Subgroups — #13061637

Order 1344.

676 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₂₁	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#4268	C₂	1	2	yes	maximal, complemented
5	C₂×#66631	C₂	1	2	yes	maximal, complemented
6	#10621376	C₂	1	1	yes	maximal, characteristic, complemented
7	#10621375	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621364	C₂	1	2	yes	maximal, complemented
9	C₂²⋊C₄×D₂₁	C₂	1	2	yes	maximal, complemented
10	C₄×D₈₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#10621355	C₂	1	1	yes	maximal, characteristic, complemented
12	#90236	—	3	1	no	maximal, Hall
13	C₂³×D₂₁	C₂²	1	2	yes	—
14	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	complemented
16	C₂×#4268	C₂²	1	2	yes	—
17	C₂²×Dic₂₁	C₂²	1	2	yes	complemented
18	D₄×D₂₁	C₄	1	8	yes	complemented
19	C₂×D₈₄	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₂₁	C₂²	1	2	yes	complemented
22	C₂×C₄×D₂₁	C₂²	1	2	yes	complemented
23	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
24	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	complemented
25	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
26	#66636	C₂²	1	2	yes	complemented
27	#66631	C₂²	1	1	yes	characteristic
28	#66631	C₂²	1	1	yes	characteristic
29	#66631	C₂²	1	2	yes	—
30	#66631	C₂²	1	2	yes	complemented
31	#66631	C₂²	1	2	yes	complemented
32	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic, complemented
33	#66629	C₂²	1	1	yes	characteristic, complemented
34	#66629	C₂²	1	1	yes	characteristic, complemented
35	C₄×Dic₂₁	C₂²	1	1	yes	characteristic, complemented
36	C₄×Dic₂₁	C₂²	1	2	yes	complemented
37	C₂×D₄×D₇	—	3	1	no	—
38	C₄×C₇×D₄	D₃	1	1	yes	characteristic, complemented
39	D₄×Dic₇	—	3	1	no	—
40	C₄×C₂²⋊D₇	—	3	2	no	—
41	C₂×(C₂×D₇)⋊C₄	—	3	2	no	—
42	#6456	—	3	1	no	—
43	#6455	—	3	1	no	—
44	(C₂²⋊D₇)⋊C₄	—	3	2	no	—
45	D₇×C₂²⋊C₄	—	3	2	no	—
46	C₄×D₂₈	—	3	1	no	—
47	#6435	—	3	1	no	—
48	#5545	—	7	1	no	maximal, Hall
49	C₂³×C₃×C₇	C₂³	1	2	yes	—
50	C₂²×D₂₁	—	4	2	no	—
51	C₂²×D₂₁	—	4	2	no	—
52	C₂²×D₂₁	C₂×C₄	1	8	yes	complemented
53	C₂²×D₂₁	C₂³	1	1	yes	characteristic
54	C₂²×D₂₁	C₂³	1	2	yes	—
55	C₃×C₇×D₄	C₂×C₄	1	4	yes	complemented
56	C₂×C₃×C₄×C₇	—	4	2	no	—
57	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
58	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
61	#4268	C₂×C₄	1	8	yes	complemented
