Subgroups — #28376701

Order 13952.

1658 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812745	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295695	C₂	1	2	yes	maximal, complemented
3	#20295687	C₂	1	2	yes	maximal, complemented
4	#20295664	C₂	1	6	yes	maximal, complemented
5	D₄×C₂²⋊D₁₀₉	C₂	1	6	yes	maximal, complemented
6	#20295604	C₂	1	3	yes	maximal, complemented
7	#20295590	C₂	1	3	yes	maximal, complemented
8	#20295537	C₂	1	3	yes	maximal, complemented
9	C₂×#16812673	C₂	1	3	yes	maximal, complemented
10	#20295509	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16812745	—	2	8	no	—
13	C₂×#13182030	C₂²	1	2	yes	complemented
14	C₂×#13182029	C₂²	1	3	yes	complemented
15	C₂²×D₄₃₆	C₂²	1	3	yes	complemented
16	C₂²×C₂²⋊D₁₀₉	C₂²	1	6	yes	complemented
17	C₂×D₄×D₁₀₉	C₂²	1	6	yes	complemented
18	#16812695	C₂²	1	6	yes	complemented
19	#16812688	C₂²	1	6	yes	complemented
20	#16812682	C₂²	1	6	yes	complemented
21	#16812678	C₂²	1	1	yes	characteristic, complemented
22	#16812678	C₂²	1	3	yes	complemented
23	#16812675	C₂²	1	6	yes	complemented
24	#16812673	C₂²	1	6	yes	complemented
25	#16812673	C₂²	1	6	yes	complemented
26	#16812673	C₂²	1	3	yes	complemented
27	#16812673	C₂²	1	6	yes	complemented
28	#16812673	C₂²	1	3	yes	complemented
29	#16812671	C₂²	1	12	yes	complemented
30	#16812670	C₂²	1	6	yes	complemented
31	#16812668	C₂²	1	6	yes	complemented
32	#16812667	C₂²	1	12	yes	complemented
33	Q₈×Dic₁₀₉	C₂²	1	2	yes	complemented
34	D₄×Dic₁₀₉	C₂²	1	6	yes	complemented
35	C₄×C₂²⋊D₁₀₉	C₂²	1	6	yes	complemented
36	C₄×C₂²⋊D₁₀₉	C₂²	1	6	yes	complemented
37	#16812658	C₂²	1	3	yes	complemented
38	C₂×C₄⋊Dic₁₀₉	C₂²	1	3	yes	complemented
39	C₂×#13182006	C₂²	1	6	yes	complemented
40	C₂×#41×C₁₀₉	—	2	8	no	—
41	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
43	C₂×#41×C₁₀₉	C₂²	1	3	yes	complemented
44	C₂×#41×C₁₀₉	C₂²	1	3	yes	complemented
45	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
46	C₂²×D₄×C₁₀₉	C₂²	1	3	yes	complemented
47	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	complemented
48	C₂³×D₁₀₉	C₂³	1	6	yes	—
49	#13182030	—	2	8	no	—
50	#13182029	—	2	12	no	—
51	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
52	C₂×D₄₃₆	—	2	12	no	—
53	C₂×D₄₃₆	C₂³	1	3	yes	—
54	C₂×D₄₃₆	C₂³	1	6	yes	—
55	C₂×C₂²⋊D₁₀₉	—	4	12	no	—
56	C₂×C₂²⋊D₁₀₉	C₂³	1	6	yes	—
57	C₂×C₂²⋊D₁₀₉	C₂³	1	6	yes	—
58	C₂×C₂²⋊D₁₀₉	C₂³	1	6	yes	—
59	C₂×C₂²⋊D₁₀₉	C₂³	1	12	yes	—
60	D₄×D₁₀₉	—	2	24	no	—
61	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
62	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
63	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
64	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
65	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
66	#13182008	C₂³	1	2	yes	—
67	#13182008	C₂³	1	6	yes	—
68	C₂²⋊Dic₁₀₉	C₂³	1	6	yes	—
69	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
70	C₂²⋊Dic₁₀₉	C₂³	1	6	yes	—
71	#13182006	C₂³	1	6	yes	—
72	#13182006	C₂³	1	12	yes	—
73	#13182006	C₂³	1	6	yes	—
74	C₄×Dic₁₀₉	C₂³	1	6	yes	—
75	C₂⁴×C₁₀₉	C₂³	1	3	yes	—
76	#41×C₁₀₉	—	2	4	no	—
