Subgroups — #137×#6923 #21462575

Order 7808.

484 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#6923	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#6923	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840802	C₂	1	2	yes	maximal, complemented
4	#16840801	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#6923	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#6923	C₂	1	2	yes	maximal, complemented
7	#16840773	C₂	1	1	yes	maximal, characteristic, complemented
8	#16840755	C₂	1	1	yes	maximal, characteristic, complemented
9	#16840753	C₂	1	1	yes	maximal, characteristic
10	M₁₆×#6923	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₆₁	C₂	1	1	yes	maximal, characteristic
12	D₆₁⋊M₁₆	C₂²	1	1	yes	characteristic
13	Q₈×#6923	—	2	1	no	—
14	Q₈×#6923	—	2	1	no	—
15	Q₈×#6923	C₂²	1	2	yes	—
16	Q₈×#6923	C₂²	1	1	yes	characteristic
17	Q₈×#6923	C₂²	1	2	yes	—
18	Q₈×#6923	C₂²	1	1	yes	characteristic
19	D₄×#6923	—	2	1	no	—
20	D₄×#6923	—	2	1	no	—
21	D₄×#6923	C₂²	1	1	yes	characteristic
22	D₄×#6923	C₂²	1	1	yes	characteristic
23	#14519856	—	2	1	no	—
24	#14519856	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#6923	—	2	1	no	—
26	C₂×C₄⋊#6923	C₂²	1	1	yes	characteristic
27	C₂×C₄×#6923	—	2	1	no	—
28	C₂×C₄×#6923	—	2	1	no	—
29	C₂×C₄×#6923	C₂²	1	1	yes	characteristic
30	#14519846	C₂²	1	2	yes	—
31	#14519845	C₂²	1	2	yes	—
32	#14519840	C₂²	1	2	yes	—
33	#14519840	C₂²	1	2	yes	—
34	#14519840	C₂²	1	2	yes	—
35	#14519839	C₂²	1	2	yes	—
36	#14519831	C₂²	1	2	yes	—
37	C₈×#6923	C₂²	1	2	yes	—
38	#41×D₆₁	C₂²	1	1	yes	characteristic
39	C₂×Q₈×D₆₁	C₂²	1	1	yes	characteristic
40	#14519810	C₄	1	1	yes	characteristic, complemented
41	#14519809	C₄	1	2	yes	complemented
42	#14519808	C₄	1	1	yes	characteristic, complemented
43	#14519807	C₄	1	1	yes	characteristic, complemented
44	#14519806	C₄	1	2	yes	complemented
45	Q₁₆×D₆₁	C₂²	1	2	yes	—
46	SD₁₆×D₆₁	C₂²	1	2	yes	—
47	M₁₆×D₆₁	C₂²	1	1	yes	characteristic
48	#137×C₆₁	C₄	1	1	yes	characteristic, complemented
49	C₂²×#6923	—	2	1	no	—
50	C₂²×#6923	—	2	1	no	—
51	C₂²×#6923	—	2	1	no	—
52	C₂²×#6923	D₄	1	1	yes	characteristic
53	C₂²×#6923	D₄	1	1	yes	characteristic
54	#41×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
56	#11758226	C₂×C₄	1	1	yes	characteristic
57	Q₈×D₆₁	—	2	1	no	—
58	Q₈×D₆₁	—	2	1	no	—
59	Q₈×D₆₁	C₂×C₄	1	2	yes	complemented
60	Q₈×D₆₁	C₂³	1	2	yes	—
61	Q₈×D₆₁	C₂³	1	1	yes	characteristic
62	#11758224	—	2	1	no	—
63	#11758224	C₂×C₄	1	1	yes	characteristic