62	C₂×Dic₂₁	—	4	2	no	—
63	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
64	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
65	C₂×Dic₂₁	C₂³	1	2	yes	—
66	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
67	D₈₄	C₂×C₄	1	4	yes	complemented
68	C₄×D₂₁	—	4	2	no	—
69	C₄×D₂₁	—	4	2	no	—
70	C₄×D₂₁	C₂×C₄	1	4	yes	complemented
71	C₂³×D₇	—	3	2	no	—
72	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂×D₃	1	2	yes	complemented
74	C₂×C₂²⋊D₇	—	3	2	no	—
75	C₂²×Dic₇	—	3	2	no	—
76	D₄×D₇	—	3	8	no	—
77	C₂×D₂₈	—	3	1	no	—
78	C₂×C₄×D₇	—	3	1	no	—
79	C₂×C₄×D₇	—	3	2	no	—
80	C₂×C₄×D₇	—	3	2	no	—
81	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
82	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
83	C₄²×C₇	C₂×D₃	1	1	yes	characteristic, complemented
84	C₂²⋊Dic₇	—	3	2	no	—
85	(C₂×D₇)⋊C₄	—	3	1	no	—
86	(C₂×D₇)⋊C₄	—	3	1	no	—
87	(C₂×D₇)⋊C₄	—	3	2	no	—
88	(C₂×D₇)⋊C₄	—	3	2	no	—
89	(C₂×D₇)⋊C₄	—	3	2	no	—
90	C₄⋊Dic₇	—	3	1	no	—
91	Dic₇⋊C₄	—	3	1	no	—
92	Dic₇⋊C₄	—	3	1	no	—
93	C₄×Dic₇	—	3	1	no	—
94	C₄×Dic₇	—	3	2	no	—
95	C₂×D₃×D₄	—	7	1	no	—
96	C₃×C₄×D₄	D₇	1	1	yes	characteristic, complemented
97	D₄×Dic₃	—	7	1	no	—
98	C₄×#67	—	7	2	no	—
99	C₂×(C₂×D₃)⋊C₄	—	7	2	no	—
100	#893	—	7	1	no	—
101	#892	—	7	1	no	—
102	#67⋊C₄	—	7	2	no	—
103	D₃×C₂²⋊C₄	—	7	2	no	—
104	C₄×D₁₂	—	7	1	no	—
105	#872	—	7	1	no	—
106	C₂²×C₃×C₇	—	2	2	no	—
107	C₂²×C₃×C₇	—	2	2	no	—
108	C₂²×C₃×C₇	C₂²×C₄	1	4	yes	—
109	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
110	C₂×D₂₁	—	2	2	no	—
111	C₂×D₂₁	—	2	2	no	—
112	C₂×D₂₁	—	2	1	no	—
113	C₂×D₂₁	—	2	1	no	—
114	C₂×D₂₁	—	4	8	no	—
115	C₂×D₂₁	C₂²×C₄	1	2	yes	—
116	C₂×D₂₁	C₂²×C₄	1	4	yes	—
117	C₃×C₄×C₇	—	2	1	no	—
118	C₃×C₄×C₇	—	2	2	no	—
119	C₃×C₄×C₇	—	2	1	no	—
120	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
121	Dic₂₁	—	2	1	no	—
122	Dic₂₁	—	2	2	no	—
123	Dic₂₁	—	2	1	no	—
124	Dic₂₁	C₂²×C₄	1	2	yes	—
125	#518	—	21	1	no	Sylow
126	C₂³×C₇	C₂²×D₃	1	2	yes	—
127	C₂²×D₇	—	3	1	no	—
128	C₂²×D₇	—	3	2	no	—
129	C₂²×D₇	—	3	8	no	—
130	C₂²×D₇	—	12	2	no	—
131	C₂²×D₇	—	12	2	no	—
132	C₇×D₄	C₄×D₃	1	4	yes	complemented
133	C₂×C₄×C₇	—	4	2	no	—
134	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
135	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
138	C₂²⋊D₇	—	3	8	no	—
139	C₂×Dic₇	—	3	1	no	—
140	C₂×Dic₇	—	3	1	no	—
141	C₂×Dic₇	—	3	1	no	—
142	C₂×Dic₇	—	3	2	no	—
