Subgroups — #13058771

Order 1344.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621373	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#892	C₂	1	1	yes	maximal, characteristic
3	C₃×#6453	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621032	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621031	C₂	1	2	yes	maximal, complemented
6	#10621030	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620998	C₂	1	2	yes	maximal
8	D₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
9	#10620979	C₂	1	2	yes	maximal
10	#10620941	C₂	1	2	yes	maximal, complemented
11	#10620933	C₂	1	1	yes	maximal, characteristic
12	#90210	—	3	1	no	maximal, Hall
13	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
14	C₂×#4248	C₂²	1	2	yes	—
15	C₂×#4248	C₂²	1	2	yes	—
16	C₂×D₃×Dic₇	C₂²	1	2	yes	—
17	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₂₁	C₂²	1	2	yes	—
19	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
20	#66629	C₂²	1	2	yes	—
21	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
22	C₇×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₄×C₇×Dic₃	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
27	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
28	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
29	Dic₃⋊Dic₇	C₂²	1	1	yes	characteristic
30	Dic₃⋊Dic₇	C₂²	1	1	yes	characteristic
31	Dic₃⋊Dic₇	C₂²	1	2	yes	—
32	#66577	C₂²	1	2	yes	—
33	Dic₇⋊Dic₃	C₂²	1	2	yes	—
34	(C₂×D₃)⋊Dic₇	C₂²	1	1	yes	characteristic
35	(C₂×D₃)⋊Dic₇	C₂²	1	1	yes	characteristic
36	C₇×#117	—	3	1	no	—
37	#6489	—	3	1	no	—
38	C₂×C₄⋊Dic₇	—	3	1	no	—
39	#6453	—	3	1	no	—
40	#6453	D₃	1	1	yes	characteristic, complemented
41	#6451	—	3	1	no	—
42	#6451	—	3	1	no	—
43	#6441	—	3	2	no	—
44	#6441	—	3	2	no	—
45	#6433	—	3	2	no	—
46	#6432	—	3	2	no	—
47	#5689	—	7	1	no	maximal, Hall
48	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁	C₂³	1	2	yes	—
54	C₂×C₇×Dic₃	C₂³	1	2	yes	—
55	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
56	C₄×D₃×C₇	Q₈	1	4	yes	complemented
57	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
58	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₇	C₂³	1	2	yes	—
60	#4248	—	4	2	no	—
61	#4248	—	4	2	no	—
62	D₃×Dic₇	—	4	2	no	—
63	C₂²×C₄×C₇	—	3	1	no	—
64	C₂²×Dic₇	—	3	2	no	—
65	C₂×Dic₁₄	—	3	2	no	—
66	C₂×Dic₁₄	—	3	2	no	—
67	C₇×C₄⋊C₄	—	3	1	no	—
68	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
69	C₇×C₂²⋊C₄	—	3	2	no	—
70	C₄²×C₇	—	3	2	no	—
71	C₂²⋊Dic₇	—	3	1	no	—
72	C₂²⋊Dic₇	—	3	1	no	—
73	C₄⋊Dic₇	—	3	2	no	—
74	C₄⋊Dic₇	—	3	1	no	—
75	C₄⋊Dic₇	—	3	1	no	—
76	C₄⋊Dic₇	—	3	2	no	—
77	C₄⋊Dic₇	—	3	1	no	—
78	C₄⋊Dic₇	C₂×D₃	1	2	yes	—
79	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
80	Dic₇⋊C₄	—	3	2	no	—
81	Dic₇⋊C₄	—	3	2	no	—
82	Dic₇⋊C₄	—	3	2	no	—
83	Dic₇⋊C₄	C₂×D₃	1	2	yes	—
84	C₄×Dic₇	—	3	1	no	—
85	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
86	C₃×#125	—	7	1	no	—
87	#897	—	7	2	no	—
88	#896	—	7	2	no	—
89	#892	D₇	1	1	yes	characteristic
