Subgroups — #13372242

Order 1792.

858 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89666	C₂	1	2	yes	maximal
2	#11736494	C₂	1	2	yes	maximal, complemented
3	C₄×#6467	C₂	1	1	yes	maximal, characteristic
4	C₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	#11728476	C₂	1	4	yes	maximal, complemented
6	#11728459	C₂	1	2	yes	maximal, complemented
7	#11728283	C₂	1	2	yes	maximal, complemented
8	#11727511	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#6470	C₂²	1	4	yes	—
10	C₂×#6467	C₂²	1	1	yes	characteristic
11	C₄×#1157	C₂²	1	2	yes	—
12	C₂×C₇×#108	C₂²	1	2	yes	—
13	C₂×C₇×#108	C₂²	1	2	yes	—
14	C₇×#423	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×#6399	C₂²	1	4	yes	—
17	#89891	C₂²	1	4	yes	—
18	#89885	C₂²	1	2	yes	—
19	#89878	C₂²	1	1	yes	characteristic
20	C₂×#6389	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
23	#89666	—	2	8	no	—
24	#89611	—	2	4	no	—
25	#89609	—	2	2	no	—
26	D₇×C₄⋊C₈	—	2	2	no	—
27	C₄×Dic₂₈	C₄	1	1	yes	characteristic, complemented
28	C₄×Dic₂₈	C₄	1	1	yes	characteristic, complemented
29	C₄×D₅₆	C₄	1	1	yes	characteristic, complemented
30	C₄×D₅₆	C₄	1	1	yes	characteristic, complemented
31	C₄×#1132	C₄	1	2	yes	complemented
32	C₄×#1132	C₄	1	2	yes	complemented
33	#89350	C₂²	1	4	yes	—
34	#89345	C₂²	1	2	yes	—
35	#22686	—	7	1	no	maximal, Sylow
36	C₂×#1157	C₂³	1	2	yes	—
37	C₂×C₇×M₁₆	C₂³	1	4	yes	—
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₄×C₂²⋊D₇	—	2	2	no	—
42	C₄×C₂²⋊D₇	—	2	2	no	—
43	#6489	C₂³	1	2	yes	—
44	C₂×C₄⋊Dic₇	—	2	1	no	—
45	C₂×C₄×Dic₇	C₂³	1	2	yes	—
46	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
47	#6470	—	2	8	no	—
48	#6470	—	4	4	no	—
49	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
50	#6467	D₄	1	4	yes	—
51	#6467	Q₈	1	4	yes	—
52	C₂×D₅₆	C₂×C₄	1	1	yes	characteristic
53	C₂×#1132	C₂×C₄	1	2	yes	—
54	C₂×#1131	—	2	4	no	—
55	C₂×C₈×D₇	—	2	4	no	—
56	C₄×D₂₈	C₂×C₄	1	2	yes	—
57	C₄×D₂₈	C₂×C₄	1	2	yes	—
58	#6435	—	2	2	no	—
59	C₄²×D₇	—	2	2	no	—
60	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
62	C₇×#108	—	2	4	no	—
63	C₇×#108	—	4	2	no	—
64	C₇×C₄⋊C₈	—	2	2	no	—
65	C₇×C₄⋊C₈	—	2	2	no	—
66	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
70	C₇⋊#95	—	2	1	no	—
71	#6399	—	2	8	no	—
72	#6395	C₂×C₄	1	2	yes	—
73	#6395	C₂×C₄	1	2	yes	—
74	#6392	C₂×C₄	1	1	yes	characteristic
75	#6392	C₂×C₄	1	1	yes	characteristic
76	#6391	C₂×C₄	1	2	yes	—
77	#6390	C₂×C₄	1	2	yes	—
78	#6390	C₂×C₄	1	2	yes	—
79	#6389	—	2	4	no	—
80	Dic₇⋊C₈	—	2	4	no	—
81	Dic₇⋊C₈	—	2	4	no	—
82	C₄⋊(C₇⋊C₈)	D₄	1	2	yes	—
83	C₄⋊(C₇⋊C₈)	D₄	1	2	yes	—
84	C₂×#444	—	7	2	no	—
85	C₄×#135	—	7	1	no	—
86	#2906	—	7	2	no	—
87	#1946	—	7	2	no	—
88	#1866	—	7	4	no	—
89	#1848	D₇	1	1	yes	characteristic, complemented
90	#1806	—	7	2	no	—
91	#1770	—	7	1	no	—
92	C₂²×C₄×C₇	#41	1	1	yes	characteristic
93	C₂²×C₄×C₇	#41	1	1	yes	characteristic
