Subgroups — C₂×#11728680 #13508176

Order 1792.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#443	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#89965	C₂	1	2	yes	maximal, complemented
3	C₂×#89467	C₂	1	1	yes	maximal, characteristic, complemented
4	#11728680	C₂	1	8	yes	maximal, direct product factor
5	C₂×#89290	C₂	1	2	yes	maximal, complemented
6	C₂×#89258	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#135	C₂²	1	1	yes	characteristic
8	C₂×C₇×#131	—	2	1	no	—
9	C₂²×D₅₆	—	2	1	no	—
10	C₂²×D₅₆	C₂²	1	2	yes	—
11	C₂×#6437	—	2	1	no	—
12	C₇×#443	—	2	2	no	—
13	C₇×#443	C₂²	1	4	yes	complemented
14	C₂×C₇×#108	C₂²	1	1	yes	characteristic
15	C₂×C₇×#104	—	2	1	no	—
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	#89965	—	2	4	no	—
18	#89965	C₂²	1	8	yes	complemented
19	C₂×#6402	—	2	2	no	—
20	C₂×#6400	—	2	2	no	—
21	C₂×#6386	C₂²	1	2	yes	—
22	#89467	—	2	2	no	—
23	#89467	C₂²	1	4	yes	complemented
24	#89290	C₂²	1	8	yes	complemented
25	#89258	C₂²	1	4	yes	complemented
26	C₂×#2213	—	7	1	no	maximal, Sylow
27	C₂×C₇×#41	—	2	1	no	—
28	C₂²×D₂₈	—	2	2	no	—
29	C₂²×D₂₈	—	4	2	no	—
30	C₇×#135	—	2	1	no	—
31	C₇×#135	—	2	1	no	—
32	C₇×#135	D₄	1	2	yes	complemented
33	C₇×#135	C₂³	1	2	yes	—
34	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic, complemented
35	C₂×C₇×SD₁₆	—	2	1	no	—
36	C₂×C₇×D₈	D₄	1	1	yes	characteristic, complemented
37	C₇×#131	—	2	4	no	—
38	C₇×#131	—	4	2	no	—
39	C₂×C₇×M₁₆	—	2	1	no	—
40	C₂×C₇×M₁₆	—	4	1	no	—
41	C₂²×C₇×C₈	—	4	1	no	—
42	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
45	C₂×D₅₆	—	2	2	no	—
46	C₂×D₅₆	—	2	2	no	—
47	C₂×D₅₆	—	2	2	no	—
48	C₂×D₅₆	—	2	2	no	—
49	C₂×D₅₆	—	2	4	no	—
50	C₂×D₅₆	—	2	2	no	—
51	C₂×D₅₆	—	4	1	no	—
52	C₂×D₅₆	—	4	1	no	—
53	C₂×D₅₆	—	4	2	no	—
54	C₂×D₅₆	C₂³	1	4	yes	—
55	#6437	—	2	4	no	—
56	#6437	—	4	2	no	—
57	C₇×#108	—	2	1	no	—
58	C₇×#108	C₂³	1	2	yes	—
59	C₇×#104	—	2	4	no	—
60	C₄×C₇×C₈	D₄	1	2	yes	—
61	C₄×C₇×C₈	C₂³	1	2	yes	—
62	#6402	—	2	8	no	—
63	#6400	—	2	8	no	—
64	#6386	—	2	2	no	—
65	#6386	C₂³	1	4	yes	—
66	C₂×C₇⋊C₁₆	—	2	2	no	—
67	C₂×#509	—	7	2	no	—
68	C₂×#493	—	7	1	no	—
69	C₂×#443	D₇	1	1	yes	characteristic, complemented
70	#2213	—	7	8	no	—
71	C₂×#364	—	7	1	no	—
72	C₂×#361	—	7	2	no	—
73	C₂³×D₇	—	4	2	no	—
74	C₇×#41	—	2	2	no	—
75	C₇×#41	—	2	2	no	—
76	C₇×#41	—	4	1	no	—
77	C₇×#41	—	4	1	no	—
78	C₂×C₇×Q₈	—	2	1	no	—
79	C₂×C₇×D₄	—	2	1	no	—
80	C₂×C₇×D₄	—	4	1	no	—
81	C₂²×C₄×C₇	—	2	1	no	—
82	C₂²×C₄×C₇	—	4	1	no	—
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂×D₂₈	—	2	4	no	—
85	C₂×D₂₈	—	4	2	no	—
86	C₂×D₂₈	—	4	2	no	—
87	C₂×D₂₈	—	4	2	no	—
88	C₂×D₂₈	—	4	2	no	—
89	C₂×D₂₈	—	4	8	no	—