77	#41×C₁₀₉	—	2	12	no	—
78	#41×C₁₀₉	—	2	12	no	—
79	#41×C₁₀₉	D₄	1	8	yes	complemented
80	C₂×Q₈×C₁₀₉	—	2	4	no	—
81	C₂×Q₈×C₁₀₉	C₂³	1	3	yes	—
82	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₂×D₄×C₁₀₉	—	2	12	no	—
84	C₂×D₄×C₁₀₉	C₂³	1	6	yes	—
85	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
86	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
87	C₂²×C₄×C₁₀₉	—	2	12	no	—
88	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
90	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
91	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
92	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
93	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
94	C₂³×C₁₀₉	—	2	6	no	—
95	C₂³×C₁₀₉	—	2	6	no	—
96	C₂³×C₁₀₉	—	4	3	no	—
97	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
98	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
99	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
100	C₂²×D₁₀₉	—	2	12	no	—
101	C₂²×D₁₀₉	—	4	6	no	—
102	C₂²×D₁₀₉	—	4	6	no	—
103	C₂²×D₁₀₉	C₂⁴	1	6	yes	—
104	Q₈×C₁₀₉	—	2	6	no	—
105	Q₈×C₁₀₉	C₂×D₄	1	4	yes	complemented
106	D₄×C₁₀₉	—	2	12	no	—
107	D₄×C₁₀₉	—	2	6	no	—
108	D₄×C₁₀₉	C₂×D₄	1	12	yes	complemented
109	C₂×C₄×C₁₀₉	—	2	6	no	—
110	C₂×C₄×C₁₀₉	—	2	2	no	—
111	C₂×C₄×C₁₀₉	—	2	6	no	—
112	C₂×C₄×C₁₀₉	—	2	6	no	—
113	C₂×C₄×C₁₀₉	—	2	6	no	—
114	C₂×C₄×C₁₀₉	C₂×D₄	1	12	yes	complemented
115	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
117	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
119	C₂²⋊D₁₀₉	—	2	12	no	—
120	C₂²⋊D₁₀₉	—	2	12	no	—
121	C₂²⋊D₁₀₉	—	4	12	no	—
122	C₂²⋊D₁₀₉	—	4	6	no	—
123	C₂×Dic₁₀₉	—	4	6	no	—
124	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
125	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
126	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
127	D₄₃₆	—	2	12	no	—
128	D₄₃₆	—	2	6	no	—
129	C₄×D₁₀₉	—	2	12	no	—
130	Dic₂₁₈	—	2	2	no	—
131	C₂²×C₁₀₉	—	2	1	no	—
132	C₂²×C₁₀₉	—	2	3	no	—
133	C₂²×C₁₀₉	—	2	3	no	—
134	C₂²×C₁₀₉	—	2	1	no	—
135	C₂²×C₁₀₉	—	2	3	no	—
136	C₂²×C₁₀₉	—	2	3	no	—
137	C₂²×C₁₀₉	—	2	3	no	—
138	C₂²×C₁₀₉	—	4	12	no	—
139	C₂²×C₁₀₉	C₂²×D₄	1	6	yes	—
140	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	complemented
141	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
142	C₂×D₁₀₉	—	2	6	no	—
143	C₂×D₁₀₉	—	2	6	no	—
144	C₂×D₁₀₉	—	2	6	no	—
145	C₂×D₁₀₉	—	8	12	no	—
146	C₄×C₁₀₉	—	2	1	no	—
147	C₄×C₁₀₉	—	2	3	no	—
148	C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
149	C₄×C₁₀₉	C₂²×D₄	1	6	yes	—
150	Dic₁₀₉	—	2	2	no	—
151	Dic₁₀₉	—	2	6	no	—
152	Dic₁₀₉	—	2	2	no	—
153	C₂×C₁₀₉	—	2	6	no	—
154	C₂×C₁₀₉	—	2	2	no	—
155	C₂×C₁₀₉	—	4	3	no	—
156	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
157	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₁₀₉	#585	1	1	yes	characteristic
159	D₁₀₉	—	4	6	no	—
160	#3488	—	109	1	no	maximal, Sylow
161	C₁₀₉	#3488	1	1	yes	characteristic, complemented, Sylow
162	C₂×H(5,𝔽₂)	—	109	1	no	—
163	C₂²×#41	D₁₀₉	1	1	yes	characteristic, complemented
164	#550	—	109	2	no	—
165	#548	—	109	2	no	—
166	#546	—	109	6	no	—
167	D₄²	—	109	6	no	—
168	#537	—	109	3	no	—
