Subgroups — #13268864

Order 1792.

494 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#349	C₂	1	1	yes	maximal, characteristic
3	C₂×#89348	C₂	1	1	yes	maximal, characteristic
4	C₂×#89307	C₂	1	2	yes	maximal, complemented
5	#11725616	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄×Dic₇	C₄	1	1	yes	characteristic
8	C₂²×#1136	C₄	1	1	yes	characteristic
9	C₂×C₇×M₃₂	C₂²	1	2	yes	—
10	M₁₆×Dic₇	—	2	4	no	—
11	C₂×#6389	C₄	1	1	yes	characteristic
12	C₂×#6389	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
15	C₂×#6386	C₂²	1	2	yes	—
16	C₇×#349	C₄	1	4	yes	complemented
17	#89348	C₄	1	4	yes	complemented
18	#89307	—	2	4	no	—
19	#12607	—	7	1	no	maximal, Sylow
20	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×Dic₇	—	2	1	no	—
22	C₂×C₇×M₁₆	—	2	2	no	—
23	C₂×C₇×M₁₆	—	2	2	no	—
24	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄×Dic₇	—	2	1	no	—
29	C₂×C₄×Dic₇	—	2	2	no	—
30	C₂×C₄×Dic₇	—	2	2	no	—
31	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
33	C₂×#1136	—	2	2	no	—
34	C₂×#1136	—	2	2	no	—
35	C₂×#1136	—	2	2	no	—
36	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₇×M₃₂	—	2	2	no	—
39	C₇×M₃₂	C₂×C₄	1	4	yes	complemented
40	C₂×C₇×C₁₆	—	2	2	no	—
41	#6389	—	2	2	no	—
42	#6389	D₄	1	4	yes	—
43	C₈×Dic₇	—	2	2	no	—
44	C₈×Dic₇	—	2	2	no	—
45	#6386	—	2	2	no	—
46	#6386	C₂×C₄	1	4	yes	complemented
47	C₂×C₇⋊C₁₆	—	2	2	no	—
48	C₂×C₄×M₁₆	—	7	1	no	—
49	C₂×#349	—	7	1	no	—
50	C₂×#349	D₇	1	1	yes	characteristic
51	C₂×#347	—	7	2	no	—
52	#1376	—	7	2	no	—
53	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	—	2	1	no	—
56	C₂²×C₄×C₇	—	2	1	no	—
57	C₂²×C₄×C₇	—	2	1	no	—
58	C₂²×C₄×C₇	C₄²	1	2	yes	—
59	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₇	M₁₆	1	2	yes	—
61	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₇	—	2	1	no	—
63	C₂²×Dic₇	—	2	4	no	—
64	C₂²×Dic₇	—	2	1	no	—
65	C₂²×Dic₇	—	4	4	no	—
66	C₇×M₁₆	—	2	4	no	—
67	C₇×M₁₆	—	2	4	no	—
68	C₂×C₇×C₈	—	2	1	no	—
69	C₂×C₇×C₈	—	2	1	no	—
70	C₂×C₇×C₈	C₄²	1	2	yes	—
71	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
74	C₂×C₇×C₈	#41	1	2	yes	—
75	C₄×Dic₇	—	2	4	no	—
76	C₄×Dic₇	—	2	1	no	—
77	C₄×Dic₇	—	2	1	no	—
78	C₄×Dic₇	C₂²⋊C₄	1	2	yes	—
79	C₄×Dic₇	M₁₆	1	2	yes	—
80	#1136	—	2	8	no	—
81	C₂×C₇⋊C₈	—	2	1	no	—
82	C₂×C₇⋊C₈	—	2	1	no	—
83	C₂×C₇⋊C₈	—	2	1	no	—
84	C₂×C₇⋊C₈	—	2	1	no	—
85	C₂×C₇⋊C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₇⋊C₈	M₁₆	1	2	yes	—
87	C₇×C₁₆	—	2	4	no	—
88	C₇⋊C₁₆	—	2	4	no	—
89	C₂²×M₁₆	—	7	1	no	—
90	C₂²×M₁₆	C₂×D₇	1	1	yes	characteristic
91	C₂²×C₄²	—	7	1	no	—
92	C₂×M₃₂	—	7	2	no	—
93	C₂×M₃₂	C₂×D₇	1	2	yes	—
94	C₄×M₁₆	—	14	4	no	—
95	C₂×#97	—	7	1	no	—
96	C₂×#97	—	7	1	no	—
97	C₂×C₄×C₈	—	7	1	no	—
98	C₂×C₄×C₈	—	7	1	no	—
99	#349	—	7	4	no	—
100	#349	Dic₇	1	4	yes	complemented
101	#347	—	14	4	no	—
102	C₂³×C₇	—	2	1	no	—
103	C₂³×C₇	—	2	1	no	—
104	C₂³×C₇	—	4	2	no	—
105	C₂³×C₇	#97	1	2	yes	—
106	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	#97	1	2	yes	—
114	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇	#117	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
120	C₂×Dic₇	—	2	4	no	—
121	C₂×Dic₇	—	2	2	no	—
122	C₂×Dic₇	—	2	2	no	—
123	C₂×Dic₇	—	4	8	no	—
124	C₂×Dic₇	—	4	2	no	—
125	C₇×C₈	—	2	2	no	—
126	C₇×C₈	C₄×D₄	1	2	yes	—
127	C₇×C₈	C₄×D₄	1	2	yes	—
128	C₇⋊C₈	—	2	2	no	—
129	C₇⋊C₈	—	2	2	no	—
130	C₂³×C₄	—	14	1	no	—
131	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	14	2	no	—
135	C₂×M₁₆	—	14	2	no	—
136	C₂×M₁₆	—	14	2	no	—
137	C₂×M₁₆	C₄×D₇	1	2	yes	—
138	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
139	C₂²×C₈	—	7	1	no	—
140	C₂²×C₈	—	7	1	no	—
141	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
143	C₂×C₄²	—	7	1	no	—
144	C₂×C₄²	—	7	1	no	—
145	C₂×C₄²	—	14	2	no	—