90	C₂×D₂₈	—	4	2	no	—
91	C₂×D₂₈	—	8	8	no	—
92	C₇×Q₁₆	—	2	1	no	—
93	C₇×Q₁₆	C₂×D₄	1	2	yes	complemented
94	C₇×SD₁₆	—	2	2	no	—
95	C₇×SD₁₆	—	2	2	no	—
96	C₇×D₈	—	2	1	no	—
97	C₇×D₈	C₂×D₄	1	2	yes	complemented
98	C₇×M₁₆	—	2	2	no	—
99	C₇×M₁₆	—	4	1	no	—
100	C₇×M₁₆	—	4	4	no	—
101	C₂×C₇×C₈	—	4	1	no	—
102	C₂×C₇×C₈	—	4	4	no	—
103	C₂×C₇×C₈	D₈	1	1	yes	characteristic
104	C₂×C₇×C₈	D₈	1	1	yes	characteristic
105	C₂×C₇×C₈	D₈	1	2	yes	—
106	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₇×C₈	#41	1	1	yes	characteristic
111	C₂×C₇×C₈	#41	1	1	yes	characteristic
112	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
113	C₄²×C₇	—	2	1	no	—
114	C₄²×C₇	C₂×D₄	1	2	yes	—
115	D₅₆	—	2	4	no	—
116	D₅₆	—	2	4	no	—
117	D₅₆	—	4	2	no	—
118	D₅₆	—	4	2	no	—
119	D₅₆	—	4	4	no	—
120	D₅₆	—	4	4	no	—
121	C₇⋊C₁₆	—	2	4	no	—
122	C₂×#135	C₂×D₇	1	1	yes	characteristic
123	C₂²×D₈	—	7	2	no	—
124	C₂²×D₈	—	14	1	no	—
125	C₂×#131	—	2	1	no	—
126	C₂×#127	—	14	1	no	—
127	#509	—	7	8	no	—
128	#509	—	14	4	no	—
129	C₂×SD₃₂	—	14	2	no	—
130	C₂×D₁₆	—	14	2	no	—
131	C₂×M₃₂	—	7	2	no	—
132	#493	—	7	4	no	—
133	#493	—	14	2	no	—
134	#443	—	2	2	no	—
135	#443	C₂×D₇	1	4	yes	complemented
136	C₂×#108	C₂×D₇	1	1	yes	characteristic
137	C₂×#104	—	2	1	no	—
138	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
139	#364	—	7	4	no	—
140	#361	—	7	8	no	—
141	C₂³×C₇	—	4	1	no	—
142	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂²×D₇	—	4	4	no	—
144	C₂²×D₇	—	8	8	no	—
145	C₂²×D₇	—	8	2	no	—
146	C₂²×D₇	—	8	2	no	—
147	C₇×Q₈	—	2	2	no	—
148	C₇×Q₈	—	4	1	no	—
149	C₇×D₄	—	2	2	no	—
150	C₇×D₄	—	4	2	no	—
151	C₇×D₄	—	4	2	no	—
152	C₇×D₄	—	4	1	no	—
153	C₂×C₄×C₇	—	2	4	no	—
154	C₂×C₄×C₇	—	2	2	no	—
155	C₂×C₄×C₇	—	4	1	no	—
156	C₂×C₄×C₇	—	4	2	no	—
157	C₂×C₄×C₇	—	4	2	no	—
158	C₂×C₄×C₇	#121	1	1	yes	characteristic
159	C₂×C₄×C₇	#121	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
164	D₂₈	—	4	4	no	—
165	D₂₈	—	4	4	no	—
166	D₂₈	—	8	16	no	—
167	D₂₈	—	8	2	no	—
168	D₂₈	—	8	2	no	—
169	C₇×C₈	—	4	2	no	—
170	C₇×C₈	#121	1	1	yes	characteristic
171	C₇×C₈	#121	1	1	yes	characteristic
172	C₇×C₈	C₂×D₈	1	2	yes	—
173	C₇×C₈	C₂×D₈	1	2	yes	—
174	C₇×C₈	C₂²×D₄	1	1	yes	characteristic
175	C₇×C₈	C₂×#41	1	1	yes	characteristic
176	C₂×#41	—	2	1	no	—
177	C₂²×D₄	—	14	2	no	—
178	C₂²×D₄	—	28	2	no	—
179	#135	—	2	1	no	—
180	#135	—	2	1	no	—
181	#135	D₂₈	1	2	yes	complemented
182	#135	C₂²×D₇	1	2	yes	—
183	C₂×Q₁₆	D₂₈	1	1	yes	characteristic, complemented
184	C₂×SD₁₆	—	2	1	no	—
185	C₂×D₈	—	7	4	no	—
186	C₂×D₈	—	14	2	no	—
187	C₂×D₈	—	14	2	no	—
188	C₂×D₈	—	14	2	no	—
189	C₂×D₈	—	14	2	no	—
190	C₂×D₈	—	14	2	no	—
191	C₂×D₈	—	14	4	no	—
192	C₂×D₈	—	28	1	no	—