143	C₂×Dic₇	—	12	2	no	—
144	D₂₈	—	3	4	no	—
145	C₄×D₇	—	3	4	no	—
146	C₄×D₇	—	12	2	no	—
147	C₄×D₇	—	12	2	no	—
148	C₂³×D₃	—	7	2	no	—
149	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₂×D₇	1	2	yes	complemented
151	C₂×#67	—	7	2	no	—
152	C₂²×Dic₃	—	7	2	no	—
153	D₃×D₄	—	7	8	no	—
154	C₂×D₁₂	—	7	1	no	—
155	C₂×C₄×D₃	—	7	1	no	—
156	C₂×C₄×D₃	—	7	2	no	—
157	C₂×C₄×D₃	—	7	2	no	—
158	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic, complemented
159	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	complemented
160	C₃×C₄²	C₂×D₇	1	1	yes	characteristic, complemented
161	#217	—	7	2	no	—
162	(C₂×D₃)⋊C₄	—	7	1	no	—
163	(C₂×D₃)⋊C₄	—	7	1	no	—
164	(C₂×D₃)⋊C₄	—	7	2	no	—
165	(C₂×D₃)⋊C₄	—	7	2	no	—
166	(C₂×D₃)⋊C₄	—	7	2	no	—
167	C₄⋊Dic₃	—	7	1	no	—
168	Dic₃⋊C₄	—	7	1	no	—
169	Dic₃⋊C₄	—	7	1	no	—
170	C₄×Dic₃	—	7	1	no	—
171	C₄×Dic₃	—	7	2	no	—
172	C₂×C₃×C₇	—	2	4	no	—
173	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
176	D₂₁	—	2	2	no	—
177	D₂₁	—	2	4	no	—
178	C₂²×D₄	—	21	1	no	—
179	C₄×D₄	—	21	1	no	—
180	C₄×D₄	—	21	2	no	—
181	C₄×D₄	—	21	1	no	—
182	C₄×D₄	—	21	2	no	—
183	C₄×D₄	—	21	1	no	—
184	C₄×D₄	D₂₁	1	1	yes	characteristic, complemented
185	#117	—	21	1	no	—
186	#117	—	21	1	no	—
187	C₂×C₂²⋊C₄	—	21	2	no	—
188	C₂×C₂²⋊C₄	—	21	2	no	—
189	C₂²×C₇	—	2	2	no	—
190	C₂²×C₇	—	2	2	no	—
191	C₂²×C₇	C₂×C₄×D₃	1	4	yes	—
192	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
193	C₂×D₇	—	3	2	no	—
194	C₂×D₇	—	3	4	no	—
195	C₂×D₇	—	6	2	no	—
196	C₂×D₇	—	6	1	no	—
197	C₂×D₇	—	6	2	no	—
198	C₂×D₇	—	6	1	no	—
199	C₂×D₇	—	12	8	no	—
200	C₄×C₇	—	2	2	no	—
201	C₄×C₇	—	2	1	no	—
202	C₄×C₇	—	2	1	no	—
203	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
204	Dic₇	—	3	2	no	—
205	Dic₇	—	6	1	no	—
206	Dic₇	—	6	2	no	—
207	Dic₇	—	6	1	no	—
208	C₂³×C₃	C₂²×D₇	1	2	yes	—
209	C₂²×D₃	—	7	1	no	—
210	C₂²×D₃	—	7	2	no	—
211	C₂²×D₃	—	7	8	no	—
212	C₂²×D₃	—	28	2	no	—
213	C₂²×D₃	—	28	2	no	—
214	C₃×D₄	C₄×D₇	1	4	yes	complemented
215	C₂×C₃×C₄	—	4	2	no	—
216	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
217	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
220	#67	—	7	8	no	—
221	C₂×Dic₃	—	7	1	no	—
222	C₂×Dic₃	—	7	1	no	—
223	C₂×Dic₃	—	7	1	no	—
224	C₂×Dic₃	—	7	2	no	—
225	C₂×Dic₃	—	28	2	no	—
226	D₁₂	—	7	4	no	—
227	C₄×D₃	—	7	4	no	—
228	C₄×D₃	—	28	2	no	—
229	C₄×D₃	—	28	2	no	—
