Subgroups — C₄×#11745278 #16828261

Order 3712.

500 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×#11745278	C₂	1	1	yes	maximal, characteristic
2	C₄×#90809	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#90807	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#90785	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271255	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#90773	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×Dic₂₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271250	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#90782	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271243	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271198	C₂	1	1	yes	maximal, characteristic, complemented
12	#14271158	C₂	1	1	yes	maximal, characteristic, complemented
13	#14271156	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₈×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#90809	C₂²	1	1	yes	characteristic
17	C₂×#90807	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₂₉	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
21	Q₈×Dic₂₉	C₂²	1	1	yes	characteristic
22	C₂×#90785	C₂²	1	1	yes	characteristic
23	D₄×Dic₂₉	C₂²	1	1	yes	characteristic
24	C₂×#90782	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₂₉	—	2	1	no	—
26	C₂×C₄×Dic₂₉	—	2	1	no	—
27	#11745278	C₄	1	8	yes	direct product factor
28	C₂×#90773	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
31	#11745255	—	2	1	no	—
32	#11745255	C₂²	1	1	yes	characteristic
33	#11745254	—	2	1	no	—
34	#11745254	C₂²	1	1	yes	characteristic
35	#11745254	C₂²	1	1	yes	characteristic
36	#11745249	C₂²	1	1	yes	characteristic
37	#11745243	—	2	1	no	—
38	#11745239	—	2	1	no	—
39	C₄×D₁₁₆	—	2	1	no	—
40	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
41	C₄²×D₂₉	—	2	1	no	—
42	C₄²×D₂₉	C₂²	1	1	yes	characteristic
43	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
44	C₂₉×#106	C₂²	1	1	yes	characteristic
45	C₂₉×#103	C₂²	1	1	yes	characteristic
46	C₂₉×#102	C₂²	1	1	yes	characteristic
47	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
48	#11745183	C₂²	1	1	yes	characteristic
49	#11745180	C₂²	1	1	yes	characteristic
50	#11745169	C₂²	1	1	yes	characteristic
51	#11745165	C₂²	1	1	yes	characteristic
52	#11745164	C₂²	1	1	yes	characteristic
53	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
54	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
55	#11745158	C₂²	1	1	yes	characteristic
56	#11745157	C₂²	1	1	yes	characteristic
57	#11745155	C₂²	1	1	yes	characteristic
58	#11745155	C₂²	1	1	yes	characteristic
59	C₄×#6597	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₉	—	2	1	no	—
63	C₂×C₂²⋊D₂₉	—	2	1	no	—
64	C₂²×Dic₂₉	—	2	1	no	—
65	C₂²×Dic₂₉	—	2	1	no	—
66	#90809	—	2	2	no	—
67	#90809	C₂×C₄	1	4	yes	complemented
68	#90807	—	2	2	no	—
69	#90807	C₂×C₄	1	4	yes	complemented
70	C₂×D₁₁₆	—	2	1	no	—
71	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₂₉	—	2	1	no	—
73	C₂×C₄×D₂₉	—	2	1	no	—
74	C₂×C₄×D₂₉	—	2	1	no	—