193	C₂×D₈	—	28	1	no	—
194	C₂×D₈	—	28	2	no	—
195	C₂×D₈	D₂₈	1	1	yes	characteristic, complemented
196	#131	—	2	4	no	—
197	#131	—	4	2	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	4	1	no	—
200	C₂²×C₈	—	4	1	no	—
201	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
203	#127	—	14	4	no	—
204	#127	—	28	2	no	—
205	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
206	SD₃₂	—	14	8	no	—
207	D₁₆	—	14	8	no	—
208	M₃₂	—	7	4	no	—
209	M₃₂	—	14	2	no	—
210	C₂×C₁₆	—	14	2	no	—
211	#108	—	2	1	no	—
212	#108	C₂²×D₇	1	2	yes	—
213	#104	—	2	4	no	—
214	C₄×C₈	C₂²⋊D₇	1	2	yes	—
215	C₄×C₈	C₂²×D₇	1	2	yes	—
216	C₂²×C₇	—	2	2	no	—
217	C₂²×C₇	—	4	2	no	—
218	C₂²×C₇	—	8	2	no	—
219	C₂²×C₇	#459	1	1	yes	characteristic
220	C₂²×C₇	#459	1	1	yes	characteristic
221	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂×D₇	—	8	8	no	—
223	C₂×D₇	—	8	4	no	—
224	C₂×D₇	—	16	4	no	—
225	C₂×D₇	—	16	4	no	—
226	C₄×C₇	—	2	4	no	—
227	C₄×C₇	—	4	2	no	—
228	C₄×C₇	#459	1	1	yes	characteristic
229	C₄×C₇	#459	1	1	yes	characteristic
230	C₄×C₇	C₂×#121	1	1	yes	characteristic
231	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
232	C₂⁴	—	28	2	no	—
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	14	4	no	—
241	C₂×D₄	—	28	8	no	—
242	C₂×D₄	—	28	2	no	—
243	C₂×D₄	—	28	2	no	—
244	C₂×D₄	—	28	2	no	—
245	C₂×D₄	—	28	2	no	—
246	C₂×D₄	—	28	2	no	—
247	C₂×D₄	—	56	8	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
251	Q₁₆	—	2	1	no	—
252	Q₁₆	C₂×D₂₈	1	2	yes	complemented
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	2	2	no	—
255	D₈	—	2	1	no	—
256	D₈	—	14	4	no	—
257	D₈	—	14	4	no	—
258	D₈	—	28	2	no	—
259	D₈	—	28	2	no	—
260	D₈	—	28	4	no	—
261	D₈	—	28	4	no	—
262	D₈	C₂×D₂₈	1	2	yes	complemented
263	M₁₆	—	2	2	no	—
264	M₁₆	—	4	1	no	—
265	M₁₆	—	4	4	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	4	no	—
268	C₂×C₈	#1141	1	1	yes	characteristic
269	C₂×C₈	#1141	1	1	yes	characteristic
270	C₂×C₈	#1141	1	2	yes	—
271	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
272	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
273	C₂×C₈	#1157	1	1	yes	characteristic
274	C₂×C₈	#1157	1	1	yes	characteristic
275	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
276	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
277	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
278	C₄²	—	2	1	no	—
279	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
280	C₁₆	—	14	4	no	—
281	C₂×C₇	—	2	1	no	—
282	C₂×C₇	—	2	1	no	—
283	C₂×C₇	—	8	2	no	—
284	C₂×C₇	#2213	1	2	yes	complemented
285	C₂×C₇	C₂×#459	1	1	yes	characteristic
286	D₇	—	16	4	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	28	4	no	—
