Subgroups — #13059483

Order 1344.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#66669	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621407	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₇×#67	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621125	C₂	1	2	yes	maximal, complemented
6	D₇×#217	C₂	1	2	yes	maximal, complemented
7	#10621018	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₃×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#10620970	C₂	1	2	yes	maximal, complemented
10	#10620969	C₂	1	2	yes	maximal, complemented
11	#10620926	C₂	1	1	yes	maximal, characteristic, complemented
12	#90241	—	3	1	no	maximal, Hall
13	C₂³×C₃×D₇	C₂²	1	2	yes	—
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₂	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₇×#67	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₃×D₂₈	—	2	4	no	—
19	C₂×C₃×D₂₈	C₂²	1	2	yes	—
20	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
22	C₂×#4245	C₂²	1	2	yes	complemented
23	C₂×#4245	C₂²	1	2	yes	complemented
24	C₂×Dic₃×D₇	C₂²	1	2	yes	complemented
25	C₂×Dic₃×D₇	C₂²	1	2	yes	complemented
26	#66669	—	2	4	no	—
27	#66636	C₂²	1	2	yes	—
28	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic, complemented
29	#66629	C₂²	1	2	yes	complemented
30	C₇×#217	C₂²	1	1	yes	characteristic
31	C₇×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic, complemented
32	C₇×Dic₃⋊C₄	C₂²	1	1	yes	characteristic, complemented
33	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic, complemented
34	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	complemented
35	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	complemented
36	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	complemented
37	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	complemented
38	C₂×#1157	—	3	1	no	—
39	C₂²×D₂₈	D₃	1	1	yes	characteristic, complemented
40	C₄×C₇×D₄	—	3	1	no	—
41	#6500	—	3	2	no	—
42	#6493	—	3	1	no	—
43	#6487	—	3	1	no	—
44	#6457	—	3	2	no	—
45	#6447	—	3	2	no	—
46	D₇×C₂²⋊C₄	—	3	2	no	—
47	#6438	—	3	1	no	—
48	C₄×D₂₈	—	3	1	no	—
49	#5806	—	7	1	no	maximal, Hall
50	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
51	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
52	C₂²×C₃×D₇	—	2	4	no	—
53	C₂²×C₃×D₇	—	2	4	no	—
54	C₂²×C₃×D₇	—	4	2	no	—
55	C₂²×C₃×D₇	C₂³	1	2	yes	—
56	C₂²×C₃×D₇	C₂³	1	2	yes	—
57	C₂×C₃×C₄×C₇	—	2	2	no	—
58	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₁	C₂³	1	2	yes	—
