Subgroups — C₂×#11729208 #13508353

Order 1792.

678 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#89914	C₂	1	2	yes	maximal, complemented
3	C₂²×D₁₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729208	C₂	1	8	yes	maximal, direct product factor
5	C₂×#89311	C₂	1	2	yes	maximal, complemented
6	C₂×#89301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#131	C₂²	1	1	yes	characteristic
8	C₂×#6512	—	2	2	no	—
9	C₂²×D₅₆	C₂²	1	2	yes	—
10	C₇×#504	—	2	2	no	—
11	C₇×#504	C₂²	1	4	yes	complemented
12	C₂×C₇×M₃₂	—	2	1	no	—
13	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₇×C₁₆	—	2	1	no	—
15	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
16	#89914	—	2	4	no	—
17	#89914	C₂²	1	8	yes	complemented
18	C₂×#6397	—	2	2	no	—
19	C₂×#6393	C₂²	1	2	yes	—
20	C₂×D₁₁₂	—	2	1	no	—
21	C₂×D₁₁₂	—	2	1	no	—
22	C₂×D₁₁₂	—	2	2	no	—
23	C₂×D₁₁₂	C₂²	1	4	yes	complemented
24	#89311	C₂²	1	8	yes	complemented
25	#89301	C₂²	1	4	yes	complemented
26	C₂×#2221	—	7	1	no	maximal, Sylow
27	C₂×C₇×#41	D₄	1	1	yes	characteristic
28	C₂²×D₂₈	—	2	2	no	—
29	C₇×#131	—	2	1	no	—
30	C₇×#131	—	2	1	no	—
31	C₇×#131	D₄	1	2	yes	—
32	C₇×#131	C₂³	1	2	yes	—
33	C₂×C₇×M₁₆	—	2	1	no	—
34	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	—	2	1	no	—
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	#6512	—	2	8	no	—
38	#6512	—	4	4	no	—
39	C₂×#1143	—	4	2	no	—
40	C₂×#1141	—	4	2	no	—
41	C₂×#1136	—	2	2	no	—
42	C₂×D₅₆	—	2	2	no	—
43	C₂×D₅₆	—	2	2	no	—
44	C₂×D₅₆	—	2	2	no	—
45	C₂×D₅₆	—	2	2	no	—
46	C₂×D₅₆	—	2	2	no	—
47	C₂×D₅₆	C₂³	1	4	yes	—
48	C₇×M₃₂	—	2	1	no	—
49	C₇×M₃₂	—	2	4	no	—
50	C₇×M₃₂	C₂³	1	2	yes	—
51	C₂×C₇×C₁₆	—	2	1	no	—
52	C₂×C₇×C₁₆	—	2	4	no	—
53	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
54	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
55	C₂×C₇×C₁₆	D₄	1	2	yes	—
56	C₂×C₇×C₁₆	C₂³	1	2	yes	—
57	#6397	—	2	8	no	—
58	#6393	—	2	2	no	—
59	#6393	C₂³	1	4	yes	—
60	D₁₁₂	—	2	4	no	—
61	D₁₁₂	—	2	4	no	—
62	C₂²×D₁₆	—	7	1	no	—
63	C₂×#504	D₇	1	1	yes	characteristic, complemented
64	C₂×#472	—	7	2	no	—
65	#2221	—	7	8	no	—
66	C₂×#368	—	7	1	no	—
67	C₂×#361	—	7	2	no	—
68	C₂³×D₇	—	4	2	no	—
69	C₇×#41	—	2	1	no	—
70	C₇×#41	—	2	1	no	—
71	C₇×#41	D₈	1	2	yes	—
72	C₇×#41	C₂×D₄	1	2	yes	—
73	C₂×C₇×Q₈	D₈	1	1	yes	characteristic
74	C₂×C₇×D₄	—	2	1	no	—
75	C₂×C₇×D₄	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₇	—	2	1	no	—
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂×D₂₈	—	2	4	no	—
79	C₂×D₂₈	—	4	2	no	—
80	C₂×D₂₈	—	4	2	no	—
81	C₂×D₂₈	—	4	2	no	—
82	C₂×D₂₈	—	4	2	no	—
83	C₂×D₂₈	—	4	2	no	—
84	C₇×M₁₆	—	2	1	no	—
85	C₇×M₁₆	—	2	2	no	—
86	C₇×M₁₆	—	2	2	no	—
87	C₇×M₁₆	C₂×D₄	1	2	yes	—
88	C₂×C₇×C₈	—	2	1	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₇×C₈	#41	1	1	yes	characteristic
95	C₂×C₇×C₈	#41	1	1	yes	characteristic
96	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
97	#1143	—	4	8	no	—
98	#1141	—	4	8	no	—
99	#1136	—	2	4	no	—
100	#1136	—	4	2	no	—
101	C₂×C₇⋊C₈	—	4	2	no	—
102	D₅₆	—	2	4	no	—
103	D₅₆	—	2	4	no	—
104	D₅₆	—	4	2	no	—
105	D₅₆	—	4	2	no	—
106	C₇×C₁₆	—	2	2	no	—
107	C₇×C₁₆	C₂×D₄	1	2	yes	—
108	C₇×C₁₆	C₂×D₄	1	2	yes	—
109	C₂×AGL(1,ℤ/8)	—	14	2	no	—