289	C₂³	—	56	8	no	—
290	C₂³	—	56	2	no	—
291	C₂³	—	56	2	no	—
292	C₂³	#6512	1	1	yes	characteristic
293	Q₈	—	2	2	no	—
294	Q₈	—	4	1	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	1	no	—
299	D₄	—	28	4	no	—
300	D₄	—	28	4	no	—
301	D₄	—	56	16	no	—
302	D₄	—	56	2	no	—
303	D₄	—	56	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	#6493	1	1	yes	characteristic
310	C₂×C₄	#6493	1	1	yes	characteristic
311	C₂×C₄	C₂×#1141	1	1	yes	characteristic
312	C₂×C₄	C₂×#1141	1	1	yes	characteristic
313	C₂×C₄	#6512	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
315	C₈	—	4	2	no	—
316	C₈	#6493	1	1	yes	characteristic
317	C₈	#6493	1	1	yes	characteristic
318	C₈	C₂×#1141	1	2	yes	—
319	C₈	C₂×#1141	1	2	yes	—
320	C₈	C₂²×D₂₈	1	1	yes	characteristic
321	C₈	C₂×#1157	1	1	yes	characteristic
322	C₇	C₂×#2213	1	1	yes	characteristic, complemented, Sylow
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	56	8	no	—
327	C₂²	—	56	4	no	—
328	C₂²	—	112	4	no	—
329	C₂²	—	112	4	no	—
330	C₂²	#89787	1	1	yes	characteristic
331	C₂²	#89787	1	1	yes	characteristic
332	C₂²	C₂×#6512	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	#89787	1	1	yes	characteristic
336	C₄	#89787	1	1	yes	characteristic
337	C₄	C₂×#6493	1	1	yes	characteristic
338	C₄	C₂²×#1141	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	—	112	4	no	—
343	C₂	#11728680	1	2	yes	direct product factor
344	C₂	C₂×#89787	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89787
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#443	C₂
socle	C₂²×C₇	#459
hypercenter	C₂×#443	D₇
hypocenter	C₇	C₂×#2213
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#443	C₂	semidirect
2. C₂×#89965	C₂	semidirect
3. C₂×#89467	C₂	semidirect
4. #11728680	C₂	direct (N × H)
5. C₂×#89290	C₂	semidirect
6. C₂×#89258	C₂	semidirect
13. C₇×#443	C₂²	semidirect
18. #89965	C₂²	semidirect
23. #89467	C₂²	semidirect
24. #89290	C₂²	semidirect
25. #89258	C₂²	semidirect
32. C₇×#135	D₄	semidirect
34. C₂×C₇×Q₁₆	D₄	semidirect
36. C₂×C₇×D₈	D₄	semidirect
69. C₂×#443	D₇	semidirect
93. C₇×Q₁₆	C₂×D₄	semidirect
97. C₇×D₈	C₂×D₄	semidirect
135. #443	C₂×D₇	semidirect
181. #135	D₂₈	semidirect
183. C₂×Q₁₆	D₂₈	semidirect
195. C₂×D₈	D₂₈	semidirect
252. Q₁₆	C₂×D₂₈	semidirect
262. D₈	C₂×D₂₈	semidirect
284. C₂×C₇	#2213	semidirect
322. C₇	C₂×#2213	semidirect
343. C₂	#11728680	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×C₇×#443	2
C₂×#89965	2
C₂×#89467	2
#11728680	2
C₂×#89290	2
C₂×#89258	2
C₂×#2213	7

Subgroups — C2×#11728680 #13508176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11728680 #13508176