64	D₄×D₆₁	—	2	1	no	—
65	D₄×D₆₁	C₂³	1	1	yes	characteristic
66	#11758222	—	2	1	no	—
67	#11758222	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₆₁	—	2	1	no	—
69	C₂×C₄×D₆₁	—	2	1	no	—
70	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
71	C₂×Dic₁₂₂	—	2	1	no	—
72	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂²⋊#6923	—	2	1	no	—
74	C₂²⋊#6923	—	2	1	no	—
75	C₂²⋊#6923	—	2	1	no	—
76	#11758217	—	2	1	no	—
77	C₄⋊#6923	—	2	2	no	—
78	C₄⋊#6923	—	2	2	no	—
79	C₄⋊#6923	—	2	1	no	—
80	C₄⋊#6923	—	2	1	no	—
81	C₄⋊#6923	—	2	1	no	—
82	C₄⋊#6923	C₂³	1	2	yes	—
83	C₄⋊#6923	C₂³	1	1	yes	characteristic
84	C₄⋊#6923	C₂³	1	1	yes	characteristic
85	C₄×#6923	—	2	1	no	—
86	C₄×#6923	—	2	1	no	—
87	C₄×#6923	—	2	2	no	—
88	C₄×#6923	—	2	2	no	—
89	C₄×#6923	—	2	1	no	—
90	C₄×#6923	D₄	1	1	yes	characteristic
91	C₄×#6923	D₄	1	1	yes	characteristic
92	C₄×#6923	C₂³	1	1	yes	characteristic
93	C₄×#6923	C₂³	1	1	yes	characteristic
94	#11758213	—	2	1	no	—
95	D₆₁⋊C₈	C₂³	1	2	yes	—
96	Q₁₆×C₆₁	C₂×C₄	1	2	yes	complemented
97	SD₁₆×C₆₁	C₂×C₄	1	2	yes	complemented
98	M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
99	#11758202	C₂×C₄	1	2	yes	complemented
100	#11758202	C₂×C₄	1	2	yes	complemented
101	#11758201	C₂×C₄	1	2	yes	complemented
102	#11758200	C₂×C₄	1	2	yes	complemented
103	#11758194	C₂×C₄	1	1	yes	characteristic, complemented
104	Dic₂₄₄	C₂×C₄	1	2	yes	complemented
105	#11758190	C₂×C₄	1	2	yes	complemented
106	#11758189	C₂×C₄	1	2	yes	complemented
107	C₈×D₆₁	C₂³	1	2	yes	—
108	C₂²×D₆₁	—	2	1	no	—
109	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
110	C₂×#6923	—	2	2	no	—
111	C₂×#6923	—	2	2	no	—
112	C₂×#6923	—	2	1	no	—
113	C₂×#6923	—	2	1	no	—
114	C₂×#6923	—	4	1	no	—
115	C₂×#6923	—	4	2	no	—
116	C₂×#6923	—	4	2	no	—
117	C₂×#6923	—	4	2	no	—
118	C₂×#6923	C₂×D₄	1	1	yes	characteristic
119	C₂×#6923	C₂×D₄	1	1	yes	characteristic
120	C₂×#6923	C₂×D₄	1	1	yes	characteristic
121	C₂×#6923	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₆₁	—	2	1	no	—
123	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
124	Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
125	D₄×C₆₁	—	2	1	no	—
126	D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₁	—	2	1	no	—
128	C₂×C₄×C₆₁	—	2	1	no	—
129	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²⋊D₆₁	—	2	1	no	—
131	C₂²⋊D₆₁	—	2	1	no	—
132	C₂²⋊D₆₁	—	2	1	no	—
133	C₂×Dic₆₁	—	2	1	no	—
134	C₂×Dic₆₁	—	2	1	no	—
135	C₂×Dic₆₁	#41	1	1	yes	characteristic
136	D₂₄₄	—	2	1	no	—
