Subgroups — C₂×#11729134 #13336427

Order 1792.

618 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×#89863	C₂	1	2	yes	maximal
3	C₂×#89451	C₂	1	1	yes	maximal, characteristic
4	C₂×#89341	C₂	1	2	yes	maximal
5	#11729134	C₂	1	8	yes	maximal, direct product factor
6	C₂×#89252	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₄	C₂²	1	1	yes	characteristic
9	C₇×#500	—	2	2	no	—
10	C₇×#500	C₂²	1	4	yes	—
11	C₂×C₇×#107	—	2	1	no	—
12	C₂×C₇×#107	—	2	1	no	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6389	—	2	1	no	—
15	C₂×Dic₇⋊C₈	—	2	1	no	—
16	#89863	—	2	4	no	—
17	#89863	C₂²	1	8	yes	—
18	C₂×#6384	—	2	2	no	—
19	C₂×#6381	—	2	2	no	—
20	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
22	#89451	—	2	2	no	—
23	#89451	C₂²	1	4	yes	—
24	#89341	C₂²	1	8	yes	—
25	#89252	C₂²	1	4	yes	—
26	C₂×#1696	—	7	1	no	maximal, Sylow
27	C₂²×C₇×Q₈	—	2	2	no	—
28	C₂²×Dic₁₄	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
31	C₇×C₄⋊Q₈	—	2	2	no	—
32	C₇×C₄⋊Q₈	—	2	2	no	—
33	C₇×C₄⋊Q₈	—	2	2	no	—
34	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
35	C₂×C₇×C₄⋊C₄	—	2	2	no	—
36	C₂×C₇×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₇	D₄	1	1	yes	characteristic
39	C₂×Dic₇⋊C₄	—	2	1	no	—
40	C₂×C₄×Dic₇	—	2	1	no	—
41	C₂²×C₇⋊C₈	—	2	1	no	—
42	C₂²×C₇⋊C₈	—	2	2	no	—
43	C₄×Dic₁₄	—	2	1	no	—
44	C₄×Dic₁₄	—	2	1	no	—
45	C₄×Dic₁₄	D₄	1	2	yes	—
46	C₄×Dic₁₄	C₂³	1	2	yes	—
47	C₇×#107	—	2	4	no	—
48	C₇×#107	—	2	4	no	—
49	C₄×C₇×C₈	D₄	1	2	yes	—
50	C₄×C₇×C₈	C₂³	1	2	yes	—
51	#6389	—	2	4	no	—
52	Dic₇⋊C₈	—	2	4	no	—
53	#6384	—	2	8	no	—
54	#6381	—	2	8	no	—
55	C₄⋊(C₇⋊C₈)	—	2	2	no	—
56	C₄⋊(C₇⋊C₈)	—	2	1	no	—
57	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
58	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
59	C₂×#500	D₇	1	1	yes	characteristic
60	C₂×#477	—	7	2	no	—
61	C₂×#446	—	7	1	no	—
62	#1696	—	7	8	no	—
63	C₂×#334	—	7	1	no	—
64	C₂×#333	—	7	2	no	—
65	C₂×C₇×Q₈	—	2	4	no	—
66	C₂×C₇×Q₈	—	4	2	no	—
67	C₂×C₇×Q₈	—	8	4	no	—
68	C₂²×C₄×C₇	—	4	2	no	—
69	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	#41	1	1	yes	characteristic
72	C₂²×Dic₇	—	2	1	no	—
73	C₂²×Dic₇	—	2	1	no	—
74	C₂×Dic₁₄	—	2	4	no	—
75	C₂×Dic₁₄	—	2	1	no	—
76	C₂×Dic₁₄	C₂×D₄	1	2	yes	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	Q₁₆	1	4	yes	—
79	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
80	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
81	C₇×C₄⋊C₄	—	2	4	no	—
82	C₇×C₄⋊C₄	—	2	4	no	—
83	C₇×C₄⋊C₄	—	4	2	no	—
84	C₇×C₄⋊C₄	—	4	2	no	—
85	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₇	#41	1	1	yes	characteristic
88	C₄²×C₇	C₂⁴	1	1	yes	characteristic
89	C₄⋊Dic₇	—	2	1	no	—
90	C₄⋊Dic₇	C₂×D₄	1	2	yes	—