94	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
95	C₂×C₂²⋊D₇	—	2	2	no	—
96	C₂²×Dic₇	—	2	2	no	—
97	#1157	—	2	2	no	—
98	#1157	—	2	2	no	—
99	#1157	C₂×D₄	1	4	yes	—
100	#1157	C₂×Q₈	1	4	yes	—
101	C₂×D₂₈	C₂²×C₄	1	2	yes	—
102	C₂×C₄×D₇	—	2	2	no	—
103	C₂×C₄×D₇	—	2	2	no	—
104	C₂×C₄×D₇	—	2	2	no	—
105	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
106	C₇×M₁₆	—	2	4	no	—
107	C₇×M₁₆	—	2	4	no	—
108	C₂×C₇×C₈	—	2	1	no	—
109	C₂×C₇×C₈	—	2	1	no	—
110	C₂×C₇×C₈	—	2	4	no	—
111	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
117	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
121	C₂²⋊Dic₇	—	2	2	no	—
122	(C₂×D₇)⋊C₄	—	2	2	no	—
123	(C₂×D₇)⋊C₄	—	2	2	no	—
124	C₄⋊Dic₇	—	4	2	no	—
125	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
126	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
127	Dic₇⋊C₄	—	2	2	no	—
128	Dic₇⋊C₄	—	2	2	no	—
129	C₄×Dic₇	—	2	2	no	—
130	C₄×Dic₇	—	2	2	no	—
131	C₄×Dic₇	—	2	2	no	—
132	C₄×Dic₇	—	2	2	no	—
133	C₄×Dic₇	—	2	2	no	—
134	C₂×C₇⋊C₈	—	2	2	no	—
135	C₂×C₇⋊C₈	—	2	2	no	—
136	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
137	Dic₂₈	C₄⋊C₄	1	4	yes	complemented
138	D₅₆	C₄⋊C₄	1	4	yes	complemented
139	#1132	C₄⋊C₄	1	8	yes	complemented
140	#1131	—	4	8	no	—
141	C₈×D₇	—	4	8	no	—
142	C₂×#135	—	7	1	no	—
143	C₂×#131	—	7	4	no	—
144	C₄×#41	—	7	2	no	—
145	#444	—	14	8	no	—
146	C₄×Q₁₆	—	7	1	no	—
147	C₄×Q₁₆	—	7	1	no	—
148	C₄×SD₁₆	—	7	2	no	—
149	C₄×SD₁₆	—	7	2	no	—
150	C₄×D₈	—	7	1	no	—
151	C₄×D₈	—	7	1	no	—
152	C₂×#108	C₂×D₇	1	2	yes	—
153	C₂×#108	C₂×D₇	1	2	yes	—
154	#427	—	7	1	no	—
155	#424	—	7	4	no	—
156	#424	—	14	4	no	—
157	#423	—	14	2	no	—
158	#423	C₂×D₇	1	2	yes	—
159	C₂×C₄⋊C₈	—	7	1	no	—
160	C₂×C₄⋊C₈	—	14	2	no	—
161	C₂×#104	—	7	4	no	—
162	#416	—	7	2	no	—
163	C₂×#97	—	7	2	no	—
164	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
165	C₄×C₄⋊C₄	—	7	1	no	—
166	#339	—	7	4	no	—
167	#336	—	7	2	no	—
168	C₂³×C₇	C₂×#41	1	1	yes	characteristic
169	C₂²×D₇	—	2	2	no	—
170	C₂×C₄×C₇	—	2	2	no	—
171	C₂×C₄×C₇	—	4	1	no	—
172	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
182	C₂²⋊D₇	—	2	4	no	—
183	C₂²⋊D₇	—	2	4	no	—
184	C₂×Dic₇	—	2	2	no	—
185	C₂×Dic₇	—	2	2	no	—
186	C₂×Dic₇	—	2	2	no	—
187	C₂×Dic₇	—	4	2	no	—
188	C₂×Dic₇	—	4	2	no	—
189	C₂×Dic₇	—	4	2	no	—
190	D₂₈	—	2	2	no	—
191	D₂₈	C₂×C₄⋊C₄	1	4	yes	—
192	C₄×D₇	—	2	4	no	—
193	C₄×D₇	—	2	4	no	—
194	C₄×D₇	—	4	4	no	—
195	C₄×D₇	—	8	2	no	—
196	Dic₁₄	—	2	2	no	—
197	Dic₁₄	C₂×C₄⋊C₄	1	4	yes	—
198	C₇×C₈	—	2	1	no	—
199	C₇×C₈	—	2	1	no	—
200	C₇×C₈	—	4	4	no	—
201	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
202	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
203	C₇⋊C₈	—	2	4	no	—
204	C₂×#41	—	7	2	no	—
205	#135	—	7	4	no	—
206	#135	—	7	4	no	—
207	C₂×Q₁₆	—	7	1	no	—
208	C₂×SD₁₆	—	7	2	no	—
209	C₂×D₈	—	7	1	no	—
210	#131	—	14	8	no	—
211	#131	—	28	4	no	—
212	C₂×M₁₆	—	14	4	no	—
