Subgroups — #27187663

Order 12928.

632 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	C₂×Q₈×#87938	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×#87938	C₂	1	1	yes	maximal, characteristic
3	(C₂×#87938)⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#20266129	C₂	1	1	yes	maximal, characteristic
5	#20266127	C₂	1	2	yes	maximal, complemented
6	#20266127	C₂	1	2	yes	maximal, complemented
7	#20266124	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊#87938	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊#87938	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×#87938	C₂	1	2	yes	maximal, complemented
11	#124×D₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×#87938	C₂²	1	2	yes	—
13	Q₈×#87938	—	2	4	no	—
14	D₄×#87938	—	2	4	no	—
15	C₂×C₄⋊#87938	C₂²	1	2	yes	—
16	C₂×C₄⋊#87938	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#87938	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#87938	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#87938	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊#87938	C₂²	1	2	yes	—
21	C₂×C₂²⋊#87938	C₂²	1	2	yes	—
22	C₂×C₂²⋊#87938	C₂²	1	2	yes	—
23	C₂×C₄×#87938	C₂²	1	1	yes	characteristic
24	C₂×C₄×#87938	C₂²	1	2	yes	—
25	C₂×C₄×#87938	C₂²	1	1	yes	characteristic
26	C₂×C₄×#87938	C₂²	1	2	yes	—
27	(C₂×#87938)⋊C₄	C₂²	1	1	yes	characteristic
28	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
29	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
30	(C₂×#87938)⋊C₄	C₂²	1	1	yes	characteristic
31	(C₂×#87938)⋊C₄	C₂²	1	2	yes	—
32	C₂×Q₈×D₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×D₄×D₁₀₁	C₂²	1	1	yes	characteristic
34	#15692244	C₄	1	1	yes	characteristic, complemented
35	#15692243	C₄	1	1	yes	characteristic, complemented
36	#15692242	C₄	1	1	yes	characteristic, complemented
37	#15692241	C₄	1	2	yes	complemented
38	#15692241	C₄	1	2	yes	complemented
39	C₂²⋊C₄×D₁₀₁	C₂²	1	2	yes	—
40	C₂²⋊C₄×D₁₀₁	C₂²	1	2	yes	—
41	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
42	#124×C₁₀₁	C₄	1	1	yes	characteristic, complemented
43	C₂²×#87938	—	2	4	no	—
44	C₂²×#87938	—	2	4	no	—
45	C₂²×#87938	—	4	2	no	—
46	C₂²×#87938	C₂³	1	1	yes	characteristic
47	C₂²×#87938	C₂³	1	2	yes	—
48	C₂²×#87938	C₂³	1	1	yes	characteristic
49	C₂²×#87938	C₂³	1	2	yes	—
50	C₂²×#87938	C₂³	1	1	yes	characteristic
51	C₂²×#87938	C₂³	1	1	yes	characteristic
52	C₄⋊#87938	—	2	4	no	—
53	C₄⋊#87938	—	2	2	no	—
54	C₄⋊#87938	—	2	2	no	—
55	C₄⋊#87938	—	2	2	no	—
56	C₄⋊#87938	D₄	1	4	yes	—
57	C₂²⋊#87938	—	2	4	no	—