169	#535	—	109	3	no	—
170	#525	—	109	3	no	—
171	C₂×#121	—	109	3	no	—
172	#520	—	109	1	no	—
173	H(5,𝔽₂)	—	218	8	no	—
174	C₂×#41	—	2	8	no	—
175	C₂×#41	—	109	2	no	—
176	C₂×#41	—	109	3	no	—
177	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
178	C₂×#41	C₂×D₁₀₉	1	3	yes	complemented
179	C₂×#41	C₂×D₁₀₉	1	3	yes	complemented
180	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic, complemented
181	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
182	C₂²×D₄	—	109	6	no	—
183	C₂²×D₄	—	109	6	no	—
184	C₂²×D₄	—	109	3	no	—
185	C₂²×D₄	C₂×D₁₀₉	1	3	yes	complemented
186	C₂³×C₄	C₂×D₁₀₉	1	3	yes	complemented
187	#127	—	109	6	no	—
188	#124	—	109	6	no	—
189	#123	—	109	6	no	—
190	C₂²⋊Q₈	—	109	6	no	—
191	C₂²⋊Q₈	—	109	1	no	—
192	C₂²⋊Q₈	—	109	3	no	—
193	#121	—	109	12	no	—
194	#121	—	109	6	no	—
195	#121	—	109	3	no	—
196	#121	—	109	6	no	—
197	#121	—	109	6	no	—
198	#121	—	109	6	no	—
199	#121	—	109	3	no	—
200	#120	—	109	6	no	—
201	#120	—	109	12	no	—
202	C₄×Q₈	—	109	2	no	—
203	C₄×D₄	—	109	6	no	—
204	C₄×D₄	—	109	6	no	—
205	C₄×D₄	—	109	6	no	—
206	#117	—	109	3	no	—
207	C₂×C₄⋊C₄	—	109	3	no	—
208	C₂×C₂²⋊C₄	—	109	6	no	—
209	C₂⁴	—	109	6	no	—
210	C₂⁴	C₂²×D₁₀₉	1	3	yes	—
211	#41	—	2	4	no	—
212	#41	—	2	12	no	—
213	#41	—	2	12	no	—
214	#41	—	218	8	no	—
215	#41	—	218	12	no	—
216	#41	C₂²⋊D₁₀₉	1	8	yes	complemented
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	109	1	no	—
219	C₂×Q₈	C₂²×D₁₀₉	1	3	yes	—
220	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
221	C₂×D₄	—	2	12	no	—
222	C₂×D₄	—	109	6	no	—
223	C₂×D₄	—	109	12	no	—
224	C₂×D₄	—	109	6	no	—
225	C₂×D₄	—	109	6	no	—
226	C₂×D₄	—	109	3	no	—
227	C₂×D₄	—	109	6	no	—
228	C₂×D₄	—	436	12	no	—
229	C₂×D₄	—	218	24	no	—
230	C₂×D₄	—	218	12	no	—
231	C₂×D₄	C₂²×D₁₀₉	1	6	yes	—
232	C₂×D₄	C₂²×D₁₀₉	1	3	yes	—
233	C₂×D₄	C₂²×D₁₀₉	1	3	yes	—
234	C₂²×C₄	—	2	12	no	—
235	C₂²×C₄	—	109	6	no	—
236	C₂²×C₄	—	109	6	no	—
237	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
238	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
239	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
240	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
242	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
243	C₄⋊C₄	—	109	2	no	—
244	C₄⋊C₄	—	109	6	no	—
245	C₄⋊C₄	—	109	3	no	—
246	C₄⋊C₄	—	109	6	no	—
247	C₄⋊C₄	—	109	1	no	—
248	C₂²⋊C₄	—	109	6	no	—
249	C₂²⋊C₄	—	109	2	no	—
250	C₂²⋊C₄	—	109	6	no	—
251	C₂²⋊C₄	—	109	6	no	—
252	C₂²⋊C₄	—	109	12	no	—
253	C₂²⋊C₄	—	109	6	no	—
254	C₄²	—	109	6	no	—
255	C₂³	—	2	6	no	—
256	C₂³	—	2	6	no	—
257	C₂³	—	4	3	no	—
258	C₂³	—	109	6	no	—
259	C₂³	—	436	6	no	—
260	C₂³	—	436	6	no	—
261	C₂³	—	218	12	no	—
262	C₂³	C₂³×D₁₀₉	1	3	yes	—
263	C₂³	C₂³×D₁₀₉	1	3	yes	—
264	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
265	Q₈	—	2	6	no	—
266	Q₈	—	218	2	no	—
267	Q₈	C₂×C₂²⋊D₁₀₉	1	4	yes	complemented
268	D₄	—	2	12	no	—
269	D₄	—	2	6	no	—
270	D₄	—	436	12	no	—
271	D₄	—	436	6	no	—
272	D₄	—	218	12	no	—
273	D₄	—	218	12	no	—
274	D₄	—	218	6	no	—
275	D₄	—	218	12	no	—
276	D₄	C₂×C₂²⋊D₁₀₉	1	12	yes	complemented