213	C₂×M₁₆	C₂²×D₇	1	4	yes	—
214	C₂²×C₈	—	7	2	no	—
215	C₂²×C₈	—	14	4	no	—
216	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
218	C₄×Q₈	—	7	2	no	—
219	C₄×Q₈	—	7	2	no	—
220	C₄×D₄	—	7	2	no	—
221	C₄×D₄	—	7	2	no	—
222	C₄×D₄	—	14	2	no	—
223	C₄×D₄	—	14	2	no	—
224	#117	—	7	2	no	—
225	#117	—	14	2	no	—
226	C₂×C₄⋊C₄	—	14	1	no	—
227	C₂×C₄²	—	7	2	no	—
228	C₂×C₄²	—	14	2	no	—
229	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
230	#108	—	2	4	no	—
231	#108	—	4	2	no	—
232	#107	—	7	1	no	—
233	#107	—	7	1	no	—
234	#106	—	7	2	no	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	—	7	2	no	—
238	C₄⋊C₈	—	7	2	no	—
239	C₄⋊C₈	—	14	4	no	—
240	C₄⋊C₈	—	14	4	no	—
241	#104	—	14	8	no	—
242	#103	—	7	2	no	—
243	#103	—	7	2	no	—
244	#102	—	7	2	no	—
245	#102	—	7	2	no	—
246	#97	—	14	4	no	—
247	C₄×C₈	C₄×D₇	1	1	yes	characteristic
248	C₄×C₈	C₄×D₇	1	1	yes	characteristic
249	C₄×C₈	C₄×D₇	1	1	yes	characteristic
250	C₄×C₈	C₄×D₇	1	1	yes	characteristic
251	#95	—	14	1	no	—
252	C₂²×C₇	—	2	1	no	—
253	C₂²×C₇	—	2	1	no	—
254	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂²×C₇	#547	1	1	yes	characteristic
256	C₂²×C₇	#554	1	1	yes	characteristic
257	C₂×D₇	—	2	4	no	—
258	C₂×D₇	—	4	2	no	—
259	C₂×D₇	—	4	2	no	—
260	C₄×C₇	—	2	1	no	—
261	C₄×C₇	—	2	1	no	—
262	C₄×C₇	—	2	1	no	—
263	C₄×C₇	#444	1	2	yes	complemented
264	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
265	C₄×C₇	#520	1	1	yes	characteristic
266	C₄×C₇	D₄²	1	1	yes	characteristic
267	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
268	Dic₇	—	2	4	no	—
269	Dic₇	—	4	2	no	—
270	Dic₇	—	4	2	no	—
271	Dic₇	—	8	2	no	—
272	#41	—	7	4	no	—
273	#41	—	7	4	no	—
274	#41	—	14	2	no	—
275	#41	—	14	2	no	—
276	C₂×Q₈	—	7	2	no	—
277	C₂×D₄	—	7	2	no	—
278	C₂×D₄	—	14	2	no	—
279	C₂²×C₄	—	14	2	no	—
280	C₂²×C₄	—	14	2	no	—
281	C₂²×C₄	—	14	2	no	—
282	C₂²×C₄	—	14	2	no	—
283	C₂²×C₄	#1161	1	1	yes	characteristic
284	C₂²×C₄	#1161	1	1	yes	characteristic
285	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
286	Q₁₆	—	7	4	no	—
287	SD₁₆	—	7	8	no	—
288	D₈	—	7	4	no	—
289	M₁₆	—	2	4	no	—
290	M₁₆	—	2	4	no	—
291	M₁₆	—	28	8	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	4	no	—
295	C₂×C₈	—	7	4	no	—
296	C₂×C₈	—	14	2	no	—
297	C₂×C₈	—	14	2	no	—
298	C₂×C₈	—	28	8	no	—
299	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
300	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
303	C₂×C₈	D₄×D₇	1	2	yes	—
304	C₂×C₈	Q₈×D₇	1	2	yes	—
305	C₄⋊C₄	—	7	2	no	—
306	C₄⋊C₄	—	7	2	no	—
307	C₄⋊C₄	—	14	2	no	—
308	C₄⋊C₄	—	14	2	no	—
309	C₄⋊C₄	—	28	2	no	—
310	C₂²⋊C₄	—	14	2	no	—
311	C₂²⋊C₄	—	14	2	no	—
312	C₂²⋊C₄	—	14	2	no	—
313	C₄²	—	14	2	no	—
314	C₄²	—	14	2	no	—
315	C₄²	—	14	2	no	—
316	C₄²	—	14	2	no	—
317	C₄²	—	14	2	no	—
318	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
319	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