75	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
76	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
77	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
78	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
79	SD₁₆×C₂₉	C₂×C₄	1	4	yes	complemented
80	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₂₉	—	2	1	no	—
83	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₂₉	—	2	1	no	—
86	C₄²×C₂₉	—	2	1	no	—
87	C₄²×C₂₉	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₂₉	—	2	1	no	—
89	#90785	C₂×C₄	1	4	yes	complemented
90	#90782	C₂×C₄	1	4	yes	complemented
91	#90781	—	2	1	no	—
92	#90781	—	2	1	no	—
93	#90781	—	2	1	no	—
94	#90781	—	2	1	no	—
95	C₄⋊Dic₂₉	—	2	1	no	—
96	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
98	#90779	—	2	1	no	—
99	#90779	—	2	1	no	—
100	C₄×Dic₂₉	—	2	1	no	—
101	C₄×Dic₂₉	—	2	1	no	—
102	C₄×Dic₂₉	—	2	1	no	—
103	C₄×Dic₂₉	—	2	1	no	—
104	C₄×Dic₂₉	—	2	1	no	—
105	C₄×Dic₂₉	D₄	1	1	yes	characteristic
106	C₄×Dic₂₉	D₄	1	1	yes	characteristic
107	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
108	C₂×#6597	C₂³	1	1	yes	characteristic
109	C₂×#6597	C₂³	1	1	yes	characteristic
110	#90773	C₂×C₄	1	4	yes	complemented
111	C₈×D₂₉	—	2	2	no	—
112	C₈×D₂₉	C₂×C₄	1	4	yes	complemented
113	C₂³×C₂₉	—	2	1	no	—
114	C₂²×D₂₉	—	2	1	no	—
115	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₂₉	—	2	1	no	—
117	Q₈×C₂₉	C₂²×C₄	1	2	yes	—
118	D₄×C₂₉	—	2	1	no	—
119	D₄×C₂₉	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₂₉	—	2	1	no	—
121	C₂×C₄×C₂₉	—	2	1	no	—
122	C₂×C₄×C₂₉	—	2	1	no	—
123	C₂×C₄×C₂₉	—	4	2	no	—
124	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₂₉	—	2	4	no	—
128	C₂×Dic₂₉	—	2	1	no	—
129	C₂×Dic₂₉	—	2	4	no	—
130	C₂×Dic₂₉	—	2	1	no	—
131	C₂×Dic₂₉	—	2	1	no	—
132	C₂×Dic₂₉	—	4	2	no	—
133	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
136	D₁₁₆	—	2	4	no	—
137	D₁₁₆	—	2	1	no	—
138	D₁₁₆	C₂²×C₄	1	2	yes	—
139	C₄×D₂₉	—	2	2	no	—
140	C₄×D₂₉	—	2	4	no	—
141	C₄×D₂₉	—	2	2	no	—
142	C₄×D₂₉	—	4	2	no	—
143	C₄×D₂₉	—	4	2	no	—
144	C₄×D₂₉	C₂²×C₄	1	2	yes	—
145	C₄×D₂₉	#41	1	2	yes	—
146	Dic₅₈	—	2	1	no	—
147	Dic₅₈	C₂²×C₄	1	2	yes	—
148	C₈×C₂₉	—	2	1	no	—
149	C₈×C₂₉	C₂²×C₄	1	2	yes	—
150	#6597	—	2	1	no	—
151	#6597	C₂²×C₄	1	2	yes	—
152	C₄×#135	—	29	1	no	maximal, Sylow
153	C₂²×C₂₉	—	2	2	no	—
154	C₂²×C₂₉	—	4	1	no	—
155	C₂²×C₂₉	—	4	1	no	—
156	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₂₉	—	2	1	no	—
158	C₂×D₂₉	—	2	1	no	—
159	C₂×D₂₉	—	2	2	no	—
160	C₂×D₂₉	—	4	1	no	—
161	C₂×D₂₉	—	4	1	no	—
162	C₂×D₂₉	C₄×D₄	1	2	yes	—
163	C₄×C₂₉	—	2	2	no	—
164	C₄×C₂₉	—	2	1	no	—
165	C₄×C₂₉	—	4	1	no	—
166	C₄×C₂₉	—	4	1	no	—
167	C₄×C₂₉	#135	1	2	yes	complemented
168	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
169	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
170	Dic₂₉	—	2	2	no	—
171	Dic₂₉	—	2	1	no	—
172	Dic₂₉	—	4	1	no	—
173	Dic₂₉	—	4	1	no	—
174	Dic₂₉	C₄×D₄	1	2	yes	—
175	Dic₂₉	#135	1	2	yes	complemented
176	C₂×#135	—	29	1	no	—
177	C₄×#41	—	29	1	no	—
178	C₄×#41	—	29	1	no	—
