Subgroups — #13061616

Order 1344.

588 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66728	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621375	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×D₂₁	C₂	1	2	yes	maximal, complemented
6	#10621364	C₂	1	4	yes	maximal, complemented
7	C₄×D₈₄	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₄₂	C₂	1	2	yes	maximal, complemented
9	#90214	—	3	1	no	maximal, Hall
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×#4268	C₂²	1	2	yes	—
12	C₂²×Dic₂₁	C₂²	1	2	yes	—
13	#66728	C₄	1	8	yes	complemented
14	C₂×D₈₄	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₁	C₂²	1	4	yes	—
16	C₂×C₄×D₂₁	C₂²	1	2	yes	—
17	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²×C₇	C₂²	1	2	yes	—
22	#66631	C₂²	1	4	yes	—
23	C₄⋊Dic₂₁	C₂²	1	2	yes	—
24	C₄⋊Dic₂₁	C₂²	1	2	yes	—
25	#66629	C₂²	1	4	yes	—
26	C₄×Dic₂₁	C₂²	1	4	yes	—
27	C₂×#1157	—	3	1	no	—
28	C₇×#117	D₃	1	1	yes	characteristic, complemented
29	C₂×C₄⋊Dic₇	—	3	1	no	—
30	#6455	—	3	2	no	—
31	D₇×C₄⋊C₄	—	3	2	no	—
32	(C₂²⋊D₇)⋊C₄	—	3	4	no	—
33	C₄×D₂₈	—	3	2	no	—
34	C₄×Dic₁₄	—	3	2	no	—
35	#5523	—	7	1	no	maximal, Hall
36	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
37	C₂²×D₂₁	C₂³	1	2	yes	—
38	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
39	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
40	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
43	#4268	C₂×C₄	1	8	yes	complemented
44	C₂×Dic₂₁	—	4	2	no	—
45	C₂×Dic₂₁	C₂³	1	4	yes	—
46	C₂×Dic₂₁	C₂³	1	2	yes	—
47	D₈₄	C₂×C₄	1	4	yes	complemented
48	C₄×D₂₁	—	4	4	no	—
49	C₄×D₂₁	C₂×C₄	1	8	yes	complemented
50	Dic₄₂	C₂×C₄	1	4	yes	complemented
51	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
52	C₂×C₂²⋊D₇	—	3	2	no	—
53	C₂²×Dic₇	—	3	2	no	—
54	#1157	—	3	8	no	—
55	C₂×D₂₈	—	3	1	no	—
56	C₂×C₄×D₇	—	3	2	no	—
57	C₂×C₄×D₇	—	3	4	no	—
58	C₂×Dic₁₄	—	3	1	no	—
59	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
60	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
61	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
62	C₄²×C₇	C₂×D₃	1	2	yes	—
63	(C₂×D₇)⋊C₄	—	3	4	no	—
64	C₄⋊Dic₇	—	3	2	no	—
65	C₄⋊Dic₇	—	3	2	no	—
66	Dic₇⋊C₄	—	3	4	no	—
67	C₄×Dic₇	—	3	4	no	—
68	C₂×#235	—	7	1	no	—
69	C₃×#117	D₇	1	1	yes	characteristic, complemented
70	C₂×C₄⋊Dic₃	—	7	1	no	—
71	#892	—	7	2	no	—
72	D₃×C₄⋊C₄	—	7	2	no	—
73	#67⋊C₄	—	7	4	no	—
74	C₄×D₁₂	—	7	2	no	—
75	C₄×Dic₆	—	7	2	no	—
76	C₂²×C₃×C₇	—	2	1	no	—
77	C₂²×C₃×C₇	—	2	1	no	—
78	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
80	C₂×D₂₁	—	2	2	no	—
81	C₂×D₂₁	—	2	2	no	—
82	C₂×D₂₁	C₂²×C₄	1	4	yes	—
83	C₃×C₄×C₇	—	2	2	no	—
84	C₃×C₄×C₇	—	2	2	no	—
85	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
86	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
87	Dic₂₁	—	2	4	no	—