90	D₃×C₄⋊C₄	—	7	1	no	—
91	#890	—	7	1	no	—
92	#889	—	7	2	no	—
93	#888	—	7	1	no	—
94	#888	—	7	2	no	—
95	#872	—	7	1	no	—
96	#869	—	7	1	no	—
97	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
98	C₂×D₃×C₇	—	2	1	no	—
99	C₂×D₃×C₇	—	2	1	no	—
100	C₂×D₃×C₇	C₂×Q₈	1	2	yes	—
101	C₃×C₄×C₇	—	2	2	no	—
102	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
103	Dic₂₁	—	2	2	no	—
104	Dic₂₁	—	2	1	no	—
105	Dic₂₁	—	2	1	no	—
106	C₃×Dic₇	—	2	2	no	—
107	C₃×Dic₇	—	2	1	no	—
108	C₃×Dic₇	—	2	1	no	—
109	C₇×Dic₃	—	2	2	no	—
110	C₇×Dic₃	C₂×Q₈	1	2	yes	—
111	#544	—	21	1	no	Sylow
112	C₂³×C₇	—	3	1	no	—
113	C₂×C₄×C₇	—	3	1	no	—
114	C₂×C₄×C₇	—	3	4	no	—
115	C₂×C₄×C₇	—	3	2	no	—
116	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
117	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
118	C₂×Dic₇	—	3	1	no	—
119	C₂×Dic₇	—	3	1	no	—
120	C₂×Dic₇	—	3	2	no	—
121	C₂×Dic₇	—	12	2	no	—
122	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
123	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
124	C₂×Dic₇	C₂²×D₃	1	2	yes	—
125	Dic₁₄	—	12	2	no	—
126	Dic₁₄	—	12	2	no	—
127	C₂×C₄×D₃	—	7	2	no	—
128	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
129	C₂×Dic₆	—	7	2	no	—
130	C₂×Dic₆	—	7	2	no	—
131	C₃×C₄⋊C₄	—	7	2	no	—
132	C₃×C₄⋊C₄	—	7	1	no	—
133	C₃×C₄⋊C₄	—	7	2	no	—
134	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
135	C₃×C₄²	—	7	1	no	—
136	(C₂×D₃)⋊C₄	—	7	1	no	—
137	(C₂×D₃)⋊C₄	—	7	1	no	—
138	(C₂×D₃)⋊C₄	C₂×D₇	1	2	yes	—
139	C₄⋊Dic₃	—	7	1	no	—
140	C₄⋊Dic₃	—	7	2	no	—
141	C₄⋊Dic₃	—	7	2	no	—
142	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
143	Dic₃⋊C₄	—	7	2	no	—
144	Dic₃⋊C₄	—	7	2	no	—
145	Dic₃⋊C₄	—	7	1	no	—
146	Dic₃⋊C₄	—	7	1	no	—
147	Dic₃⋊C₄	—	7	2	no	—
148	C₄×Dic₃	—	7	1	no	—
149	C₄×Dic₃	C₂×D₇	1	2	yes	—
150	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
151	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
153	D₃×C₇	—	2	2	no	—
154	C₄⋊Q₈	—	21	2	no	—
155	C₄⋊Q₈	—	21	1	no	—
156	C₄⋊Q₈	—	21	1	no	—
157	#125	—	7	1	no	—
158	#125	—	21	2	no	—
159	#125	—	21	1	no	—
160	C₂²⋊Q₈	—	21	2	no	—
161	C₂²⋊Q₈	—	21	2	no	—
162	#117	—	3	1	no	—
163	#117	—	21	1	no	—
164	C₂×C₄⋊C₄	—	21	1	no	—
165	C₂²×C₇	—	3	2	no	—
166	C₂²×C₇	—	6	1	no	—
167	C₂²×C₇	—	6	1	no	—
168	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
169	C₄×C₇	—	2	2	no	—
170	C₄×C₇	—	3	2	no	—
171	C₄×C₇	—	6	2	no	—
172	C₄×C₇	D₃×Q₈	1	2	yes	—
173	Dic₇	—	2	1	no	—
174	Dic₇	—	2	2	no	—
175	Dic₇	—	2	1	no	—
176	Dic₇	—	6	1	no	—
177	Dic₇	—	6	2	no	—
178	Dic₇	—	6	1	no	—
179	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
180	C₂×C₃×C₄	—	7	2	no	—
181	C₂×C₃×C₄	—	7	1	no	—
182	C₂×C₃×C₄	—	7	1	no	—
183	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
184	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
185	C₂×Dic₃	—	7	1	no	—
186	C₂×Dic₃	—	7	1	no	—
187	C₂×Dic₃	—	7	2	no	—
188	C₂×Dic₃	C₂²×D₇	1	2	yes	—
189	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
190	C₄×D₃	—	28	2	no	—
191	C₄×D₃	Dic₁₄	1	4	yes	complemented
192	Dic₆	—	28	2	no	—
