Subgroups — #28372653

Order 13952.

548 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812724	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16812673	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295277	C₂	1	2	yes	maximal, complemented
5	#20295274	C₂	1	2	yes	maximal, complemented
6	#20295260	C₂	1	2	yes	maximal, complemented
7	#20295257	C₂	1	2	yes	maximal, complemented
8	#20295069	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295030	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295027	C₂	1	1	yes	maximal, characteristic, complemented
11	#525×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₀₉	—	2	1	no	—
14	#16812724	—	2	4	no	—
15	C₂×#13182016	C₂²	1	2	yes	complemented
16	C₂×#13182014	C₂²	1	2	yes	complemented
17	C₂×#13182013	—	2	4	no	—
18	C₂×#13182013	C₂²	1	1	yes	characteristic
19	C₂×#13182013	C₂²	1	1	yes	characteristic
20	C₂×#13182013	C₂²	1	2	yes	complemented
21	C₂²×#11724784	C₂²	1	2	yes	complemented
22	#16812673	—	2	1	no	—
23	#16812673	—	2	1	no	—
24	#16812673	C₂²	1	2	yes	—
25	#16812673	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×#13182006	—	2	1	no	—
28	#16812632	C₂²	1	2	yes	complemented
29	#16812630	C₂²	1	2	yes	complemented
30	#16812623	C₂²	1	2	yes	complemented
31	#16812620	C₂²	1	2	yes	complemented
32	#16812619	C₂²	1	2	yes	complemented
33	#16812619	C₂²	1	2	yes	complemented
34	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
36	#123×C₁₀₉	—	2	1	no	—
37	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
38	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
39	#121×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
40	#121×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
41	#120×C₁₀₉	—	2	1	no	—
42	C₄×D₄×C₁₀₉	—	2	1	no	—
43	C₄×D₄×C₁₀₉	—	2	1	no	—
44	#117×C₁₀₉	C₂²	1	1	yes	characteristic
45	C₂³×D₁₀₉	—	2	1	no	—
46	C₂×D₄₃₆	—	2	2	no	—
47	C₂×D₄₃₆	—	2	2	no	—
48	C₂×D₄₃₆	—	2	1	no	—
49	C₂×D₄₃₆	C₂³	1	2	yes	—
50	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
51	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
52	C₂×C₂²⋊D₁₀₉	—	4	4	no	—
53	C₂²×Dic₁₀₉	—	2	1	no	—
54	#13182016	—	2	4	no	—
55	#13182014	—	2	4	no	—
56	#13182013	—	2	2	no	—
57	#13182013	—	2	4	no	—
58	#13182013	D₄	1	4	yes	complemented
59	C₂×#11724784	—	2	4	no	—
60	C₂×#11724784	C₂³	1	2	yes	—
61	C₂×#11724784	C₂³	1	2	yes	—
62	C₄⋊Dic₁₀₉	—	2	1	no	—
63	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
64	#13182006	—	2	2	no	—
65	#13182006	—	2	2	no	—
66	C₂⁴×C₁₀₉	D₄	1	1	yes	characteristic
67	#41×C₁₀₉	—	2	2	no	—
68	#41×C₁₀₉	—	2	2	no	—
69	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₂×D₄×C₁₀₉	—	2	1	no	—
71	C₂×D₄×C₁₀₉	—	2	1	no	—
72	C₂×D₄×C₁₀₉	—	2	1	no	—
73	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₉	—	2	2	no	—
75	C₂²×C₄×C₁₀₉	—	2	2	no	—
76	C₂²×C₄×C₁₀₉	—	2	1	no	—
77	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₀₉	—	2	1	no	—
84	C₄⋊C₄×C₁₀₉	—	2	1	no	—
85	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
88	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
89	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
90	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
91	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
92	C₄²×C₁₀₉	—	2	1	no	—
93	C₂³×C₁₀₉	—	2	2	no	—
94	C₂³×C₁₀₉	—	2	2	no	—
95	C₂³×C₁₀₉	—	2	1	no	—
96	C₂³×C₁₀₉	—	4	1	no	—
97	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×D₁₀₉	—	2	2	no	—
101	C₂²×D₁₀₉	—	4	2	no	—
102	C₂²×D₁₀₉	—	4	2	no	—
