Subgroups — C₂×#11727069 #13508525

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×#89667	C₂	1	1	yes	maximal, characteristic
2	C₂×#89627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#89313	C₂	1	1	yes	maximal, characteristic
5	C₂×#89306	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#89289	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#89283	C₂	1	1	yes	maximal, characteristic
8	#11727069	C₂	1	8	yes	maximal, direct product factor
9	C₂×#6470	—	2	1	no	—
10	C₂×#6467	C₂²	1	1	yes	characteristic
11	C₂×#6451	—	2	1	no	—
12	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₇×#108	C₂²	1	1	yes	characteristic
14	C₂×C₇×#106	C₂²	1	1	yes	characteristic
15	C₂×#6399	—	2	1	no	—
16	C₂×#6391	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×#6386	C₂²	1	1	yes	characteristic
19	C₂×#6382	—	2	1	no	—
20	#89667	—	2	2	no	—
21	#89667	C₂²	1	4	yes	—
22	#89627	—	2	2	no	—
23	#89627	C₂²	1	4	yes	complemented
24	C₇×#365	C₂²	1	4	yes	complemented
25	#89313	C₂²	1	4	yes	—
26	#89306	C₂²	1	4	yes	complemented
27	#89289	C₂²	1	4	yes	complemented
28	#89283	C₂²	1	4	yes	—
29	C₂×#2230	—	7	1	no	maximal, Sylow
30	C₂×#1157	—	2	1	no	—
31	C₂²×Dic₁₄	—	4	1	no	—
32	C₂×C₇×M₁₆	—	2	1	no	—
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₇×C₄⋊C₄	—	2	1	no	—
35	C₂×C₄⋊Dic₇	—	2	1	no	—
36	C₂×Dic₇⋊C₄	—	4	1	no	—
37	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
38	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
39	#6470	—	2	4	no	—
40	#6470	—	4	2	no	—
41	C₂×Dic₂₈	Q₈	1	1	yes	characteristic, complemented
42	#6467	—	2	1	no	—
43	#6467	—	2	1	no	—
44	#6467	Q₈	1	2	yes	complemented
45	#6467	C₂³	1	2	yes	—
46	C₂×D₅₆	Q₈	1	1	yes	characteristic, complemented
47	C₂×#1132	—	2	1	no	—
48	C₂×#1131	—	4	1	no	—
49	C₂×C₈×D₇	—	4	1	no	—
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×#108	—	2	1	no	—
56	C₇×#108	C₂³	1	2	yes	—
57	C₇×#106	—	2	1	no	—
58	C₇×#106	C₂³	1	2	yes	—
59	#6399	—	2	4	no	—
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	D₇	1	1	yes	characteristic, complemented
73	C₂×#364	—	7	1	no	—
74	C₂×#360	—	7	1	no	—
75	C₂×#360	—	7	1	no	—
76	C₂²×C₄×C₇	—	4	1	no	—
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₇	—	4	1	no	—
79	C₂²×Dic₇	—	2	1	no	—
80	C₂²×Dic₇	—	4	1	no	—
81	#1157	—	2	2	no	—
82	#1157	—	2	2	no	—
83	#1157	—	4	1	no	—
84	#1157	—	4	1	no	—
85	C₂×D₂₈	—	2	1	no	—
86	C₂×C₄×D₇	—	4	1	no	—
87	C₂×Dic₁₄	—	2	1	no	—
88	C₂×Dic₁₄	—	4	4	no	—
89	C₂×Dic₁₄	—	8	4	no	—
90	C₇×M₁₆	—	2	2	no	—
91	C₇×M₁₆	—	4	1	no	—
92	C₂×C₇×C₈	—	4	1	no	—
93	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₇×C₈	#41	1	1	yes	characteristic
97	C₂×C₇×C₈	#41	1	1	yes	characteristic
98	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
99	C₇×C₄⋊C₄	—	2	2	no	—
100	C₇×C₄⋊C₄	—	4	1	no	—
101	C₄⋊Dic₇	—	2	2	no	—
102	C₄⋊Dic₇	—	4	1	no	—
103	Dic₇⋊C₄	—	4	4	no	—
104	C₄×Dic₇	—	2	1	no	—
105	C₄×Dic₇	C₂×D₄	1	2	yes	—
106	C₂×C₇⋊C₈	SD₁₆	1	1	yes	characteristic