110	C₂²×D₈	—	7	2	no	—
111	C₂×#131	C₂×D₇	1	1	yes	characteristic
112	C₂×D₁₆	—	7	4	no	—
113	C₂×D₁₆	—	14	1	no	—
114	C₂×D₁₆	—	14	1	no	—
115	C₂×D₁₆	—	14	2	no	—
116	#504	—	2	2	no	—
117	#504	C₂×D₇	1	4	yes	complemented
118	C₂×M₃₂	—	2	1	no	—
119	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
120	C₂²×C₁₆	—	2	1	no	—
121	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
122	#472	—	7	8	no	—
123	#472	—	14	4	no	—
124	C₂×#108	—	7	2	no	—
125	C₂×#100	—	14	2	no	—
126	#368	—	7	4	no	—
127	#361	—	7	8	no	—
128	C₂³×C₇	—	2	1	no	—
129	C₂³×C₇	#135	1	1	yes	characteristic
130	C₂²×D₇	—	4	4	no	—
131	C₂²×D₇	—	8	2	no	—
132	C₂²×D₇	—	8	2	no	—
133	C₂²×D₇	—	16	4	no	—
134	C₇×Q₈	—	2	1	no	—
135	C₇×Q₈	C₂×D₈	1	2	yes	—
136	C₇×D₄	—	2	2	no	—
137	C₇×D₄	—	2	2	no	—
138	C₇×D₄	—	2	1	no	—
139	C₇×D₄	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₇	—	2	1	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₇	#135	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
149	D₂₈	—	4	4	no	—
150	D₂₈	—	4	4	no	—
151	D₂₈	—	8	2	no	—
152	D₂₈	—	8	2	no	—
153	C₇×C₈	—	2	2	no	—
154	C₇×C₈	#121	1	1	yes	characteristic
155	C₇×C₈	#121	1	1	yes	characteristic
156	C₇×C₈	C₂²×D₄	1	1	yes	characteristic
157	C₇×C₈	C₂×#41	1	1	yes	characteristic
158	C₇⋊C₈	—	4	4	no	—
159	C₂×#41	D₂₈	1	1	yes	characteristic
160	C₂²×D₄	—	14	2	no	—
161	AGL(1,ℤ/8)	—	14	8	no	—
162	AGL(1,ℤ/8)	—	28	4	no	—
163	C₂×SD₁₆	—	28	2	no	—
164	C₂×D₈	—	7	4	no	—
165	C₂×D₈	—	14	2	no	—
166	C₂×D₈	—	14	2	no	—
167	C₂×D₈	—	14	2	no	—
168	C₂×D₈	—	14	2	no	—
169	C₂×D₈	—	14	2	no	—
170	C₂×D₈	—	28	2	no	—
171	#131	—	2	1	no	—
172	#131	—	2	1	no	—
173	#131	D₂₈	1	2	yes	—
174	#131	C₂²×D₇	1	2	yes	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	14	2	no	—
177	C₂×M₁₆	D₂₈	1	1	yes	characteristic
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
180	D₁₆	—	14	4	no	—
181	D₁₆	—	14	4	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	2	4	no	—
184	M₃₂	C₂²×D₇	1	2	yes	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	—	2	4	no	—
187	C₂×C₁₆	C₂²⋊D₇	1	1	yes	characteristic
188	C₂×C₁₆	C₂²⋊D₇	1	1	yes	characteristic
189	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
190	C₂×C₁₆	C₂²×D₇	1	2	yes	—
191	#108	—	7	4	no	—
192	#108	—	14	2	no	—
193	#100	—	14	8	no	—
194	C₂²×C₇	—	2	2	no	—
195	C₂²×C₇	—	2	2	no	—
196	C₂²×C₇	—	4	2	no	—
197	C₂²×C₇	#466	1	1	yes	characteristic
198	C₂²×C₇	#466	1	1	yes	characteristic
199	C₂²×C₇	C₂×#135	1	1	yes	characteristic
200	C₂×D₇	—	8	4	no	—
201	C₂×D₇	—	16	4	no	—
202	C₂×D₇	—	16	4	no	—
203	C₂×D₇	—	16	4	no	—
204	C₄×C₇	—	2	2	no	—
205	C₄×C₇	#466	1	1	yes	characteristic
206	C₄×C₇	#466	1	1	yes	characteristic
207	C₄×C₇	C₂×#121	1	1	yes	characteristic
208	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
209	C₂⁴	—	28	2	no	—
210	#41	—	2	1	no	—
211	#41	—	2	1	no	—
212	#41	D₅₆	1	2	yes	—
213	#41	C₂×D₂₈	1	2	yes	—
214	C₂×Q₈	D₅₆	1	1	yes	characteristic
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	14	4	no	—
217	C₂×D₄	—	28	2	no	—
218	C₂×D₄	—	28	2	no	—
219	C₂×D₄	—	28	2	no	—
220	C₂×D₄	—	28	2	no	—
221	C₂×D₄	—	28	2	no	—
222	C₂×D₄	D₅₆	1	1	yes	characteristic
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