103	C₂²×D₁₀₉	—	4	2	no	—
104	Q₈×C₁₀₉	—	2	1	no	—
105	Q₈×C₁₀₉	—	2	1	no	—
106	D₄×C₁₀₉	—	2	1	no	—
107	D₄×C₁₀₉	—	2	1	no	—
108	D₄×C₁₀₉	—	4	2	no	—
109	D₄×C₁₀₉	—	4	2	no	—
110	D₄×C₁₀₉	—	4	2	no	—
111	C₂×C₄×C₁₀₉	—	2	2	no	—
112	C₂×C₄×C₁₀₉	—	2	2	no	—
113	C₂×C₄×C₁₀₉	—	2	2	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	4	2	no	—
122	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	#41	1	2	yes	—
127	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₁₀₉	—	4	4	no	—
129	C₂²⋊D₁₀₉	—	4	4	no	—
130	C₂×Dic₁₀₉	—	2	2	no	—
131	C₂×Dic₁₀₉	—	4	2	no	—
132	D₄₃₆	—	2	2	no	—
133	D₄₃₆	—	2	2	no	—
134	D₄₃₆	—	4	2	no	—
135	#11724784	—	2	2	no	—
136	#11724784	C₂×D₄	1	4	yes	complemented
137	C₂²×C₁₀₉	—	2	1	no	—
138	C₂²×C₁₀₉	—	2	1	no	—
139	C₂²×C₁₀₉	—	2	1	no	—
140	C₂²×C₁₀₉	—	2	1	no	—
141	C₂²×C₁₀₉	—	2	1	no	—
142	C₂²×C₁₀₉	—	2	1	no	—
143	C₂²×C₁₀₉	—	2	1	no	—
144	C₂²×C₁₀₉	—	4	4	no	—
145	C₂²×C₁₀₉	—	4	1	no	—
146	C₂²×C₁₀₉	—	4	1	no	—
147	C₂²×C₁₀₉	—	4	1	no	—
148	C₂²×C₁₀₉	#121	1	2	yes	—
149	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	complemented
150	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₀₉	—	4	2	no	—
152	C₂×D₁₀₉	—	4	2	no	—
153	C₂×D₁₀₉	—	4	2	no	—
154	C₂×D₁₀₉	—	8	4	no	—
155	C₄×C₁₀₉	—	2	1	no	—
156	C₄×C₁₀₉	—	2	1	no	—
157	C₄×C₁₀₉	—	4	1	no	—
158	C₄×C₁₀₉	—	4	1	no	—
159	C₄×C₁₀₉	—	4	1	no	—
160	C₄×C₁₀₉	#121	1	2	yes	—
161	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
162	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
163	Dic₁₀₉	—	4	2	no	—
164	C₂×C₁₀₉	—	2	2	no	—
165	C₂×C₁₀₉	—	2	2	no	—
166	C₂×C₁₀₉	—	4	1	no	—
167	C₂×C₁₀₉	—	8	1	no	—
168	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
169	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₁₀₉	#575	1	1	yes	characteristic
171	D₁₀₉	—	8	2	no	—
172	#3055	—	109	1	no	maximal, Sylow
173	C₁₀₉	#3055	1	1	yes	characteristic, complemented, Sylow
174	C₂×AGL(1,ℤ/8)	—	109	1	no	—
175	C₂²×M₁₆	—	109	1	no	—
176	#525	D₁₀₉	1	1	yes	characteristic, complemented
177	C₂×#121	—	109	1	no	—
178	#469	—	109	2	no	—
179	#468	—	109	2	no	—
180	#466	—	109	2	no	—
181	#465	—	109	2	no	—
182	#428	—	109	1	no	—
183	#418	—	109	1	no	—
184	#417	—	109	1	no	—
185	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
186	C₂²×D₄	—	109	1	no	—
187	C₂²×D₄	—	218	1	no	—
188	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
189	AGL(1,ℤ/8)	—	218	4	no	—
190	C₂×SD₁₆	—	109	2	no	—
191	C₂×D₈	—	109	2	no	—
192	C₂×M₁₆	—	109	1	no	—
193	C₂×M₁₆	—	109	1	no	—
194	C₂×M₁₆	—	109	2	no	—
195	C₂×M₁₆	—	218	4	no	—
196	C₂²×C₈	—	109	2	no	—
197	#123	—	2	1	no	—
198	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
199	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
200	#121	—	109	2	no	—
201	#121	—	109	2	no	—
202	#121	—	218	1	no	—
203	#121	—	218	1	no	—
204	#121	C₂×D₁₀₉	1	1	yes	characteristic, complemented
205	#121	C₂×D₁₀₉	1	1	yes	characteristic, complemented
206	#120	—	2	1	no	—
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	#117	C₂×D₁₀₉	1	1	yes	characteristic
210	C₂×C₄⋊C₄	—	109	1	no	—
211	C₂×C₂²⋊C₄	—	218	1	no	—
212	#107	—	109	2	no	—
213	#106	—	109	2	no	—
214	#103	—	109	2	no	—
215	#102	—	109	2	no	—
216	#102	—	109	2	no	—
217	#102	—	109	2	no	—
218	C₂⁴	—	218	1	no	—
219	C₂⁴	C₂²⋊D₁₀₉	1	1	yes	characteristic
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	109	2	no	—
227	C₂×D₄	—	436	4	no	—
228	C₂×D₄	—	218	2	no	—
229	C₂×D₄	—	218	2	no	—
230	C₂×D₄	—	218	2	no	—
231	C₂×D₄	—	218	1	no	—
232	C₂×D₄	—	218	2	no	—
