4	no	—
287	C₂×C₄	—	14	2	no	—
288	C₂×C₄	—	28	2	no	—
289	C₂×C₄	—	56	2	no	—
290	C₂×C₄	#6441	1	1	yes	characteristic
291	C₂×C₄	#6441	1	1	yes	characteristic
292	C₂×C₄	#6474	1	1	yes	characteristic
293	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
294	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
296	C₈	—	7	2	no	—
297	C₈	—	7	2	no	—
298	C₈	—	28	2	no	—
299	C₈	#6441	1	1	yes	characteristic
300	C₈	#6441	1	1	yes	characteristic
301	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
302	C₈	C₂×#1159	1	1	yes	characteristic
303	C₇	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	#89533	1	1	yes	characteristic
308	C₂²	#89533	1	1	yes	characteristic
309	C₂²	C₂×#6474	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	14	4	no	—
313	C₄	—	56	2	no	—
314	C₄	#89533	1	1	yes	characteristic
315	C₄	#89533	1	1	yes	characteristic
316	C₄	C₂×#6441	1	1	yes	characteristic
317	C₄	C₂×D₇×SD₁₆	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	8	2	no	—
321	C₂	#11727000	1	2	yes	direct product factor
322	C₂	C₂×#89533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89533
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#360	C₂
socle	C₂²×C₇	#476
prime radical	C₂×C₇×#135	C₂²
hypercenter	C₂×#360	D₇
hypocenter	C₇	C₂×#2230
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
2. C₂×#89627	C₂	semidirect
5. C₂×#89308	C₂	semidirect
6. C₂×#89306	C₂	semidirect
7. C₂×#89282	C₂	semidirect
8. #11727000	C₂	direct (N × H)
23. #89627	C₂²	semidirect
26. #89308	C₂²	semidirect
27. #89306	C₂²	semidirect
28. #89282	C₂²	semidirect
34. C₇×#135	Q₈	semidirect
36. C₂×C₇×Q₁₆	Q₈	semidirect
38. C₂×C₇×D₈	Q₈	semidirect
91. C₇×Q₁₆	C₂×Q₈	semidirect
95. C₇×D₈	C₂×Q₈	semidirect
175. #135	Dic₁₄	semidirect
177. C₂×Q₁₆	Dic₁₄	semidirect
179. C₂×D₈	Dic₁₄	semidirect
238. Q₁₆	C₂×Dic₁₄	semidirect
242. D₈	C₂×Dic₁₄	semidirect
270. C₂×C₇	#2230	semidirect
303. C₇	C₂×#2230	semidirect
321. C₂	#11727000	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×#89968	2
C₂×#89627	2
C₂×C₇×#360	2
C₂×#89361	2
C₂×#89308	2
C₂×#89306	2
C₂×#89282	2
#11727000	2
C₂×#2230	7

Subgroups — C2×#11727000 #13508524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11727000 #13508524