61	C₂×Dic₂₁	C₂³	1	2	yes	—
62	Dic₄₂	—	2	2	no	—
63	C₇×#67	—	2	2	no	—
64	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
65	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
66	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
67	C₄×D₃×C₇	—	2	2	no	—
68	C₃×D₂₈	—	2	4	no	—
69	C₃×D₂₈	—	2	2	no	—
70	C₃×D₂₈	D₄	1	4	yes	complemented
71	#4245	—	2	4	no	—
72	#4245	—	4	2	no	—
73	Dic₃×D₇	—	2	4	no	—
74	Dic₃×D₇	—	4	2	no	—
75	C₂³×D₇	C₂×D₃	1	2	yes	—
76	C₂×C₇×D₄	—	3	1	no	—
77	C₂²×C₄×C₇	—	3	1	no	—
78	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
79	C₂×C₂²⋊D₇	—	3	2	no	—
80	C₂×C₂²⋊D₇	—	3	2	no	—
81	#1157	—	6	4	no	—
82	C₂×D₂₈	—	2	4	no	—
83	C₂×D₂₈	C₂×D₃	1	1	yes	characteristic
84	C₂×D₂₈	C₂×D₃	1	2	yes	—
85	C₂×D₂₈	C₂×D₃	1	1	yes	characteristic
86	C₂×C₄×D₇	—	3	2	no	—
87	C₂×C₄×D₇	—	3	2	no	—
88	C₂×Dic₁₄	—	3	1	no	—
89	C₇×C₄⋊C₄	—	3	1	no	—
90	C₇×C₂²⋊C₄	—	3	1	no	—
91	C₇×C₂²⋊C₄	—	3	1	no	—
92	C₄²×C₇	—	3	1	no	—
93	C₂²⋊Dic₇	—	3	2	no	—
94	(C₂×D₇)⋊C₄	—	3	2	no	—
95	(C₂×D₇)⋊C₄	—	3	2	no	—
96	(C₂×D₇)⋊C₄	—	3	2	no	—
97	(C₂×D₇)⋊C₄	—	3	2	no	—
98	C₄⋊Dic₇	—	3	1	no	—
99	Dic₇⋊C₄	—	3	2	no	—
100	C₂²×C₃×D₄	—	7	1	no	—
101	C₂×#237	—	7	1	no	—
102	#953	—	7	2	no	—
103	C₂×#217	—	7	2	no	—
104	#939	—	7	2	no	—
105	#938	—	7	1	no	—
106	#936	—	7	1	no	—
107	#935	—	7	2	no	—
108	D₄×Dic₃	—	7	1	no	—
109	C₄×#67	D₇	1	1	yes	characteristic, complemented
110	#924	—	7	1	no	—
111	C₂²×C₃×C₇	—	2	1	no	—
112	C₂²×C₃×C₇	—	2	1	no	—
113	C₂²×C₃×C₇	#41	1	2	yes	complemented
114	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
115	C₂×D₃×C₇	—	2	1	no	—
116	C₂×D₃×C₇	—	2	1	no	—
117	C₂×D₃×C₇	—	2	1	no	—
118	C₂×C₃×D₇	—	2	2	no	—
119	C₂×C₃×D₇	—	2	2	no	—
120	C₂×C₃×D₇	—	2	2	no	—
121	C₂×C₃×D₇	—	2	2	no	—
122	C₂×C₃×D₇	—	2	2	no	—
123	C₂×C₃×D₇	—	4	8	no	—
124	C₂×C₃×D₇	C₂×D₄	1	4	yes	—
125	C₃×C₄×C₇	—	2	1	no	—
126	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
127	Dic₂₁	—	2	2	no	—
128	Dic₂₁	—	2	2	no	—
129	C₇×Dic₃	—	2	1	no	—
130	C₇×Dic₃	—	2	1	no	—
131	C₇×Dic₃	—	2	1	no	—
132	#546	—	21	1	no	Sylow
133	C₂³×C₇	—	3	1	no	—
134	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
135	C₂²×D₇	—	2	4	no	—
136	C₂²×D₇	—	2	4	no	—
137	C₂²×D₇	—	4	2	no	—
138	C₂²×D₇	C₂²×D₃	1	2	yes	—
139	C₂²×D₇	C₂²×D₃	1	2	yes	—
140	C₇×D₄	—	6	2	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	3	1	no	—
143	C₂×C₄×C₇	—	3	1	no	—
144	C₂×C₄×C₇	—	3	1	no	—
145	C₂×C₄×C₇	—	6	2	no	—
146	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