230	C₃×C₇	#518	1	1	yes	characteristic, complemented, Hall
231	C₂⁴	—	21	2	no	—
232	C₂×D₄	—	21	2	no	—
233	C₂×D₄	—	21	8	no	—
234	C₂×D₄	—	21	1	no	—
235	C₂×D₄	C₂×D₂₁	1	1	yes	characteristic
236	C₂²×C₄	—	21	1	no	—
237	C₂²×C₄	—	21	2	no	—
238	C₂²×C₄	—	21	2	no	—
239	C₂²×C₄	—	21	2	no	—
240	C₂²×C₄	C₂×D₂₁	1	2	yes	complemented
241	C₄⋊C₄	—	21	1	no	—
242	C₄⋊C₄	—	21	1	no	—
243	C₄⋊C₄	—	21	1	no	—
244	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic, complemented
245	C₂²⋊C₄	—	21	2	no	—
246	C₂²⋊C₄	—	21	2	no	—
247	C₂²⋊C₄	—	21	1	no	—
248	C₂²⋊C₄	—	21	1	no	—
249	C₂²⋊C₄	—	21	2	no	—
250	C₂²⋊C₄	—	21	2	no	—
251	C₂²⋊C₄	C₂×D₂₁	1	2	yes	complemented
252	C₄²	—	21	2	no	—
253	C₄²	—	21	1	no	—
254	C₄²	C₂×D₂₁	1	1	yes	characteristic, complemented
255	C₂×C₇	—	2	4	no	—
256	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
257	C₂×C₇	#1004	1	1	yes	characteristic
258	C₂×C₇	#1009	1	1	yes	characteristic
259	D₇	—	6	2	no	—
260	D₇	—	6	4	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	C₂×C₄×D₇	1	4	yes	—
264	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
265	C₂×D₃	—	7	2	no	—
266	C₂×D₃	—	7	4	no	—
267	C₂×D₃	—	14	2	no	—
268	C₂×D₃	—	14	1	no	—
269	C₂×D₃	—	14	2	no	—
270	C₂×D₃	—	14	1	no	—
271	C₂×D₃	—	28	8	no	—
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
276	Dic₃	—	7	2	no	—
277	Dic₃	—	14	1	no	—
278	Dic₃	—	14	2	no	—
279	Dic₃	—	14	1	no	—
280	C₂³	—	21	1	no	—
281	C₂³	—	21	2	no	—
282	C₂³	—	21	8	no	—
283	C₂³	—	84	2	no	—
284	C₂³	—	84	2	no	—
285	C₂³	C₂²×D₂₁	1	2	yes	—
286	D₄	—	21	8	no	—
287	D₄	—	21	4	no	—
288	D₄	C₄×D₂₁	1	4	yes	complemented
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	21	1	no	—
291	C₂×C₄	—	21	4	no	—
292	C₂×C₄	—	21	1	no	—
293	C₂×C₄	—	21	2	no	—
294	C₂×C₄	—	21	1	no	—
295	C₂×C₄	—	84	2	no	—
296	C₂×C₄	—	84	2	no	—
297	C₂×C₄	—	84	2	no	—
298	C₂×C₄	C₂²×D₂₁	1	2	yes	—
299	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
302	C₇	#5545	1	1	yes	characteristic, complemented, Sylow
303	C₂×C₃	—	2	4	no	—
304	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
305	C₂×C₃	#6548	1	1	yes	characteristic
306	C₂×C₃	#6553	1	1	yes	characteristic
307	D₃	—	14	2	no	—
308	D₃	—	14	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	21	2	no	—
312	C₂²	—	21	4	no	—
313	C₂²	—	42	2	no	—
314	C₂²	—	42	1	no	—
315	C₂²	—	42	2	no	—
316	C₂²	—	42	1	no	—
317	C₂²	—	84	8	no	—
318	C₂²	C₂×C₄×D₂₁	1	4	yes	—