107	C₂×C₇⋊C₈	SD₁₆	1	1	yes	characteristic
108	C₂×C₇⋊C₈	SD₁₆	1	2	yes	—
109	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
110	Dic₂₈	—	2	1	no	—
111	Dic₂₈	C₂×Q₈	1	2	yes	complemented
112	D₅₆	—	2	1	no	—
113	D₅₆	C₂×Q₈	1	2	yes	complemented
114	#1132	—	2	2	no	—
115	#1132	—	2	2	no	—
116	#1131	—	4	4	no	—
117	C₈×D₇	—	4	4	no	—
118	C₇×C₁₆	—	2	2	no	—
119	C₇⋊C₁₆	—	2	2	no	—
120	C₂×#135	—	7	1	no	—
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	C₂×D₇	1	1	yes	characteristic
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	C₂×D₇	1	4	yes	complemented
137	#364	—	7	4	no	—
138	#360	—	7	4	no	—
139	#360	—	7	4	no	—
140	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂²×D₇	—	4	1	no	—
142	C₂×C₄×C₇	—	4	2	no	—
143	C₂×C₄×C₇	—	8	2	no	—
144	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₇	—	4	2	no	—
151	C₂²⋊D₇	—	4	2	no	—
152	C₂×Dic₇	—	2	4	no	—
153	C₂×Dic₇	—	2	2	no	—
154	C₂×Dic₇	—	4	1	no	—
155	C₂×Dic₇	—	4	2	no	—
156	C₂×Dic₇	—	8	2	no	—
157	D₂₈	—	2	2	no	—
158	D₂₈	—	4	1	no	—
159	C₄×D₇	—	4	2	no	—
160	C₄×D₇	—	4	2	no	—
161	Dic₁₄	—	2	2	no	—
162	Dic₁₄	—	4	1	no	—
163	Dic₁₄	—	8	8	no	—
164	C₇×C₈	—	4	2	no	—
165	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
166	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
167	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
168	C₇×C₈	C₂×#41	1	1	yes	characteristic
169	C₇⋊C₈	C₂×SD₁₆	1	2	yes	—
170	C₇⋊C₈	C₂×SD₁₆	1	2	yes	—
171	C₂×#41	—	14	1	no	—
172	C₂²×Q₈	—	28	1	no	—
173	#135	—	7	2	no	—
174	#135	—	7	2	no	—
175	#135	—	14	1	no	—
176	#135	—	14	1	no	—
177	C₂×Q₁₆	—	7	1	no	—
178	C₂×SD₁₆	—	14	1	no	—
179	C₂×D₈	—	7	1	no	—
180	#131	—	14	4	no	—
181	#131	—	28	2	no	—
182	C₂×M₁₆	—	2	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	—	2	1	no	—
200	#108	C₂²×D₇	1	2	yes	—
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	#476	1	1	yes	characteristic
211	C₂²×C₇	#476	1	1	yes	characteristic
212	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₂×D₇	—	4	2	no	—
214	C₂×D₇	—	8	2	no	—
215	C₄×C₇	—	8	2	no	—
216	C₄×C₇	#476	1	1	yes	characteristic
217	C₄×C₇	#476	1	1	yes	characteristic
218	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
219	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
220	Dic₇	—	2	4	no	—
221	Dic₇	—	4	2	no	—
222	Dic₇	—	8	2	no	—
223	#41	—	14	2	no	—
224	#41	—	14	2	no	—
225	#41	—	28	1	no	—
226	#41	—	28	1	no	—
227	C₂×Q₈	—	14	1	no	—
228	C₂×Q₈	—	28	4	no	—
229	C₂×Q₈	—	56	4	no	—
230	C₂×D₄	—	14	1	no	—
231	C₂×D₄	—	28	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	14	1	no	—
234	C₂²×C₄	—	28	1	no	—
235	C₂²×C₄	—	28	1	no	—
236	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
237	Q₁₆	—	7	2	no	—
238	Q₁₆	—	14	1	no	—
239	SD₁₆	—	14	2	no	—
240	SD₁₆	—	14	2	no	—
241	D₈	—	7	2	no	—
242	D₈	—	14	1	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	4	1	no	—
245	M₁₆	—	28	4	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	7	1	no	—
248	C₂×C₈	—	7	1	no	—