320	C₄²	D₄×D₇	1	1	yes	characteristic
321	C₄²	D₄×D₇	1	1	yes	characteristic
322	C₂×C₇	—	2	1	no	—
323	C₂×C₇	—	2	1	no	—
324	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
325	C₂×C₇	C₂×#444	1	1	yes	characteristic
326	C₂×C₇	#2957	1	1	yes	characteristic
327	D₇	—	4	4	no	—
328	C₂³	—	14	2	no	—
329	C₂³	C₂×#1161	1	1	yes	characteristic
330	Q₈	—	7	4	no	—
331	Q₈	—	14	2	no	—
332	D₄	—	7	4	no	—
333	D₄	—	14	4	no	—
334	D₄	—	14	4	no	—
335	D₄	—	14	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	14	2	no	—
339	C₂×C₄	—	14	4	no	—
340	C₂×C₄	—	14	4	no	—
341	C₂×C₄	—	14	2	no	—
342	C₂×C₄	—	14	2	no	—
343	C₂×C₄	—	28	4	no	—
344	C₂×C₄	—	28	2	no	—
345	C₂×C₄	—	28	2	no	—
346	C₂×C₄	—	28	2	no	—
347	C₂×C₄	—	56	2	no	—
348	C₂×C₄	#6456	1	1	yes	characteristic
349	C₂×C₄	#6456	1	1	yes	characteristic
350	C₂×C₄	#6456	1	1	yes	characteristic
351	C₂×C₄	#6456	1	1	yes	characteristic
352	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
355	C₂×C₄	#6548	1	1	yes	characteristic
356	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
357	C₂×C₄	#6551	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	4	4	no	—
361	C₈	—	14	4	no	—
362	C₈	D₇×C₄⋊C₄	1	2	yes	—
363	C₈	D₇×C₄⋊C₄	1	2	yes	—
364	C₇	#22686	1	1	yes	characteristic, complemented, Sylow
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	14	4	no	—
368	C₂²	—	28	2	no	—
369	C₂²	—	28	2	no	—
370	C₂²	C₂×#6456	1	1	yes	characteristic
371	C₂²	#90417	1	1	yes	characteristic
372	C₂²	#90421	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	14	4	no	—
377	C₄	—	28	2	no	—
378	C₄	—	28	2	no	—
379	C₄	—	56	2	no	—
380	C₄	#89666	1	2	yes	complemented
381	C₄	C₂×D₇×C₄⋊C₄	1	1	yes	characteristic
382	C₄	#90195	1	1	yes	characteristic
383	C₄	#90408	1	1	yes	characteristic
384	C₄	#90418	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	28	4	no	—
388	C₂	#11731805	1	1	yes	characteristic
389	C₂	C₂×#89666	1	1	yes	characteristic
390	C₂	#11736694	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₇×#1848	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂×#6467	C₂²
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	#1848	D₇
hypocenter	C₇	#22686
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. #11736494	C₂	semidirect
4. C₇×#1848	C₂	semidirect
5. #11728476	C₂	semidirect
6. #11728459	C₂	semidirect
7. #11728283	C₂	semidirect
8. #11727511	C₂	semidirect
27. C₄×Dic₂₈	C₄	semidirect
28. C₄×Dic₂₈	C₄	semidirect
29. C₄×D₅₆	C₄	semidirect
30. C₄×D₅₆	C₄	semidirect
31. C₄×#1132	C₄	semidirect
32. C₄×#1132	C₄	semidirect
89. #1848	D₇	semidirect
137. Dic₂₈	C₄⋊C₄	semidirect
138. D₅₆	C₄⋊C₄	semidirect
139. #1132	C₄⋊C₄	semidirect
263. C₄×C₇	#444	semidirect
364. C₇	#22686	semidirect
380. C₄	#89666	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89666	2
#11736494	2
C₄×#6467	2
C₇×#1848	2
#11728476	2
#11728459	2
#11728283	2
#11727511	2
#22686	7