193	Dic₆	—	28	2	no	—
194	C₃×C₇	#544	1	1	yes	characteristic, complemented, Hall
195	C₂×Q₈	—	21	2	no	—
196	C₂×Q₈	—	21	2	no	—
197	C₂²×C₄	—	3	1	no	—
198	C₂²×C₄	—	21	2	no	—
199	C₄⋊C₄	—	3	1	no	—
200	C₄⋊C₄	—	7	2	no	—
201	C₄⋊C₄	—	7	1	no	—
202	C₄⋊C₄	—	7	2	no	—
203	C₄⋊C₄	—	21	2	no	—
204	C₄⋊C₄	—	21	2	no	—
205	C₄⋊C₄	—	21	2	no	—
206	C₄⋊C₄	—	21	1	no	—
207	C₄⋊C₄	—	21	1	no	—
208	C₄⋊C₄	—	21	2	no	—
209	C₄⋊C₄	—	21	1	no	—
210	C₄⋊C₄	—	21	2	no	—
211	C₄⋊C₄	D₃×D₇	1	1	yes	characteristic
212	C₂²⋊C₄	—	3	2	no	—
213	C₂²⋊C₄	—	21	1	no	—
214	C₂²⋊C₄	—	21	1	no	—
215	C₄²	—	3	2	no	—
216	C₄²	—	7	1	no	—
217	C₄²	—	21	1	no	—
218	C₂×C₇	—	6	2	no	—
219	C₂×C₇	C₂×D₃×Q₈	1	1	yes	characteristic
220	C₂×C₇	#1009	1	1	yes	characteristic
221	C₂×C₇	#1010	1	1	yes	characteristic
222	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
223	C₂×D₃	—	2	1	no	—
224	C₂×D₃	—	2	1	no	—
225	C₂×D₃	C₂×Dic₁₄	1	2	yes	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	14	1	no	—
228	C₃×C₄	—	14	2	no	—
229	C₃×C₄	—	14	1	no	—
230	C₃×C₄	C₂×Dic₁₄	1	2	yes	—
231	Dic₃	—	2	2	no	—
232	Dic₃	—	14	1	no	—
233	Dic₃	—	14	2	no	—
234	Dic₃	—	14	1	no	—
235	Dic₃	C₂×Dic₁₄	1	2	yes	—
236	C₂³	—	3	1	no	—
237	Q₈	—	84	2	no	—
238	Q₈	—	84	2	no	—
239	C₂×C₄	—	3	1	no	—
240	C₂×C₄	—	3	4	no	—
241	C₂×C₄	—	3	2	no	—
242	C₂×C₄	—	7	1	no	—
243	C₂×C₄	—	7	1	no	—
244	C₂×C₄	—	7	2	no	—
245	C₂×C₄	—	21	1	no	—
246	C₂×C₄	—	21	1	no	—
247	C₂×C₄	—	21	2	no	—
248	C₂×C₄	—	84	2	no	—
249	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
250	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
251	C₇	#5689	1	1	yes	characteristic, complemented, Sylow
252	C₂×C₃	C₂²×Dic₁₄	1	1	yes	characteristic
253	C₂×C₃	#6553	1	1	yes	characteristic
254	C₂×C₃	#6554	1	1	yes	characteristic
255	D₃	—	2	2	no	—
256	C₂²	—	3	2	no	—
257	C₂²	—	6	1	no	—
258	C₂²	—	6	1	no	—
259	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	3	2	no	—
262	C₄	—	6	2	no	—
263	C₄	—	14	1	no	—
264	C₄	—	14	2	no	—
265	C₄	—	14	1	no	—
266	C₄	—	42	1	no	—
267	C₄	—	42	2	no	—
268	C₄	—	42	1	no	—
269	C₄	D₃×Dic₁₄	1	2	yes	—
270	C₃	#90210	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	6	2	no	—
272	C₂	C₂×D₃×Dic₁₄	1	1	yes	characteristic
273	C₂	#10621567	1	1	yes	characteristic
274	C₂	#10621582	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₇×C₄⋊C₄	C₂²
prime radical	C₂²×D₃×C₇	C₂³
hypercenter	C₄⋊C₄	D₃×D₇
hypocenter	C₃×C₇	#544
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #10621373	C₂	semidirect
3. C₃×#6453	C₂	semidirect
4. #10621032	C₂	semidirect
5. #10621031	C₂	semidirect
6. #10621030	C₂	semidirect
10. #10620941	C₂	semidirect
40. #6453	D₃	semidirect
56. C₄×D₃×C₇	Q₈	semidirect
191. C₄×D₃	Dic₁₄	semidirect
194. C₃×C₇	#544	semidirect
251. C₇	#5689	semidirect
270. C₃	#90210	semidirect

Maximal subgroups

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

subgroup	index
#10621373	2
C₇×#892	2
C₃×#6453	2
#10621032	2
#10621031	2
#10621030	2
#10620998	2
D₃×C₄⋊Dic₇	2
#10620979	2
#10620941	2
#10620933	2
#90210	3
#5689	7