146	C₂×C₄²	—	14	2	no	—
147	C₂×C₄²	—	14	1	no	—
148	M₃₂	—	2	2	no	—
149	M₃₂	—	7	4	no	—
150	M₃₂	—	14	2	no	—
151	M₃₂	C₂×Dic₇	1	4	yes	complemented
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	14	2	no	—
154	#97	—	7	4	no	—
155	#97	—	14	2	no	—
156	C₄×C₈	—	14	2	no	—
157	C₄×C₈	—	14	2	no	—
158	C₂²×C₇	—	2	2	no	—
159	C₂²×C₇	—	2	2	no	—
160	C₂²×C₇	—	4	2	no	—
161	C₂²×C₇	—	4	2	no	—
162	C₂²×C₇	—	4	2	no	—
163	C₂²×C₇	C₂×#97	1	1	yes	characteristic
164	C₂²×C₇	#407	1	1	yes	characteristic
165	C₂²×C₇	#432	1	1	yes	characteristic
166	C₄×C₇	—	2	2	no	—
167	C₄×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
168	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
169	C₄×C₇	#407	1	1	yes	characteristic
170	C₄×C₇	#432	1	1	yes	characteristic
171	Dic₇	—	2	4	no	—
172	Dic₇	—	4	2	no	—
173	C₂⁴	#1131	1	1	yes	characteristic
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	14	1	no	—
179	C₂²×C₄	—	14	4	no	—
180	C₂²×C₄	—	14	1	no	—
181	C₂²×C₄	—	28	4	no	—
182	C₂²×C₄	#1131	1	2	yes	—
183	C₂²×C₄	#1131	1	1	yes	characteristic
184	C₂²×C₄	C₄×Dic₇	1	2	yes	—
185	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
186	M₁₆	—	2	4	no	—
187	M₁₆	—	2	4	no	—
188	M₁₆	—	14	8	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	7	2	no	—
192	C₂×C₈	—	7	2	no	—
193	C₂×C₈	—	14	1	no	—
194	C₂×C₈	—	14	1	no	—
195	C₂×C₈	—	14	1	no	—
196	C₂×C₈	—	14	1	no	—
197	C₂×C₈	C₄×Dic₇	1	2	yes	—
198	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
199	C₂×C₈	D₄×D₇	1	2	yes	—
200	C₂×C₈	#1159	1	2	yes	—
201	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
202	C₄²	—	7	2	no	—
203	C₄²	—	7	2	no	—
204	C₄²	—	14	4	no	—
205	C₄²	—	14	1	no	—
206	C₄²	—	14	1	no	—
207	C₁₆	—	2	4	no	—
208	C₁₆	—	14	4	no	—
209	C₂×C₇	—	2	1	no	—
210	C₂×C₇	—	2	1	no	—
211	C₂×C₇	—	4	2	no	—
212	C₂×C₇	#1749	1	1	yes	characteristic
213	C₂×C₇	#2165	1	2	yes	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	4	2	no	—
217	C₂³	#6389	1	2	yes	—
218	C₂³	C₂×#1131	1	1	yes	characteristic
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	14	4	no	—
226	C₂×C₄	—	14	2	no	—
227	C₂×C₄	—	14	2	no	—
228	C₂×C₄	—	28	8	no	—
229	C₂×C₄	—	28	2	no	—
230	C₂×C₄	#6389	1	2	yes	—
231	C₂×C₄	#6440	1	1	yes	characteristic
232	C₂×C₄	D₇×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₂×#1131	1	1	yes	characteristic
234	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
235	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	14	2	no	—
239	C₈	—	14	2	no	—
240	C₈	(C₂²⋊D₇)⋊C₄	1	2	yes	—
241	C₈	D₄×Dic₇	1	2	yes	—
242	C₇	#12607	1	1	yes	characteristic, complemented, Sylow
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#89491	1	1	yes	characteristic
249	C₂²	#89498	1	1	yes	characteristic
250	C₂²	C₂×#6389	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	14	4	no	—
253	C₄	—	28	2	no	—
254	C₄	#89491	1	1	yes	characteristic
255	C₄	#89498	1	1	yes	characteristic
256	C₄	C₂²⋊C₄×Dic₇	1	1	yes	characteristic
257	C₄	M₁₆×Dic₇	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	#11726868	1	2	yes	—
262	C₂	#11727660	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6440
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#349	C₂
socle	C₂²×C₇	#432
prime radical	C₂⁴×C₇	M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#349	D₇
hypocenter	C₇	#12607
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
4. C₂×#89307	C₂	semidirect
5. #11725616	C₂	semidirect
16. C₇×#349	C₄	semidirect
17. #89348	C₄	semidirect
39. C₇×M₃₂	C₂×C₄	semidirect
46. #6386	C₂×C₄	semidirect
100. #349	Dic₇	semidirect
151. M₃₂	C₂×Dic₇	semidirect
242. C₇	#12607	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×M₁₆×Dic₇	2
C₂×C₇×#349	2
C₂×#89348	2
C₂×#89307	2
#11725616	2
#12607	7