91	Dic₇⋊C₄	—	2	2	no	—
92	Dic₇⋊C₄	—	2	2	no	—
93	C₄×Dic₇	—	2	2	no	—
94	C₄×Dic₇	—	2	2	no	—
95	C₂×C₇⋊C₈	—	2	4	no	—
96	C₂×C₇⋊C₈	—	2	2	no	—
97	C₂×C₇⋊C₈	—	4	4	no	—
98	C₂×C₇⋊C₈	—	4	2	no	—
99	C₂×C₄⋊Q₈	C₂×D₇	1	2	yes	—
100	C₂×C₄×Q₈	—	7	1	no	—
101	#500	—	2	2	no	—
102	#500	C₂×D₇	1	4	yes	—
103	#477	—	7	8	no	—
104	#477	—	14	4	no	—
105	#446	—	7	4	no	—
106	#446	—	14	2	no	—
107	C₂×#107	—	2	1	no	—
108	C₂×#107	—	2	1	no	—
109	C₂×#107	—	14	2	no	—
110	C₂×C₄⋊C₈	—	7	2	no	—
111	C₂×C₄⋊C₈	—	7	1	no	—
112	C₂×C₄⋊C₈	—	14	1	no	—
113	C₂×#103	—	14	2	no	—
114	C₂×#97	—	14	1	no	—
115	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
116	#334	—	7	4	no	—
117	#333	—	7	8	no	—
118	C₂³×C₇	#121	1	1	yes	characteristic
119	C₇×Q₈	—	8	4	no	—
120	C₇×Q₈	—	8	4	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	4	4	no	—
123	C₂×C₄×C₇	—	8	4	no	—
124	C₂×C₄×C₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
128	C₂×C₄×C₇	#137	1	4	yes	—
129	C₂×C₄×C₇	#137	1	2	yes	—
130	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₇	—	2	2	no	—
134	C₂×Dic₇	—	2	2	no	—
135	C₂×Dic₇	—	4	2	no	—
136	C₂×Dic₇	—	4	2	no	—
137	Dic₁₄	—	2	4	no	—
138	Dic₁₄	—	4	2	no	—
139	C₇×C₈	—	2	2	no	—
140	C₇×C₈	C₂×Q₁₆	1	4	yes	—
141	C₇⋊C₈	—	4	4	no	—
142	C₇⋊C₈	—	4	2	no	—
143	C₂²×Q₈	—	2	2	no	—
144	C₂²×Q₈	—	7	1	no	—
145	C₂²×C₈	—	14	2	no	—
146	C₂²×C₈	—	14	1	no	—
147	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
148	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	C₂²×D₇	1	4	yes	—
153	C₄×Q₈	—	7	2	no	—
154	C₄×Q₈	—	7	2	no	—
155	C₄×Q₈	—	14	1	no	—
156	C₄×Q₈	—	14	1	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	7	1	no	—
160	C₂×C₄⋊C₄	—	14	1	no	—
161	C₂×C₄²	—	14	1	no	—
162	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
163	#107	—	2	4	no	—
164	#107	—	2	4	no	—
165	#107	—	14	8	no	—
166	C₄⋊C₈	—	7	2	no	—
167	C₄⋊C₈	—	7	4	no	—
168	C₄⋊C₈	—	14	4	no	—
169	C₄⋊C₈	—	14	2	no	—
170	C₄⋊C₈	—	14	1	no	—
171	#103	—	14	8	no	—
172	#97	—	14	4	no	—
173	C₄×C₈	C₂²⋊D₇	1	2	yes	—
174	C₄×C₈	C₂²×D₇	1	2	yes	—
175	C₂²×C₇	#462	1	2	yes	—
176	C₂²×C₇	#470	1	2	yes	—
177	C₂²×C₇	#472	1	2	yes	—
178	C₂²×C₇	C₂×#121	1	1	yes	characteristic
179	C₄×C₇	—	2	1	no	—
180	C₄×C₇	—	2	1	no	—
181	C₄×C₇	—	8	4	no	—
182	C₄×C₇	#462	1	2	yes	—
183	C₄×C₇	#470	1	2	yes	—
184	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
185	C₄×C₇	C₂×#137	1	2	yes	—
186	C₄×C₇	C₂×#137	1	1	yes	characteristic
187	Dic₇	—	4	2	no	—
188	Dic₇	—	4	2	no	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	7	2	no	—
192	C₂×Q₈	—	8	4	no	—
193	C₂×Q₈	—	14	1	no	—
194	C₂×Q₈	—	14	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	14	1	no	—
197	C₂²×C₄	—	14	1	no	—