225	SD₁₆	—	28	8	no	—
226	D₈	—	14	4	no	—
227	D₈	—	14	4	no	—
228	D₈	—	28	8	no	—
229	D₈	—	28	2	no	—
230	D₈	—	28	2	no	—
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	14	4	no	—
235	M₁₆	—	28	2	no	—
236	M₁₆	C₂×D₂₈	1	2	yes	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	28	2	no	—
241	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
242	C₂×C₈	#1157	1	1	yes	characteristic
243	C₂×C₈	#1157	1	1	yes	characteristic
244	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
246	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
247	C₁₆	—	2	2	no	—
248	C₁₆	C₂×C₂²⋊D₇	1	2	yes	—
249	C₁₆	C₂×C₂²⋊D₇	1	2	yes	—
250	C₂×C₇	—	2	1	no	—
251	C₂×C₇	—	2	1	no	—
252	C₂×C₇	—	4	2	no	—
253	C₂×C₇	#2221	1	2	yes	complemented
254	C₂×C₇	C₂×#466	1	1	yes	characteristic
255	D₇	—	16	4	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	28	4	no	—
258	C₂³	—	56	2	no	—
259	C₂³	—	56	2	no	—
260	C₂³	—	112	4	no	—
261	C₂³	#6467	1	1	yes	characteristic
262	Q₈	—	2	1	no	—
263	Q₈	C₂×D₅₆	1	2	yes	—
264	D₄	—	2	2	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	1	no	—
267	D₄	—	28	4	no	—
268	D₄	—	28	4	no	—
269	D₄	—	56	2	no	—
270	D₄	—	56	2	no	—
271	D₄	C₂×D₅₆	1	2	yes	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
276	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
277	C₂×C₄	#6467	1	1	yes	characteristic
278	C₂×C₄	#6493	1	1	yes	characteristic
279	C₂×C₄	#6493	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	28	4	no	—
283	C₈	#6493	1	1	yes	characteristic
284	C₈	#6493	1	1	yes	characteristic
285	C₈	C₂×#1157	1	1	yes	characteristic
286	C₈	C₂²×C₂²⋊D₇	1	1	yes	characteristic
287	C₇	C₂×#2221	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	56	4	no	—
292	C₂²	—	112	4	no	—
293	C₂²	—	112	4	no	—
294	C₂²	—	112	4	no	—
295	C₂²	#89887	1	1	yes	characteristic
296	C₂²	#89887	1	1	yes	characteristic
297	C₂²	C₂×#6467	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	#89887	1	1	yes	characteristic
300	C₄	#89887	1	1	yes	characteristic
301	C₄	C₂²×D₅₆	1	1	yes	characteristic
302	C₄	C₂×#6493	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	2	no	—
306	C₂	—	112	4	no	—
307	C₂	#11729208	1	2	yes	direct product factor
308	C₂	C₂×#89887	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89887
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#504	C₂
socle	C₂²×C₇	#466
hypercenter	C₂×#504	D₇
hypocenter	C₇	C₂×#2221
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#504	C₂	semidirect
2. C₂×#89914	C₂	semidirect
3. C₂²×D₁₁₂	C₂	semidirect
4. #11729208	C₂	direct (N × H)
5. C₂×#89311	C₂	semidirect
6. C₂×#89301	C₂	semidirect
11. C₇×#504	C₂²	semidirect
17. #89914	C₂²	semidirect
23. C₂×D₁₁₂	C₂²	semidirect
24. #89311	C₂²	semidirect
25. #89301	C₂²	semidirect
63. C₂×#504	D₇	semidirect
117. #504	C₂×D₇	semidirect
253. C₂×C₇	#2221	semidirect
287. C₇	C₂×#2221	semidirect
307. C₂	#11729208	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×#504	2
C₂×#89914	2
C₂²×D₁₁₂	2
#11729208	2
C₂×#89311	2
C₂×#89301	2
C₂×#2221	7

Subgroups — C2×#11729208 #13508353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11729208 #13508353