319	C₂²	C₂³×D₂₁	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	1	no	—
323	C₄	—	21	2	no	—
324	C₄	—	42	1	no	—
325	C₄	—	42	2	no	—
326	C₄	—	42	1	no	—
327	C₄	C₂×C₄×D₂₁	1	2	yes	—
328	C₃	#90236	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	4	no	—
330	C₂	—	42	2	no	—
331	C₂	—	42	4	no	—
332	C₂	C₂²×C₄×D₂₁	1	1	yes	characteristic
333	C₂	#10621657	1	1	yes	characteristic
334	C₂	#10621662	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator	C₂×C₃×C₇	C₂³×C₄
Fitting	C₃×C₄×C₇×D₄	C₂
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×D₄×D₂₁	C₂
hypercenter	C₄×D₄	D₂₁
hypocenter	C₃×C₇	#518
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇×D₄	C₂	semidirect
3. D₄×Dic₂₁	C₂	semidirect
4. C₄×#4268	C₂	semidirect
5. C₂×#66631	C₂	semidirect
6. #10621376	C₂	semidirect
7. #10621375	C₂	semidirect
8. #10621364	C₂	semidirect
9. C₂²⋊C₄×D₂₁	C₂	semidirect
10. C₄×D₈₄	C₂	semidirect
11. #10621355	C₂	semidirect
15. C₂²×C₃×C₄×C₇	C₂²	semidirect
17. C₂²×Dic₂₁	C₂²	semidirect
18. D₄×D₂₁	C₄	semidirect
21. C₂×C₄×D₂₁	C₂²	semidirect
22. C₂×C₄×D₂₁	C₂²	semidirect
23. C₃×C₇×C₄⋊C₄	C₂²	semidirect
24. C₃×C₇×C₂²⋊C₄	C₂²	semidirect
25. C₃×C₄²×C₇	C₂²	semidirect
26. #66636	C₂²	semidirect
30. #66631	C₂²	semidirect
31. #66631	C₂²	semidirect
32. C₄⋊Dic₂₁	C₂²	semidirect
33. #66629	C₂²	semidirect
34. #66629	C₂²	semidirect
35. C₄×Dic₂₁	C₂²	semidirect
36. C₄×Dic₂₁	C₂²	semidirect
38. C₄×C₇×D₄	D₃	semidirect
52. C₂²×D₂₁	C₂×C₄	semidirect
55. C₃×C₇×D₄	C₂×C₄	semidirect
61. #4268	C₂×C₄	semidirect
67. D₈₄	C₂×C₄	semidirect
70. C₄×D₂₁	C₂×C₄	semidirect
73. C₂²×C₄×C₇	C₂×D₃	semidirect
81. C₇×C₄⋊C₄	C₂×D₃	semidirect
82. C₇×C₂²⋊C₄	C₂×D₃	semidirect
83. C₄²×C₇	C₂×D₃	semidirect
96. C₃×C₄×D₄	D₇	semidirect
132. C₇×D₄	C₄×D₃	semidirect
150. C₂²×C₃×C₄	C₂×D₇	semidirect
158. C₃×C₄⋊C₄	C₂×D₇	semidirect
159. C₃×C₂²⋊C₄	C₂×D₇	semidirect
160. C₃×C₄²	C₂×D₇	semidirect
184. C₄×D₄	D₂₁	semidirect
214. C₃×D₄	C₄×D₇	semidirect
230. C₃×C₇	#518	semidirect
240. C₂²×C₄	C₂×D₂₁	semidirect
244. C₄⋊C₄	C₂×D₂₁	semidirect
251. C₂²⋊C₄	C₂×D₂₁	semidirect
254. C₄²	C₂×D₂₁	semidirect
288. D₄	C₄×D₂₁	semidirect
302. C₇	#5545	semidirect
328. C₃	#90236	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₂₁	2
C₃×C₄×C₇×D₄	2
D₄×Dic₂₁	2
C₄×#4268	2
C₂×#66631	2
#10621376	2
#10621375	2
#10621364	2
C₂²⋊C₄×D₂₁	2
C₄×D₈₄	2
#10621355	2
#90236	3
#5545	7