198	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
199	C₂²×C₄	#1159	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	14	4	no	—
203	C₂×C₈	—	14	2	no	—
204	C₂×C₈	—	28	4	no	—
205	C₂×C₈	—	28	2	no	—
206	C₂×C₈	#1144	1	4	yes	—
207	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
208	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	7	2	no	—
214	C₄⋊C₄	—	14	2	no	—
215	C₄⋊C₄	—	14	1	no	—
216	C₄⋊C₄	—	14	2	no	—
217	C₄²	—	14	2	no	—
218	C₄²	—	14	2	no	—
219	C₄²	D₄×D₇	1	1	yes	characteristic
220	C₄²	#1159	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
222	C₄²	C₂³×D₇	1	1	yes	characteristic
223	C₂×C₇	#1696	1	4	yes	complemented
224	C₂×C₇	C₂×#462	1	1	yes	characteristic
225	C₂×C₇	C₂×#470	1	1	yes	characteristic
226	C₂×C₇	C₂×#472	1	1	yes	characteristic
227	C₂³	#6501	1	1	yes	characteristic
228	Q₈	—	8	4	no	—
229	Q₈	—	8	4	no	—
230	Q₈	—	14	4	no	—
231	Q₈	—	28	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	8	4	no	—
235	C₂×C₄	—	14	2	no	—
236	C₂×C₄	—	14	2	no	—
237	C₂×C₄	—	28	2	no	—
238	C₂×C₄	—	28	2	no	—
239	C₂×C₄	#6481	1	4	yes	—
240	C₂×C₄	#6501	1	1	yes	characteristic
241	C₂×C₄	#6501	1	1	yes	characteristic
242	C₂×C₄	#6501	1	1	yes	characteristic
243	C₂×C₄	#6505	1	2	yes	—
244	C₂×C₄	C₂×#1144	1	2	yes	—
245	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
246	C₂×C₄	C₂×#1159	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	28	4	no	—
250	C₈	—	28	2	no	—
251	C₈	C₂×#1144	1	4	yes	—
252	C₇	C₂×#1696	1	1	yes	characteristic, complemented, Sylow
253	C₂²	#89862	1	2	yes	—
254	C₂²	#89932	1	2	yes	—
255	C₂²	#89961	1	2	yes	—
256	C₂²	C₂×#6501	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	8	4	no	—
260	C₄	—	28	2	no	—
261	C₄	—	28	2	no	—
262	C₄	#89862	1	2	yes	—
263	C₄	#89961	1	2	yes	—
264	C₄	C₂×#6481	1	2	yes	—
265	C₄	C₂×#6505	1	1	yes	characteristic
266	C₄	C₂²×#1144	1	1	yes	characteristic
267	C₂	#11729134	1	4	yes	direct product factor
268	C₂	C₂×#89862	1	1	yes	characteristic
269	C₂	C₂×#89932	1	1	yes	characteristic
270	C₂	C₂×#89961	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6501
commutator, cosocle	C₄²×C₇	C₂⁴
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×#500	C₂
socle, prime radical	C₂³×C₇	#121
hypercenter	C₂×#500	D₇
hypocenter	C₇	C₂×#1696
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
5. #11729134	C₂	direct (N × H)
223. C₂×C₇	#1696	semidirect
252. C₇	C₂×#1696	semidirect
267. C₂	#11729134	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×#500	2
C₂×#89863	2
C₂×#89451	2
C₂×#89341	2
#11729134	2
C₂×#89252	2
C₂×#1696	7

Subgroups — C2×#11729134 #13336427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11729134 #13336427