58	C₂²⋊#87938	—	2	4	no	—
59	C₂²⋊#87938	—	4	2	no	—
60	C₄×#87938	—	2	4	no	—
61	C₄×#87938	—	2	2	no	—
62	C₄×#87938	—	2	2	no	—
63	C₄×#87938	—	4	2	no	—
64	C₂³×D₁₀₁	C₂³	1	2	yes	—
65	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	complemented
66	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic, complemented
67	Q₈×D₁₀₁	—	2	2	no	—
68	Q₈×D₁₀₁	—	2	2	no	—
69	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
71	D₄×D₁₀₁	—	2	2	no	—
72	D₄×D₁₀₁	—	2	2	no	—
73	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
74	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
77	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	complemented
78	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	complemented
79	#13108051	C₂×C₄	1	2	yes	complemented
80	#13108051	C₂×C₄	1	2	yes	complemented
81	#13108051	C₂×C₄	1	2	yes	complemented
82	#13108051	C₂×C₄	1	2	yes	complemented
83	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
84	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
85	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
86	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
87	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	complemented
89	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	complemented
90	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
91	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
92	C₂²×D₁₀₁	—	2	2	no	—
93	C₂²×D₁₀₁	—	2	2	no	—
94	C₂²×D₁₀₁	—	2	2	no	—
95	C₂²×D₁₀₁	—	2	2	no	—
96	C₂²×D₁₀₁	—	2	2	no	—
97	C₂²×D₁₀₁	—	2	2	no	—
98	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
99	C₂²×D₁₀₁	C₂⁴	1	1	yes	characteristic
100	C₂×#87938	—	2	1	no	—
101	C₂×#87938	—	2	1	no	—
102	C₂×#87938	—	2	2	no	—
103	C₂×#87938	—	2	2	no	—
104	C₂×#87938	—	2	1	no	—
105	C₂×#87938	—	2	1	no	—
106	C₂×#87938	—	2	1	no	—
107	C₂×#87938	—	2	1	no	—
108	C₂×#87938	—	2	2	no	—
109	C₂×#87938	—	2	1	no	—
110	C₂×#87938	—	2	1	no	—
111	C₂×#87938	—	2	2	no	—
112	C₂×#87938	—	2	2	no	—
113	C₂×#87938	—	2	1	no	—
114	C₂×#87938	—	2	1	no	—
115	C₂×#87938	—	4	8	no	—
116	C₂×#87938	C₂×D₄	1	4	yes	—
117	C₂×#87938	#41	1	2	yes	—
118	C₂×#87938	#41	1	2	yes	—
119	Q₈×C₁₀₁	—	2	2	no	—
120	D₄×C₁₀₁	—	2	2	no	—
121	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
124	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
125	C₂²⋊D₁₀₁	—	2	4	no	—
126	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
129	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
130	D₄₀₄	—	2	2	no	—
131	C₄×D₁₀₁	—	2	2	no	—
132	C₄×D₁₀₁	—	2	2	no	—
133	C₄×D₁₀₁	—	2	1	no	—