137	D₂₄₄	C₂²×C₄	1	1	yes	characteristic
138	C₄×D₆₁	—	2	2	no	—
139	C₄×D₆₁	—	2	1	no	—
140	C₄×D₆₁	—	2	2	no	—
141	C₄×D₆₁	—	2	1	no	—
142	C₄×D₆₁	—	2	1	no	—
143	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
144	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
145	C₄×D₆₁	#41	1	1	yes	characteristic
146	C₄×D₆₁	C₂⁴	1	1	yes	characteristic
147	Dic₁₂₂	—	2	1	no	—
148	Dic₁₂₂	—	2	2	no	—
149	Dic₁₂₂	—	2	2	no	—
150	Dic₁₂₂	—	2	1	no	—
151	Dic₁₂₂	C₂²×C₄	1	2	yes	—
152	Dic₁₂₂	C₂²×C₄	1	1	yes	characteristic
153	#92420	—	2	2	no	—
154	C₈×C₆₁	C₂²×C₄	1	2	yes	—
155	#92418	C₂²×C₄	1	2	yes	—
156	C₂²×C₆₁	—	2	1	no	—
157	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
158	C₂×D₆₁	—	2	1	no	—
159	C₂×D₆₁	—	2	1	no	—
160	C₂×D₆₁	—	2	1	no	—
161	C₂×D₆₁	—	4	1	no	—
162	C₂×D₆₁	—	4	1	no	—
163	C₂×D₆₁	C₄×D₄	1	1	yes	characteristic
164	C₂×D₆₁	C₂²×D₄	1	1	yes	characteristic
165	#6923	—	2	1	no	—
166	#6923	—	2	1	no	—
167	#6923	—	2	1	no	—
168	#6923	—	4	1	no	—
169	#6923	—	4	2	no	—
170	#6923	—	4	1	no	—
171	#6923	#137	1	2	yes	direct product factor
172	C₄×C₆₁	—	2	2	no	—
173	C₄×C₆₁	—	2	1	no	—
174	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
175	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
176	Dic₆₁	—	2	2	no	—
177	Dic₆₁	—	2	1	no	—
178	Dic₆₁	C₄×D₄	1	1	yes	characteristic
179	Dic₆₁	C₂×#41	1	1	yes	characteristic
180	C₄×#137	—	61	1	no	maximal, Sylow
181	C₂×C₆₁	—	2	1	no	—
182	C₂×C₆₁	—	4	1	no	—
183	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
184	D₆₁	—	2	1	no	—
185	D₆₁	—	4	1	no	—
186	D₆₁	C₂×#137	1	1	yes	characteristic
187	D₆₁	#575	1	1	yes	characteristic
188	C₂×#137	—	61	1	no	—
189	C₄×#41	—	61	1	no	—
190	C₂×C₄×Q₈	—	61	1	no	—
191	#441	—	61	2	no	—
192	#440	—	61	2	no	—
193	C₄×Q₁₆	—	61	2	no	—
194	C₄×SD₁₆	—	61	2	no	—
195	#428	—	61	1	no	—
196	#419	—	61	1	no	—
197	#417	—	61	1	no	—
198	C₄×M₁₆	—	61	1	no	—
199	C₆₁	C₄×#137	1	1	yes	characteristic, complemented, Sylow
200	C₂×#41	—	61	1	no	—
201	C₂²×Q₈	—	61	1	no	—
202	#137	—	61	1	no	—
203	#137	—	61	1	no	—
204	#137	—	61	2	no	—
205	#137	—	61	2	no	—
206	#137	—	61	1	no	—
207	#137	#6923	1	1	yes	characteristic, direct product factor
208	C₂×Q₁₆	—	61	2	no	—
209	C₂×SD₁₆	—	61	2	no	—
210	C₂×M₁₆	—	61	1	no	—
211	C₂×M₁₆	—	61	1	no	—
212	C₄×Q₈	—	61	2	no	—
213	C₄×Q₈	—	61	2	no	—
214	C₄×Q₈	—	61	1	no	—
215	C₄×Q₈	—	61	1	no	—
216	C₄×Q₈	—	122	1	no	—
217	C₄×Q₈	—	122	1	no	—
218	C₄×D₄	—	61	1	no	—
219	C₄×D₄	—	61	1	no	—
220	C₄×D₄	—	122	1	no	—
221	C₄×D₄	—	122	1	no	—