233	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	218	1	no	—
238	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
244	SD₁₆	—	218	4	no	—
245	D₈	—	218	4	no	—
246	M₁₆	—	109	4	no	—
247	M₁₆	—	218	4	no	—
248	M₁₆	—	218	2	no	—
249	C₂×C₈	—	109	2	no	—
250	C₂×C₈	—	109	2	no	—
251	C₂×C₈	—	218	4	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	109	2	no	—
255	C₄⋊C₄	—	218	1	no	—
256	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	218	2	no	—
264	C₂²⋊C₄	—	218	2	no	—
265	C₄²	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	—	436	2	no	—
271	C₂³	—	436	2	no	—
272	C₂³	—	436	2	no	—
273	C₂³	—	218	2	no	—
274	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
275	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
276	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	436	4	no	—
285	D₄	—	436	4	no	—
286	D₄	—	436	2	no	—
287	D₄	—	218	2	no	—
288	D₄	—	218	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	436	2	no	—
301	C₂×C₄	—	218	2	no	—
302	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
303	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
304	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
306	C₂×C₄	#13182028	1	2	yes	—
307	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
308	C₈	—	109	4	no	—
309	C₈	—	218	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	872	4	no	—
322	C₂²	—	436	2	no	—
323	C₂²	—	436	2	no	—
324	C₂²	—	436	2	no	—
325	C₂²	#16812671	1	2	yes	—
326	C₂²	#16812724	1	2	yes	complemented
327	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	436	2	no	—
334	C₄	#16812671	1	2	yes	—
335	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
336	C₄	C₂×#13182028	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	2	2	no	—
339	C₂	—	4	1	no	—
340	C₂	—	8	1	no	—
341	C₂	—	872	2	no	—
342	C₂	C₂×#16812671	1	1	yes	characteristic
343	C₂	C₂×#16812724	1	1	yes	characteristic
344	C₂	#20295824	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#525×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#525	D₁₀₉
hypocenter	C₁₀₉	#3055
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#16812724	C₂	semidirect
2. C₂²×#13182013	C₂	semidirect
3. C₂×#16812673	C₂	semidirect
4. #20295277	C₂	semidirect
5. #20295274	C₂	semidirect
6. #20295260	C₂	semidirect
7. #20295257	C₂	semidirect
8. #20295069	C₂	semidirect
9. #20295030	C₂	semidirect
10. #20295027	C₂	semidirect
11. #525×C₁₀₉	C₂	semidirect
15. C₂×#13182016	C₂²	semidirect
16. C₂×#13182014	C₂²	semidirect
20. C₂×#13182013	C₂²	semidirect
21. C₂²×#11724784	C₂²	semidirect
28. #16812632	C₂²	semidirect
29. #16812630	C₂²	semidirect
30. #16812623	C₂²	semidirect
31. #16812620	C₂²	semidirect
32. #16812619	C₂²	semidirect
33. #16812619	C₂²	semidirect
37. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
38. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
39. #121×C₁₀₉	C₂²	semidirect
40. #121×C₁₀₉	C₂²	semidirect
58. #13182013	D₄	semidirect
136. #11724784	C₂×D₄	semidirect
149. C₂²×C₁₀₉	AGL(1,ℤ/8)	semidirect
173. C₁₀₉	#3055	semidirect
176. #525	D₁₀₉	semidirect
198. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
199. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
204. #121	C₂×D₁₀₉	semidirect
205. #121	C₂×D₁₀₉	semidirect
326. C₂²	#16812724	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812724	2
C₂²×#13182013	2
C₂×#16812673	2
#20295277	2
#20295274	2
#20295260	2
#20295257	2
#20295069	2
#20295030	2
#20295027	2
#525×C₁₀₉	2
#3055	109