134	C₄×D₁₀₁	—	2	1	no	—
135	C₄×D₁₀₁	—	2	1	no	—
136	C₄×D₁₀₁	—	2	1	no	—
137	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
138	C₄×D₁₀₁	#41	1	2	yes	—
139	Dic₂₀₂	—	2	2	no	—
140	Dic₂₀₂	—	2	4	no	—
141	C₂²×C₁₀₁	—	2	2	no	—
142	C₂²×C₁₀₁	—	2	2	no	—
143	C₂²×C₁₀₁	—	2	2	no	—
144	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
145	C₂×D₁₀₁	—	2	2	no	—
146	C₂×D₁₀₁	—	2	2	no	—
147	C₂×D₁₀₁	—	2	2	no	—
148	C₂×D₁₀₁	—	4	2	no	—
149	C₂×D₁₀₁	—	4	2	no	—
150	C₂×D₁₀₁	—	4	2	no	—
151	C₂×D₁₀₁	—	4	2	no	—
152	C₂×D₁₀₁	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₀₁	C₂×#41	1	1	yes	characteristic
154	C₂×D₁₀₁	C₂×#41	1	1	yes	characteristic
155	C₂×D₁₀₁	C₂×#41	1	1	yes	characteristic
156	C₂×D₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×D₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
158	#87938	—	2	2	no	—
159	#87938	—	2	4	no	—
160	#87938	—	2	2	no	—
161	#87938	—	4	1	no	—
162	#87938	—	4	1	no	—
163	#87938	—	4	2	no	—
164	C₄×C₁₀₁	—	2	2	no	—
165	C₄×C₁₀₁	—	2	1	no	—
166	C₄×C₁₀₁	—	2	1	no	—
167	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
168	Dic₁₀₁	—	2	2	no	—
169	Dic₁₀₁	—	2	1	no	—
170	Dic₁₀₁	—	2	1	no	—
171	Dic₁₀₁	C₄×D₄	1	2	yes	—
172	C₂×C₁₀₁	—	4	2	no	—
173	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
175	C₂×C₁₀₁	#520	1	1	yes	characteristic
176	D₁₀₁	—	4	2	no	—
177	D₁₀₁	#546	1	1	yes	characteristic
178	D₁₀₁	#548	1	1	yes	characteristic
179	D₁₀₁	#556	1	1	yes	characteristic
180	D₁₀₁	#559	1	1	yes	characteristic
181	#2367	—	101	1	no	maximal, Sylow
182	C₁₀₁	#2367	1	1	yes	characteristic, complemented, Sylow
183	C₂×#124	—	101	1	no	—
184	C₂×C₄×Q₈	—	101	1	no	—
185	C₂×C₄×D₄	—	101	1	no	—
186	#391	—	101	1	no	—
187	#390	—	101	1	no	—
188	#388	—	101	2	no	—
189	#388	—	101	2	no	—
190	#385	—	101	2	no	—
191	C₄×C₄⋊C₄	—	101	1	no	—
192	C₄×C₄⋊C₄	—	101	1	no	—
193	C₄×C₂²⋊C₄	—	101	2	no	—
194	C₂²×Q₈	—	101	1	no	—
195	C₂²×D₄	—	101	1	no	—
196	C₂³×C₄	—	101	2	no	—
197	#124	—	101	1	no	—
198	#124	—	101	1	no	—
199	#124	—	101	2	no	—
200	#124	—	101	2	no	—
201	#124	—	101	1	no	—
202	#124	#87938	1	1	yes	characteristic, complemented
203	C₄×Q₈	—	202	4	no	—
204	C₄×D₄	—	202	4	no	—
205	C₂×C₄⋊C₄	—	101	1	no	—
206	C₂×C₄⋊C₄	—	101	2	no	—
207	C₂×C₄⋊C₄	—	101	1	no	—
208	C₂×C₄⋊C₄	—	101	1	no	—
209	C₂×C₄⋊C₄	—	101	1	no	—
210	C₂×C₂²⋊C₄	—	101	2	no	—
211	C₂×C₂²⋊C₄	—	101	2	no	—
212	C₂×C₂²⋊C₄	—	101	2	no	—
213	C₂×C₂²⋊C₄	—	101	2	no	—
214	C₂×C₂²⋊C₄	—	101	2	no	—
215	C₂×C₄²	—	101	1	no	—
216	C₂×C₄²	—	101	2	no	—
217	C₂×C₄²	—	101	1	no	—
218	C₂×C₄²	—	101	2	no	—
219	C₂×C₄²	—	101	1	no	—