222	#117	—	61	1	no	—
223	#117	—	122	1	no	—
224	C₂×C₄⋊C₄	—	61	1	no	—
225	C₂×C₄⋊C₄	—	122	1	no	—
226	C₂×C₄²	—	61	1	no	—
227	C₂×C₄²	—	122	1	no	—
228	C₂×C₄²	—	122	1	no	—
229	#107	—	61	2	no	—
230	#106	—	61	2	no	—
231	#103	—	61	2	no	—
232	#103	—	61	2	no	—
233	#103	—	61	2	no	—
234	#102	—	61	2	no	—
235	#97	—	61	2	no	—
236	C₄×C₈	—	61	2	no	—
237	#41	—	61	1	no	—
238	#41	—	61	1	no	—
239	#41	—	61	1	no	—
240	#41	—	122	1	no	—
241	#41	—	122	1	no	—
242	#41	C₂×#6923	1	1	yes	characteristic
243	C₂×Q₈	—	61	2	no	—
244	C₂×Q₈	—	61	1	no	—
245	C₂×Q₈	—	61	2	no	—
246	C₂×Q₈	—	61	1	no	—
247	C₂×Q₈	—	122	1	no	—
248	C₂×Q₈	—	122	1	no	—
249	C₂×Q₈	—	122	1	no	—
250	C₂×Q₈	C₂×#6923	1	1	yes	characteristic, complemented
251	C₂×D₄	—	61	1	no	—
252	C₂×D₄	—	122	1	no	—
253	C₂²×C₄	—	61	1	no	—
254	C₂²×C₄	—	61	1	no	—
255	C₂²×C₄	—	61	1	no	—
256	C₂²×C₄	—	122	1	no	—
257	C₂²×C₄	—	122	1	no	—
258	C₂²×C₄	—	122	1	no	—
259	C₂²×C₄	—	122	1	no	—
260	C₂²×C₄	—	122	1	no	—
261	Q₁₆	—	61	2	no	—
262	Q₁₆	—	61	2	no	—
263	Q₁₆	—	61	2	no	—
264	Q₁₆	C₂×#6923	1	2	yes	complemented
265	SD₁₆	—	61	2	no	—
266	SD₁₆	—	61	2	no	—
267	SD₁₆	—	61	2	no	—
268	SD₁₆	C₂×#6923	1	2	yes	complemented
269	M₁₆	—	61	1	no	—
270	M₁₆	—	61	2	no	—
271	M₁₆	—	122	1	no	—
272	M₁₆	—	122	1	no	—
273	M₁₆	C₂×#6923	1	1	yes	characteristic, complemented
274	C₂×C₈	—	61	2	no	—
275	C₂×C₈	—	61	2	no	—
276	C₄⋊C₄	—	61	2	no	—
277	C₄⋊C₄	—	61	1	no	—
278	C₄⋊C₄	—	61	1	no	—
279	C₄⋊C₄	—	122	1	no	—
280	C₄⋊C₄	—	122	2	no	—
281	C₄⋊C₄	—	122	2	no	—
282	C₄⋊C₄	—	122	1	no	—
283	C₄⋊C₄	—	122	1	no	—
284	C₂²⋊C₄	—	122	1	no	—
285	C₂²⋊C₄	—	122	1	no	—
286	C₂²⋊C₄	—	122	1	no	—
287	C₄²	—	61	1	no	—
288	C₄²	—	61	1	no	—
289	C₄²	—	61	1	no	—
290	C₄²	—	61	1	no	—
291	C₄²	—	122	1	no	—
292	C₄²	—	122	1	no	—
293	C₄²	—	122	1	no	—
294	C₄²	—	122	2	no	—
295	C₄²	—	122	2	no	—
296	C₂³	—	61	1	no	—
297	C₂³	—	122	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	61	2	no	—
300	Q₈	—	61	1	no	—
301	Q₈	—	122	2	no	—
302	Q₈	—	122	2	no	—
303	Q₈	—	122	1	no	—
304	Q₈	—	122	1	no	—
305	Q₈	C₂²×#6923	1	2	yes	—
306	Q₈	C₂²×#6923	1	1	yes	characteristic
307	D₄	—	2	1	no	—
308	D₄	—	61	1	no	—
309	D₄	—	122	1	no	—
310	D₄	—	122	1	no	—
311	D₄	—	122	1	no	—
312	D₄	—	122	1	no	—
313	D₄	C₂²×#6923	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	61	1	no	—
317	C₂×C₄	—	61	1	no	—
318	C₂×C₄	—	61	1	no	—
319	C₂×C₄	—	61	1	no	—
320	C₂×C₄	—	61	1	no	—
321	C₂×C₄	—	61	1	no	—
322	C₂×C₄	—	61	1	no	—
323	C₂×C₄	—	61	1	no	—
324	C₂×C₄	—	61	1	no	—
325	C₂×C₄	—	122	1	no	—