88	Dic₂₁	C₂²×C₄	1	4	yes	—
89	#520	—	21	1	no	Sylow
90	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
91	C₂²×D₇	—	3	2	no	—
92	C₂×C₄×C₇	C₄×D₃	1	4	yes	complemented
93	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
94	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
95	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
97	C₂²⋊D₇	—	3	8	no	—
98	C₂×Dic₇	—	3	2	no	—
99	C₂×Dic₇	—	3	4	no	—
100	C₂×Dic₇	—	12	2	no	—
101	D₂₈	—	3	4	no	—
102	C₄×D₇	—	3	8	no	—
103	C₄×D₇	—	12	4	no	—
104	Dic₁₄	—	3	4	no	—
105	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
106	C₂×#67	—	7	2	no	—
107	C₂²×Dic₃	—	7	2	no	—
108	#235	—	7	8	no	—
109	C₂×D₁₂	—	7	1	no	—
110	C₂×C₄×D₃	—	7	2	no	—
111	C₂×C₄×D₃	—	7	4	no	—
112	C₂×Dic₆	—	7	1	no	—
113	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
114	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
115	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
116	C₃×C₄²	C₂×D₇	1	2	yes	—
117	(C₂×D₃)⋊C₄	—	7	4	no	—
118	C₄⋊Dic₃	—	7	2	no	—
119	C₄⋊Dic₃	—	7	2	no	—
120	Dic₃⋊C₄	—	7	4	no	—
121	C₄×Dic₃	—	7	4	no	—
122	C₂×C₃×C₇	—	2	2	no	—
123	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
126	D₂₁	—	2	4	no	—
127	C₂×#41	—	21	1	no	—
128	C₄×Q₈	—	21	2	no	—
129	C₄×D₄	—	21	4	no	—
130	C₄×D₄	—	21	2	no	—
131	#117	—	21	2	no	—
132	#117	D₂₁	1	1	yes	characteristic, complemented
133	C₂×C₄⋊C₄	—	21	1	no	—
134	C₂×C₄⋊C₄	—	21	2	no	—
135	C₂²×C₇	—	2	1	no	—
136	C₂²×C₇	—	2	1	no	—
137	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
138	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
139	C₂×D₇	—	3	4	no	—
140	C₂×D₇	—	6	2	no	—
141	C₂×D₇	—	6	2	no	—
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	2	2	no	—
144	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
145	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
146	Dic₇	—	3	4	no	—
147	Dic₇	—	6	4	no	—
148	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
149	C₂²×D₃	—	7	2	no	—
150	C₂×C₃×C₄	C₄×D₇	1	4	yes	complemented
151	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
152	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
153	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
155	#67	—	7	8	no	—
156	C₂×Dic₃	—	7	2	no	—
157	C₂×Dic₃	—	7	4	no	—
158	C₂×Dic₃	—	28	2	no	—
159	D₁₂	—	7	4	no	—
160	C₄×D₃	—	7	8	no	—
161	C₄×D₃	—	28	4	no	—
162	Dic₆	—	7	4	no	—
163	C₃×C₇	#520	1	1	yes	characteristic, complemented, Hall
164	#41	—	21	8	no	—
165	C₂×Q₈	—	21	1	no	—
166	C₂×D₄	—	21	2	no	—
167	C₂×D₄	—	21	1	no	—
168	C₂²×C₄	—	21	2	no	—
169	C₂²×C₄	—	21	2	no	—
170	C₂²×C₄	—	21	4	no	—
171	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
172	C₄⋊C₄	—	21	4	no	—
173	C₄⋊C₄	—	21	2	no	—
174	C₄⋊C₄	—	21	2	no	—
175	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
176	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