148	C₂²⋊D₇	—	6	4	no	—
149	C₂²⋊D₇	—	12	2	no	—
150	C₂×Dic₇	—	3	2	no	—
151	C₂×Dic₇	—	3	2	no	—
152	D₂₈	—	2	4	no	—
153	D₂₈	—	2	2	no	—
154	D₂₈	#67	1	4	yes	complemented
155	C₄×D₇	—	6	4	no	—
156	C₄×D₇	—	12	2	no	—
157	Dic₁₄	—	6	2	no	—
158	C₂⁴×C₃	—	7	2	no	—
159	C₂×C₃×D₄	—	7	1	no	—
160	C₂×C₃×D₄	—	7	2	no	—
161	C₂×C₃×D₄	—	7	1	no	—
162	C₂×C₃×D₄	—	14	4	no	—
163	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
164	C₂×#67	—	7	2	no	—
165	C₂×#67	—	7	2	no	—
166	C₂×#67	C₂×D₇	1	1	yes	characteristic
167	C₂²×Dic₃	—	7	2	no	—
168	C₂²×Dic₃	—	7	2	no	—
169	#237	—	14	4	no	—
170	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic, complemented
171	C₂×Dic₆	—	7	1	no	—
172	#217	—	7	2	no	—
173	#217	—	7	2	no	—
174	#217	—	7	2	no	—
175	#217	—	7	2	no	—
176	#217	—	7	2	no	—
177	#217	C₂×D₇	1	1	yes	characteristic
178	(C₂×D₃)⋊C₄	C₂×D₇	1	1	yes	characteristic, complemented
179	C₄⋊Dic₃	—	7	1	no	—
180	Dic₃⋊C₄	—	7	2	no	—
181	Dic₃⋊C₄	C₂×D₇	1	1	yes	characteristic, complemented
182	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic, complemented
183	C₂×C₃×C₇	—	2	2	no	—
184	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
186	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
187	C₃×D₇	—	2	4	no	—
188	C₃×D₇	—	4	2	no	—
189	D₃×C₇	—	4	1	no	—
190	C₂×#41	—	21	1	no	—
191	C₂²×D₄	—	7	1	no	—
192	#124	—	21	1	no	—
193	#123	—	21	2	no	—
194	C₂²⋊Q₈	—	21	1	no	—
195	#121	—	21	2	no	—
196	#121	—	21	1	no	—
197	#120	—	21	2	no	—
198	C₄×D₄	—	3	1	no	—
199	C₄×D₄	—	21	1	no	—
200	C₂×C₂²⋊C₄	—	21	2	no	—
201	C₂²×C₇	—	2	1	no	—
202	C₂²×C₇	—	2	1	no	—
203	C₂²×C₇	—	6	1	no	—
204	C₂²×C₇	—	6	1	no	—
205	C₂²×C₇	—	6	1	no	—
206	C₂²×C₇	#237	1	2	yes	complemented
207	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
208	C₂×D₇	—	2	2	no	—
209	C₂×D₇	—	2	2	no	—
210	C₂×D₇	—	2	2	no	—
211	C₂×D₇	—	2	2	no	—
212	C₂×D₇	—	2	2	no	—
213	C₂×D₇	—	4	8	no	—
214	C₂×D₇	C₂×#67	1	4	yes	—
215	C₄×C₇	—	2	1	no	—
216	C₄×C₇	—	6	1	no	—
217	C₄×C₇	—	6	1	no	—
218	C₄×C₇	—	6	1	no	—
219	C₄×C₇	C₂×#67	1	2	yes	—
220	Dic₇	—	6	2	no	—
221	Dic₇	—	6	2	no	—
222	C₂³×C₃	—	7	2	no	—
223	C₂³×C₃	—	7	2	no	—
224	C₂³×C₃	—	14	4	no	—
225	C₂³×C₃	—	14	4	no	—
226	C₂³×C₃	—	28	2	no	—
227	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
228	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
229	C₃×D₄	—	7	4	no	—
230	C₃×D₄	—	14	4	no	—
231	C₃×D₄	—	14	2	no	—
232	C₂×C₃×C₄	—	2	2	no	—
233	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
235	#67	—	2	2	no	—
236	#67	—	14	4	no	—