179	C₄×Q₁₆	—	29	1	no	—
180	C₄×Q₁₆	—	29	1	no	—
181	C₄×SD₁₆	—	29	1	no	—
182	C₄×SD₁₆	—	29	1	no	—
183	C₄×SD₁₆	—	29	1	no	—
184	C₄×SD₁₆	D₂₉	1	1	yes	characteristic, complemented
185	C₄×D₈	—	29	1	no	—
186	C₄×D₈	—	29	1	no	—
187	#427	—	29	1	no	—
188	#416	—	29	1	no	—
189	#416	—	29	1	no	—
190	C₂×C₄×C₈	—	29	1	no	—
191	C₂×C₂₉	—	4	2	no	—
192	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂×C₂₉	C₂×#135	1	1	yes	characteristic
194	C₂×C₂₉	C₂×#135	1	1	yes	characteristic
195	D₂₉	—	2	2	no	—
196	D₂₉	—	4	2	no	—
197	C₂×#41	—	29	1	no	—
198	C₂×#41	—	29	1	no	—
199	#135	—	29	8	no	—
200	C₂×Q₁₆	—	29	1	no	—
201	C₂×SD₁₆	—	29	1	no	—
202	C₂×SD₁₆	C₂×D₂₉	1	1	yes	characteristic
203	C₂×D₈	—	29	1	no	—
204	C₂²×C₈	—	29	1	no	—
205	C₂²×C₈	—	29	1	no	—
206	C₄×Q₈	—	29	1	no	—
207	C₄×Q₈	—	29	1	no	—
208	C₄×Q₈	—	29	1	no	—
209	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
210	C₄×D₄	—	29	1	no	—
211	C₄×D₄	—	29	1	no	—
212	C₄×D₄	—	29	1	no	—
213	C₄×D₄	—	58	1	no	—
214	C₄×D₄	—	58	1	no	—
215	C₄×D₄	—	58	1	no	—
216	C₄×D₄	—	58	1	no	—
217	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
218	#117	—	29	1	no	—
219	#117	—	29	1	no	—
220	#117	—	58	1	no	—
221	#117	—	58	1	no	—
222	C₂×C₄²	—	29	1	no	—
223	C₂×C₄²	—	58	1	no	—
224	C₂×C₄²	—	58	1	no	—
225	#107	—	29	1	no	—
226	#107	—	29	1	no	—
227	#106	—	29	1	no	—
228	#106	C₂×D₂₉	1	1	yes	characteristic
229	#103	—	29	1	no	—
230	#103	—	29	1	no	—
231	#103	—	29	1	no	—
232	#103	C₂×D₂₉	1	1	yes	characteristic
233	#102	—	29	1	no	—
234	#102	—	29	1	no	—
235	#102	—	29	1	no	—
236	#102	C₂×D₂₉	1	1	yes	characteristic
237	C₄×C₈	—	29	1	no	—
238	C₄×C₈	—	29	1	no	—
239	C₄×C₈	—	29	1	no	—
240	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
241	C₂₉	C₄×#135	1	1	yes	characteristic, complemented, Sylow
242	#41	—	29	4	no	—
243	#41	—	29	4	no	—
244	#41	—	58	2	no	—
245	#41	—	58	2	no	—
246	C₂×Q₈	—	29	1	no	—
247	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
248	C₂×D₄	—	29	1	no	—
249	C₂×D₄	—	58	1	no	—
250	C₂×D₄	—	58	1	no	—
251	C₂×D₄	C₂²×D₂₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	29	1	no	—
254	C₂²×C₄	—	29	1	no	—
255	C₂²×C₄	—	29	1	no	—
256	C₂²×C₄	—	58	1	no	—
257	C₂²×C₄	—	58	1	no	—
258	C₂²×C₄	—	58	1	no	—
259	C₂²×C₄	—	58	1	no	—
260	C₂²×C₄	—	58	1	no	—
261	Q₁₆	—	29	4	no	—
262	SD₁₆	—	29	4	no	—
263	SD₁₆	C₄×D₂₉	1	4	yes	complemented
264	D₈	—	29	4	no	—
265	C₂×C₈	—	29	1	no	—
266	C₂×C₈	—	29	4	no	—
267	C₂×C₈	—	29	1	no	—
268	C₂×C₈	—	58	2	no	—
269	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	29	1	no	—
273	C₄⋊C₄	—	29	1	no	—
274	C₄⋊C₄	—	58	1	no	—
275	C₄⋊C₄	—	58	1	no	—
276	C₄⋊C₄	—	58	1	no	—
277	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	58	1	no	—
281	C₂²⋊C₄	—	58	1	no	—
282	C₂²⋊C₄	—	58	1	no	—
283	C₂²⋊C₄	—	58	1	no	—
284	C₂²⋊C₄	—	58	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	29	1	no	—
287	C₄²	—	29	1	no	—
288	C₄²	—	29	1	no	—
289	C₄²	—	58	1	no	—
290	C₄²	—	58	1	no	—
291	C₄²	—	58	1	no	—
292	C₄²	—	58	1	no	—
293	C₄²	—	58	1	no	—
294	C₄²	C₂²×D₂₉	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	29	1	no	—