177	C₂²⋊C₄	—	21	4	no	—
178	C₂²⋊C₄	C₂×D₂₁	1	2	yes	—
179	C₄²	—	21	4	no	—
180	C₄²	C₂×D₂₁	1	2	yes	—
181	C₂×C₇	—	2	2	no	—
182	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
183	C₂×C₇	#1009	1	1	yes	characteristic
184	C₂×C₇	#1010	1	1	yes	characteristic
185	D₇	—	6	4	no	—
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
189	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
190	C₂×D₃	—	7	4	no	—
191	C₂×D₃	—	14	2	no	—
192	C₂×D₃	—	14	2	no	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
196	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
197	Dic₃	—	7	4	no	—
198	Dic₃	—	14	4	no	—
199	C₂³	—	21	2	no	—
200	C₂³	C₂²×D₂₁	1	1	yes	characteristic
201	Q₈	—	21	4	no	—
202	D₄	—	21	8	no	—
203	D₄	—	21	4	no	—
204	C₂×C₄	—	21	2	no	—
205	C₂×C₄	—	21	8	no	—
206	C₂×C₄	—	21	4	no	—
207	C₂×C₄	—	84	4	no	—
208	C₂×C₄	—	84	2	no	—
209	C₂×C₄	C₄×D₂₁	1	4	yes	complemented
210	C₂×C₄	C₂²×D₂₁	1	2	yes	—
211	C₂×C₄	C₂²×D₂₁	1	2	yes	—
212	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
214	C₇	#5523	1	1	yes	characteristic, complemented, Sylow
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
217	C₂×C₃	#6553	1	1	yes	characteristic
218	C₂×C₃	#6554	1	1	yes	characteristic
219	D₃	—	14	4	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	21	4	no	—
223	C₂²	—	42	2	no	—
224	C₂²	—	42	2	no	—
225	C₂²	C₂×C₄×D₂₁	1	2	yes	—
226	C₂²	C₂³×D₂₁	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	21	4	no	—
230	C₄	—	42	4	no	—
231	C₄	C₂×C₄×D₂₁	1	2	yes	—
232	C₄	C₂×C₄×D₂₁	1	2	yes	—
233	C₃	#90214	1	1	yes	characteristic, complemented, Sylow
234	C₂	—	2	2	no	—
235	C₂	—	42	4	no	—
236	C₂	C₂²×C₄×D₂₁	1	1	yes	characteristic
237	C₂	#10621662	1	1	yes	characteristic
238	C₂	#10621663	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator	C₂×C₃×C₇	C₂³×C₄
Fitting	C₃×C₇×#117	C₂
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×#66728	C₂
hypercenter	#117	D₂₁
hypocenter	C₃×C₇	#520
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×#117	C₂	semidirect
3. C₂×C₄⋊Dic₂₁	C₂	semidirect
4. #10621375	C₂	semidirect
5. C₄⋊C₄×D₂₁	C₂	semidirect
6. #10621364	C₂	semidirect
7. C₄×D₈₄	C₂	semidirect
8. C₄×Dic₄₂	C₂	semidirect
13. #66728	C₄	semidirect
28. C₇×#117	D₃	semidirect
38. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
43. #4268	C₂×C₄	semidirect
47. D₈₄	C₂×C₄	semidirect
49. C₄×D₂₁	C₂×C₄	semidirect
50. Dic₄₂	C₂×C₄	semidirect
69. C₃×#117	D₇	semidirect
92. C₂×C₄×C₇	C₄×D₃	semidirect
132. #117	D₂₁	semidirect
150. C₂×C₃×C₄	C₄×D₇	semidirect
163. C₃×C₇	#520	semidirect
209. C₂×C₄	C₄×D₂₁	semidirect
214. C₇	#5523	semidirect
233. C₃	#90214	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66728	2
C₃×C₇×#117	2
C₂×C₄⋊Dic₂₁	2
#10621375	2
C₄⋊C₄×D₂₁	2
#10621364	2
C₄×D₈₄	2
C₄×Dic₄₂	2
#90214	3
#5523	7