237	#67	—	28	2	no	—
238	C₂×Dic₃	—	7	2	no	—
239	C₂×Dic₃	—	7	2	no	—
240	C₂×Dic₃	—	14	4	no	—
241	C₂×Dic₃	—	28	2	no	—
242	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
243	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
244	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
245	C₄×D₃	—	2	2	no	—
246	Dic₆	—	14	2	no	—
247	C₃×C₇	#546	1	1	yes	characteristic, complemented, Hall
248	C₂⁴	—	7	2	no	—
249	#41	—	42	4	no	—
250	C₂×Q₈	—	21	1	no	—
251	C₂×D₄	—	3	1	no	—
252	C₂×D₄	—	7	1	no	—
253	C₂×D₄	—	7	2	no	—
254	C₂×D₄	—	7	1	no	—
255	C₂×D₄	—	14	4	no	—
256	C₂×D₄	—	21	2	no	—
257	C₂×D₄	—	21	2	no	—
258	C₂²×C₄	—	3	1	no	—
259	C₂²×C₄	—	21	2	no	—
260	C₂²×C₄	—	21	2	no	—
261	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
262	C₄⋊C₄	—	3	1	no	—
263	C₄⋊C₄	—	21	2	no	—
264	C₄⋊C₄	—	21	1	no	—
265	C₂²⋊C₄	—	3	1	no	—
266	C₂²⋊C₄	—	3	1	no	—
267	C₂²⋊C₄	—	21	2	no	—
268	C₂²⋊C₄	—	21	2	no	—
269	C₂²⋊C₄	—	21	2	no	—
270	C₂²⋊C₄	—	21	2	no	—
271	C₂²⋊C₄	—	21	2	no	—
272	C₄²	—	3	1	no	—
273	C₂×C₇	—	2	2	no	—
274	C₂×C₇	—	12	1	no	—
275	C₂×C₇	C₂×#237	1	1	yes	characteristic
276	C₂×C₇	#1004	1	1	yes	characteristic
277	C₂×C₇	C₂²×#67	1	1	yes	characteristic
278	D₇	—	2	4	no	—
279	D₇	—	4	2	no	—
280	C₂²×C₃	—	2	1	no	—
281	C₂²×C₃	—	2	1	no	—
282	C₂²×C₃	—	7	4	no	—
283	C₂²×C₃	—	14	2	no	—
284	C₂²×C₃	—	14	2	no	—
285	C₂²×C₃	—	14	2	no	—
286	C₂²×C₃	—	14	2	no	—
287	C₂²×C₃	—	14	2	no	—
288	C₂²×C₃	—	28	8	no	—
289	C₂²×C₃	#1157	1	2	yes	complemented
290	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
291	C₂×D₃	—	2	1	no	—
292	C₂×D₃	—	2	1	no	—
293	C₂×D₃	—	2	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	D₄×D₇	1	2	yes	—
296	Dic₃	—	2	1	no	—
297	Dic₃	—	2	1	no	—
298	Dic₃	—	2	1	no	—
299	Dic₃	—	14	2	no	—
300	Dic₃	—	14	2	no	—
301	C₂³	—	3	1	no	—
302	C₂³	—	7	2	no	—
303	C₂³	—	7	2	no	—
304	C₂³	—	14	4	no	—
305	C₂³	—	14	4	no	—
306	C₂³	—	28	2	no	—
307	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
308	Q₈	—	42	2	no	—
309	D₄	—	6	2	no	—
310	D₄	—	7	4	no	—
311	D₄	—	14	4	no	—
312	D₄	—	14	2	no	—
313	D₄	—	42	4	no	—
314	D₄	—	84	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	3	1	no	—
317	C₂×C₄	—	3	1	no	—
318	C₂×C₄	—	3	1	no	—
319	C₂×C₄	—	6	2	no	—
320	C₂×C₄	—	21	2	no	—
321	C₂×C₄	—	21	2	no	—
322	C₂×C₄	—	42	4	no	—
323	C₂×C₄	—	84	2	no	—
324	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
325	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
326	C₇	#5806	1	1	yes	characteristic, complemented, Sylow
327	C₂×C₃	—	2	2	no	—