249	C₂×C₈	—	7	2	no	—
250	C₂×C₈	—	7	2	no	—
251	C₂×C₈	—	28	4	no	—
252	C₂×C₈	#1157	1	1	yes	characteristic
253	C₂×C₈	#1157	1	1	yes	characteristic
254	C₂×C₈	D₄×D₇	1	1	yes	characteristic
255	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
256	C₂×C₈	Q₈×D₇	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	#2230	1	2	yes	complemented
270	C₂×C₇	C₂×#476	1	1	yes	characteristic
271	D₇	—	8	2	no	—
272	C₂³	—	28	1	no	—
273	C₂³	#6477	1	1	yes	characteristic
274	Q₈	—	14	2	no	—
275	Q₈	—	28	1	no	—
276	Q₈	—	56	8	no	—
277	D₄	—	14	2	no	—
278	D₄	—	28	2	no	—
279	D₄	—	28	2	no	—
280	D₄	—	28	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	14	4	no	—
284	C₂×C₄	—	14	2	no	—
285	C₂×C₄	—	28	1	no	—
286	C₂×C₄	—	28	2	no	—
287	C₂×C₄	—	28	2	no	—
288	C₂×C₄	—	28	2	no	—
289	C₂×C₄	—	56	2	no	—
290	C₂×C₄	#6459	1	1	yes	characteristic
291	C₂×C₄	#6459	1	1	yes	characteristic
292	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
293	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
294	C₂×C₄	#6477	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
296	C₈	—	4	2	no	—
297	C₈	—	7	2	no	—
298	C₈	—	7	2	no	—
299	C₈	#6459	1	1	yes	characteristic
300	C₈	#6459	1	1	yes	characteristic
301	C₈	C₂×#1157	1	1	yes	characteristic
302	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
303	C₇	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
304	C₂²	—	2	2	no	—
305	C₂²	—	28	2	no	—
306	C₂²	—	56	2	no	—
307	C₂²	#89642	1	1	yes	characteristic
308	C₂²	#89642	1	1	yes	characteristic
309	C₂²	C₂×#6477	1	1	yes	characteristic
310	C₄	—	8	2	no	—
311	C₄	—	14	4	no	—
312	C₄	—	28	2	no	—
313	C₄	—	56	2	no	—
314	C₄	#89642	1	1	yes	characteristic
315	C₄	#89642	1	1	yes	characteristic
316	C₄	C₂×#6459	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	—	56	2	no	—
321	C₂	#11727069	1	2	yes	direct product factor
322	C₂	C₂×#89642	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89642
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#365	C₂
socle	C₂²×C₇	#476
prime radical	C₂×#6467	C₂²
hypercenter	C₂×#365	D₇
hypocenter	C₇	C₂×#2230
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
2. C₂×#89627	C₂	semidirect
3. C₂×C₇×#365	C₂	semidirect
5. C₂×#89306	C₂	semidirect
6. C₂×#89289	C₂	semidirect
8. #11727069	C₂	direct (N × H)
23. #89627	C₂²	semidirect
24. C₇×#365	C₂²	semidirect
26. #89306	C₂²	semidirect
27. #89289	C₂²	semidirect
41. C₂×Dic₂₈	Q₈	semidirect
44. #6467	Q₈	semidirect
46. C₂×D₅₆	Q₈	semidirect
72. C₂×#365	D₇	semidirect
111. Dic₂₈	C₂×Q₈	semidirect
113. D₅₆	C₂×Q₈	semidirect
136. #365	C₂×D₇	semidirect
269. C₂×C₇	#2230	semidirect
303. C₇	C₂×#2230	semidirect
321. C₂	#11727069	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×#89667	2
C₂×#89627	2
C₂×C₇×#365	2
C₂×#89313	2
C₂×#89306	2
C₂×#89289	2
C₂×#89283	2
#11727069	2
C₂×#2230	7

Subgroups — C2×#11727069 #13508525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11727069 #13508525