326	C₂×C₄	—	122	1	no	—
327	C₂×C₄	—	122	2	no	—
328	C₂×C₄	—	122	2	no	—
329	C₂×C₄	—	122	1	no	—
330	C₂×C₄	—	122	2	no	—
331	C₂×C₄	—	122	1	no	—
332	C₂×C₄	—	122	1	no	—
333	C₂×C₄	—	122	1	no	—
334	C₂×C₄	—	122	1	no	—
335	C₂×C₄	—	122	2	no	—
336	C₂×C₄	—	244	2	no	—
337	C₂×C₄	—	244	1	no	—
338	C₂×C₄	—	244	2	no	—
339	C₂×C₄	—	244	2	no	—
340	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
341	C₈	—	61	2	no	—
342	C₈	—	122	2	no	—
343	C₈	C₂²×#6923	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	61	1	no	—
346	C₂²	—	61	1	no	—
347	C₂²	—	122	1	no	—
348	C₂²	—	122	1	no	—
349	C₂²	—	122	1	no	—
350	C₂²	—	244	1	no	—
351	C₂²	—	244	1	no	—
352	C₂²	D₄×#6923	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	61	2	no	—
356	C₄	—	61	1	no	—
357	C₄	—	61	1	no	—
358	C₄	—	122	1	no	—
359	C₄	—	122	1	no	—
360	C₄	—	122	1	no	—
361	C₄	—	122	1	no	—
362	C₄	—	122	2	no	—
363	C₄	—	244	1	no	—
364	C₄	—	244	1	no	—
365	C₄	—	244	2	no	—
366	C₄	D₄×#6923	1	1	yes	characteristic
367	C₄	C₂³×#6923	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	4	1	no	—
370	C₂	—	61	1	no	—
371	C₂	—	61	1	no	—
372	C₂	—	122	1	no	—
373	C₂	—	244	1	no	—
374	C₂	C₂×D₄×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#6923
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₄	C₂³×#6923
Fitting	#137×C₆₁	C₄
socle	C₂×C₆₁	C₂×C₄×D₄
hypercenter	#137	#6923
hypocenter	C₆₁	C₄×#137
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×#6923	C₂	semidirect
3. #16840802	C₂	semidirect
4. #16840801	C₂	semidirect
5. Q₁₆×#6923	C₂	semidirect
6. SD₁₆×#6923	C₂	semidirect
7. #16840773	C₂	semidirect
8. #16840755	C₂	semidirect
10. M₁₆×#6923	C₂	semidirect
40. #14519810	C₄	semidirect
41. #14519809	C₄	semidirect
42. #14519808	C₄	semidirect
43. #14519807	C₄	semidirect
44. #14519806	C₄	semidirect
48. #137×C₆₁	C₄	semidirect
55. C₂×Q₈×C₆₁	C₂×C₄	semidirect
59. Q₈×D₆₁	C₂×C₄	semidirect
72. C₂×Dic₁₂₂	C₂×C₄	semidirect
96. Q₁₆×C₆₁	C₂×C₄	semidirect
97. SD₁₆×C₆₁	C₂×C₄	semidirect
98. M₁₆×C₆₁	C₂×C₄	semidirect
99. #11758202	C₂×C₄	semidirect
100. #11758202	C₂×C₄	semidirect
101. #11758201	C₂×C₄	semidirect
102. #11758200	C₂×C₄	semidirect
103. #11758194	C₂×C₄	semidirect
104. Dic₂₄₄	C₂×C₄	semidirect
105. #11758190	C₂×C₄	semidirect
106. #11758189	C₂×C₄	semidirect
171. #6923	#137	direct (N × H)
199. C₆₁	C₄×#137	semidirect
207. #137	#6923	direct (N × H)
250. C₂×Q₈	C₂×#6923	semidirect
264. Q₁₆	C₂×#6923	semidirect
268. SD₁₆	C₂×#6923	semidirect
273. M₁₆	C₂×#6923	semidirect