328	C₂×C₃	—	14	4	no	—
329	C₂×C₃	—	28	2	no	—
330	C₂×C₃	C₂×#1157	1	1	yes	characteristic
331	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
332	C₂×C₃	#6553	1	1	yes	characteristic
333	D₃	—	4	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	6	1	no	—
337	C₂²	—	6	1	no	—
338	C₂²	—	6	1	no	—
339	C₂²	—	7	4	no	—
340	C₂²	—	14	2	no	—
341	C₂²	—	14	2	no	—
342	C₂²	—	14	2	no	—
343	C₂²	—	14	2	no	—
344	C₂²	—	14	2	no	—
345	C₂²	—	28	8	no	—
346	C₂²	#66669	1	2	yes	complemented
347	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	6	1	no	—
350	C₄	—	6	1	no	—
351	C₄	—	6	1	no	—
352	C₄	—	42	2	no	—
353	C₄	—	42	2	no	—
354	C₄	D₇×#67	1	2	yes	—
355	C₃	#90241	1	1	yes	characteristic, complemented, Sylow
356	C₂	—	2	2	no	—
357	C₂	—	12	1	no	—
358	C₂	—	14	4	no	—
359	C₂	—	28	2	no	—
360	C₂	C₂×#66669	1	1	yes	characteristic
361	C₂	#10621564	1	1	yes	characteristic
362	C₂	C₂×D₇×#67	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₂²×C₃×C₄×C₇	C₂²
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#546
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₂₈	C₂	semidirect
2. C₂×#66669	C₂	semidirect
3. #10621407	C₂	semidirect
4. C₄×C₇×#67	C₂	semidirect
5. #10621125	C₂	semidirect
6. D₇×#217	C₂	semidirect
7. #10621018	C₂	semidirect
8. Dic₃×D₂₈	C₂	semidirect
9. #10620970	C₂	semidirect
10. #10620969	C₂	semidirect
11. #10620926	C₂	semidirect
15. C₂×Dic₄₂	C₂²	semidirect
17. C₂×C₄×D₃×C₇	C₂²	semidirect
22. C₂×#4245	C₂²	semidirect
23. C₂×#4245	C₂²	semidirect
24. C₂×Dic₃×D₇	C₂²	semidirect
25. C₂×Dic₃×D₇	C₂²	semidirect
28. C₄⋊Dic₂₁	C₂²	semidirect
29. #66629	C₂²	semidirect
31. C₇×(C₂×D₃)⋊C₄	C₂²	semidirect
32. C₇×Dic₃⋊C₄	C₂²	semidirect
33. C₄×C₇×Dic₃	C₂²	semidirect
34. (C₂×D₇)⋊Dic₃	C₂²	semidirect
35. (C₂×D₇)⋊Dic₃	C₂²	semidirect
36. (C₂×D₇)⋊Dic₃	C₂²	semidirect
37. (C₂×D₇)⋊Dic₃	C₂²	semidirect
39. C₂²×D₂₈	D₃	semidirect
70. C₃×D₂₈	D₄	semidirect
109. C₄×#67	D₇	semidirect
113. C₂²×C₃×C₇	#41	semidirect
154. D₂₈	#67	semidirect
170. C₂×C₄×D₃	C₂×D₇	semidirect
178. (C₂×D₃)⋊C₄	C₂×D₇	semidirect
181. Dic₃⋊C₄	C₂×D₇	semidirect
182. C₄×Dic₃	C₂×D₇	semidirect
206. C₂²×C₇	#237	semidirect
247. C₃×C₇	#546	semidirect
289. C₂²×C₃	#1157	semidirect
326. C₇	#5806	semidirect
346. C₂²	#66669	semidirect
355. C₃	#90241	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₂₈	2
C₂×#66669	2
#10621407	2
C₄×C₇×#67	2
#10621125	2
D₇×#217	2
#10621018	2
Dic₃×D₂₈	2
#10620970	2
#10620969	2
#10620926	2
#90241	3
#5806	7