220	#95	—	101	1	no	—
221	#95	—	101	2	no	—
222	#95	—	101	2	no	—
223	#95	—	101	1	no	—
224	#95	—	101	2	no	—
225	C₂⁴	—	101	2	no	—
226	C₂×Q₈	—	101	2	no	—
227	C₂×Q₈	—	101	1	no	—
228	C₂×Q₈	—	202	2	no	—
229	C₂×Q₈	—	202	2	no	—
230	C₂×Q₈	C₂×#87938	1	1	yes	characteristic, complemented
231	C₂×D₄	—	101	2	no	—
232	C₂×D₄	—	101	1	no	—
233	C₂×D₄	—	202	2	no	—
234	C₂×D₄	—	202	2	no	—
235	C₂×D₄	C₂×#87938	1	1	yes	characteristic
236	C₂²×C₄	—	101	1	no	—
237	C₂²×C₄	—	101	1	no	—
238	C₂²×C₄	—	101	1	no	—
239	C₂²×C₄	—	101	2	no	—
240	C₂²×C₄	—	101	1	no	—
241	C₂²×C₄	—	101	1	no	—
242	C₂²×C₄	—	101	2	no	—
243	C₂²×C₄	—	101	2	no	—
244	C₂²×C₄	—	101	1	no	—
245	C₂²×C₄	—	101	1	no	—
246	C₂²×C₄	—	404	2	no	—
247	C₂²×C₄	—	202	4	no	—
248	C₂²×C₄	—	202	4	no	—
249	C₄⋊C₄	—	101	4	no	—
250	C₄⋊C₄	—	202	2	no	—
251	C₄⋊C₄	—	202	2	no	—
252	C₄⋊C₄	—	202	2	no	—
253	C₄⋊C₄	—	202	4	no	—
254	C₂²⋊C₄	—	101	2	no	—
255	C₂²⋊C₄	—	101	2	no	—
256	C₂²⋊C₄	—	101	2	no	—
257	C₂²⋊C₄	—	101	2	no	—
258	C₂²⋊C₄	—	101	2	no	—
259	C₂²⋊C₄	—	101	2	no	—
260	C₂²⋊C₄	—	404	2	no	—
261	C₂²⋊C₄	—	202	4	no	—
262	C₂²⋊C₄	—	202	4	no	—
263	C₂²⋊C₄	C₂×#87938	1	2	yes	complemented
264	C₂²⋊C₄	C₂×#87938	1	2	yes	complemented
265	C₄²	—	101	1	no	—
266	C₄²	—	101	1	no	—
267	C₄²	—	101	1	no	—
268	C₄²	—	404	2	no	—
269	C₄²	—	202	2	no	—
270	C₄²	—	202	2	no	—
271	C₄²	—	202	4	no	—
272	C₄²	C₂×#87938	1	1	yes	characteristic, complemented
273	C₂³	—	101	1	no	—
274	C₂³	—	101	2	no	—
275	C₂³	—	202	2	no	—
276	C₂³	—	202	2	no	—
277	C₂³	—	202	2	no	—
278	C₂³	—	202	2	no	—
279	C₂³	—	202	2	no	—
280	C₂³	—	202	2	no	—
281	C₂³	C₂²×#87938	1	2	yes	—
282	Q₈	—	2	2	no	—
283	Q₈	—	202	4	no	—
284	Q₈	—	202	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	202	4	no	—
287	D₄	—	202	2	no	—
288	C₂×C₄	—	101	1	no	—
289	C₂×C₄	—	101	2	no	—
290	C₂×C₄	—	101	2	no	—
291	C₂×C₄	—	101	4	no	—
292	C₂×C₄	—	101	2	no	—
293	C₂×C₄	—	101	2	no	—
294	C₂×C₄	—	101	1	no	—
295	C₂×C₄	—	101	1	no	—
296	C₂×C₄	—	101	2	no	—
297	C₂×C₄	—	404	8	no	—
298	C₂×C₄	—	202	1	no	—
299	C₂×C₄	—	202	2	no	—
300	C₂×C₄	—	202	1	no	—
301	C₂×C₄	—	202	1	no	—
302	C₂×C₄	—	202	2	no	—
303	C₂×C₄	—	202	1	no	—
304	C₂×C₄	—	202	2	no	—
305	C₂×C₄	—	202	1	no	—
306	C₂×C₄	—	202	1	no	—
307	C₂×C₄	—	202	1	no	—
308	C₂×C₄	—	202	2	no	—
309	C₂×C₄	—	202	2	no	—
310	C₂×C₄	—	202	1	no	—
311	C₂×C₄	—	202	1	no	—
312	C₂×C₄	—	202	1	no	—
313	C₂×C₄	—	202	2	no	—
314	C₂×C₄	—	202	1	no	—
315	C₂×C₄	—	202	1	no	—
316	C₂×C₄	—	202	1	no	—
317	C₂×C₄	—	202	2	no	—
318	C₂×C₄	—	202	1	no	—
319	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