297	C₂³	—	58	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	29	2	no	—
300	Q₈	—	58	1	no	—
301	Q₈	C₂×C₄×D₂₉	1	2	yes	—
302	D₄	—	2	1	no	—
303	D₄	—	29	2	no	—
304	D₄	—	58	4	no	—
305	D₄	—	58	4	no	—
306	D₄	—	58	1	no	—
307	D₄	C₂×C₄×D₂₉	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	29	1	no	—
313	C₂×C₄	—	29	1	no	—
314	C₂×C₄	—	29	2	no	—
315	C₂×C₄	—	29	2	no	—
316	C₂×C₄	—	29	1	no	—
317	C₂×C₄	—	58	2	no	—
318	C₂×C₄	—	58	4	no	—
319	C₂×C₄	—	58	4	no	—
320	C₂×C₄	—	58	1	no	—
321	C₂×C₄	—	58	2	no	—
322	C₂×C₄	—	58	1	no	—
323	C₂×C₄	—	58	1	no	—
324	C₂×C₄	—	116	2	no	—
325	C₂×C₄	—	116	2	no	—
326	C₂×C₄	—	116	2	no	—
327	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
328	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
329	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	29	2	no	—
332	C₈	—	58	1	no	—
333	C₈	C₂×C₄×D₂₉	1	2	yes	—
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	29	2	no	—
338	C₂²	—	58	1	no	—
339	C₂²	—	58	1	no	—
340	C₂²	—	58	2	no	—
341	C₂²	—	116	1	no	—
342	C₂²	—	116	1	no	—
343	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	29	2	no	—
349	C₄	—	29	2	no	—
350	C₄	—	58	2	no	—
351	C₄	—	58	1	no	—
352	C₄	—	116	1	no	—
353	C₄	—	116	1	no	—
354	C₄	#11745278	1	2	yes	direct product factor
355	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
356	C₄	#41×D₂₉	1	1	yes	characteristic
357	C₂	—	4	2	no	—
358	C₂	—	58	2	no	—
359	C₂	—	116	2	no	—
360	C₂	C₄×D₄×D₂₉	1	1	yes	characteristic
361	C₂	#135×D₂₉	1	1	yes	characteristic
362	C₂	C₂×#11745278	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₉
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₄×SD₁₆×C₂₉	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×#11745278	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₄×SD₁₆	D₂₉
hypocenter	C₂₉	C₄×#135
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#90809	C₂	semidirect
3. C₄×#90807	C₂	semidirect
4. C₄×#90785	C₂	semidirect
5. #14271255	C₂	semidirect
6. C₄×#90773	C₂	semidirect
7. SD₁₆×Dic₂₉	C₂	semidirect
8. #14271250	C₂	semidirect
9. C₄×#90782	C₂	semidirect
10. #14271243	C₂	semidirect
11. #14271198	C₂	semidirect
12. #14271158	C₂	semidirect
13. #14271156	C₂	semidirect
14. C₄×C₈×D₂₉	C₂	semidirect
15. C₄×SD₁₆×C₂₉	C₂	semidirect
27. #11745278	C₄	direct (N × H)
67. #90809	C₂×C₄	semidirect
69. #90807	C₂×C₄	semidirect
79. SD₁₆×C₂₉	C₂×C₄	semidirect
89. #90785	C₂×C₄	semidirect
90. #90782	C₂×C₄	semidirect
110. #90773	C₂×C₄	semidirect
112. C₈×D₂₉	C₂×C₄	semidirect
167. C₄×C₂₉	#135	semidirect
175. Dic₂₉	#135	semidirect
184. C₄×SD₁₆	D₂₉	semidirect
241. C₂₉	C₄×#135	semidirect
263. SD₁₆	C₄×D₂₉	semidirect
354. C₄	#11745278	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11745278	2
C₄×#90809	2
C₄×#90807	2
C₄×#90785	2
#14271255	2
C₄×#90773	2
SD₁₆×Dic₂₉	2
#14271250	2
C₄×#90782	2
#14271243	2
#14271198	2
#14271158	2
#14271156	2
C₄×C₈×D₂₉	2
C₄×SD₁₆×C₂₉	2
C₄×#135	29

Subgroups — C4×#11745278 #16828261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11745278 #16828261