320	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
321	C₂×C₄	C₂²×#87938	1	2	yes	—
322	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	101	1	no	—
327	C₂²	—	101	1	no	—
328	C₂²	—	101	1	no	—
329	C₂²	—	101	1	no	—
330	C₂²	—	101	1	no	—
331	C₂²	—	101	1	no	—
332	C₂²	—	404	2	no	—
333	C₂²	—	404	2	no	—
334	C₂²	—	404	2	no	—
335	C₂²	—	404	2	no	—
336	C₂²	—	202	2	no	—
337	C₂²	—	202	2	no	—
338	C₂²	—	202	2	no	—
339	C₂²	C₂³×#87938	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	101	2	no	—
344	C₄	—	404	2	no	—
345	C₄	—	404	1	no	—
346	C₄	—	404	1	no	—
347	C₄	—	202	2	no	—
348	C₄	—	202	4	no	—
349	C₄	—	202	2	no	—
350	C₄	—	202	2	no	—
351	C₄	—	202	1	no	—
352	C₄	—	202	1	no	—
353	C₄	D₄×#87938	1	2	yes	—
354	C₂	—	4	2	no	—
355	C₂	—	101	1	no	—
356	C₂	—	101	1	no	—
357	C₂	—	101	1	no	—
358	C₂	—	101	1	no	—
359	C₂	—	404	2	no	—
360	C₂	C₂×D₄×#87938	1	1	yes	characteristic
361	C₂	#41×#87938	1	1	yes	characteristic
362	C₂	#20266246	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#124×C₁₀₁	C₄
prime radical	C₂×D₄×D₁₀₁	C₂²
cosocle	C₂²×D₁₀₁	C₂⁴
hypercenter	#124	#87938
hypocenter	C₁₀₁	#2367
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×#87938	C₂	semidirect
3. (C₂×#87938)⋊Q₈	C₂	semidirect
5. #20266127	C₂	semidirect
6. #20266127	C₂	semidirect
7. #20266124	C₂	semidirect
8. C₄×C₄⋊#87938	C₂	semidirect
9. C₄×C₄⋊#87938	C₂	semidirect
10. C₂²⋊C₄×#87938	C₂	semidirect
34. #15692244	C₄	semidirect
35. #15692243	C₄	semidirect
36. #15692242	C₄	semidirect
37. #15692241	C₄	semidirect
38. #15692241	C₄	semidirect
42. #124×C₁₀₁	C₄	semidirect
65. C₂×Dic₂₀₂	C₂×C₄	semidirect
66. C₂×Dic₂₀₂	C₂×C₄	semidirect
77. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
78. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
79. #13108051	C₂×C₄	semidirect
80. #13108051	C₂×C₄	semidirect
81. #13108051	C₂×C₄	semidirect
82. #13108051	C₂×C₄	semidirect
83. C₄×Dic₁₀₁	C₂×C₄	semidirect
84. C₄×Dic₁₀₁	C₂×C₄	semidirect
85. C₄×Dic₁₀₁	C₂×C₄	semidirect
86. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
88. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
89. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
90. C₄²×C₁₀₁	C₂×C₄	semidirect
182. C₁₀₁	#2367	semidirect
202. #124	#87938	semidirect
230. C₂×Q₈	C₂×#87938	semidirect
263. C₂²⋊C₄	C₂×#87938	semidirect
264. C₂²⋊C₄	C₂×#87938	semidirect
272. C₄²	C₂×#87938	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#87938	2
C₂×D₄×#87938	2
(C₂×#87938)⋊Q₈	2
#20266129	2
#20266127	2
#20266124	2
C₄×C₄⋊#87938	2
C₂²⋊C₄×#87938	2
#124×D₁₀₁	2
#2367	101