277	C₂×C₄	—	2	6	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	6	no	—
280	C₂×C₄	—	2	6	no	—
281	C₂×C₄	—	2	6	no	—
282	C₂×C₄	—	109	2	no	—
283	C₂×C₄	—	109	6	no	—
284	C₂×C₄	—	109	2	no	—
285	C₂×C₄	—	436	6	no	—
286	C₂×C₄	—	218	12	no	—
287	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	12	yes	complemented
288	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
290	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
291	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	3	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	3	no	—
296	C₂²	—	2	3	no	—
297	C₂²	—	2	3	no	—
298	C₂²	—	2	3	no	—
299	C₂²	—	4	12	no	—
300	C₂²	—	872	12	no	—
301	C₂²	—	218	6	no	—
302	C₂²	—	218	6	no	—
303	C₂²	—	218	6	no	—
304	C₂²	C₂²×C₂²⋊D₁₀₉	1	6	yes	—
305	C₂²	#16812745	1	2	yes	complemented
306	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	3	no	—
309	C₄	—	218	2	no	—
310	C₄	—	218	6	no	—
311	C₄	—	218	2	no	—
312	C₄	C₂²×C₂²⋊D₁₀₉	1	2	yes	—
313	C₄	C₂²×C₂²⋊D₁₀₉	1	6	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	2	6	no	—
316	C₂	—	4	3	no	—
317	C₂	—	436	6	no	—
318	C₂	C₂³×C₂²⋊D₁₀₉	1	1	yes	characteristic
319	C₂	C₂×#16812745	1	1	yes	characteristic
320	C₂	#20295862	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	C₂²×#41×C₁₀₉	C₂
hypercenter	C₂²×#41	D₁₀₉
hypocenter	C₁₀₉	#3488
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#16812745	C₂	semidirect
2. #20295695	C₂	semidirect
3. #20295687	C₂	semidirect
4. #20295664	C₂	semidirect
5. D₄×C₂²⋊D₁₀₉	C₂	semidirect
6. #20295604	C₂	semidirect
7. #20295590	C₂	semidirect
8. #20295537	C₂	semidirect
9. C₂×#16812673	C₂	semidirect
10. #20295509	C₂	semidirect
11. C₂²×#41×C₁₀₉	C₂	semidirect
13. C₂×#13182030	C₂²	semidirect
14. C₂×#13182029	C₂²	semidirect
15. C₂²×D₄₃₆	C₂²	semidirect
16. C₂²×C₂²⋊D₁₀₉	C₂²	semidirect
17. C₂×D₄×D₁₀₉	C₂²	semidirect
18. #16812695	C₂²	semidirect
19. #16812688	C₂²	semidirect
20. #16812682	C₂²	semidirect
21. #16812678	C₂²	semidirect
22. #16812678	C₂²	semidirect
23. #16812675	C₂²	semidirect
24. #16812673	C₂²	semidirect
25. #16812673	C₂²	semidirect
26. #16812673	C₂²	semidirect
27. #16812673	C₂²	semidirect
28. #16812673	C₂²	semidirect
29. #16812671	C₂²	semidirect
30. #16812670	C₂²	semidirect
31. #16812668	C₂²	semidirect
32. #16812667	C₂²	semidirect
33. Q₈×Dic₁₀₉	C₂²	semidirect
34. D₄×Dic₁₀₉	C₂²	semidirect
35. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
36. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
37. #16812658	C₂²	semidirect
38. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
39. C₂×#13182006	C₂²	semidirect
42. C₂×#41×C₁₀₉	C₂²	semidirect
43. C₂×#41×C₁₀₉	C₂²	semidirect
44. C₂×#41×C₁₀₉	C₂²	semidirect
45. C₂²×Q₈×C₁₀₉	C₂²	semidirect
46. C₂²×D₄×C₁₀₉	C₂²	semidirect
47. C₂³×C₄×C₁₀₉	C₂²	semidirect
79. #41×C₁₀₉	D₄	semidirect
105. Q₈×C₁₀₉	C₂×D₄	semidirect
108. D₄×C₁₀₉	C₂×D₄	semidirect
114. C₂×C₄×C₁₀₉	C₂×D₄	semidirect
140. C₂²×C₁₀₉	H(5,𝔽₂)	semidirect
161. C₁₀₉	#3488	semidirect
163. C₂²×#41	D₁₀₉	semidirect
178. C₂×#41	C₂×D₁₀₉	semidirect
179. C₂×#41	C₂×D₁₀₉	semidirect
180. C₂×#41	C₂×D₁₀₉	semidirect
181. C₂²×Q₈	C₂×D₁₀₉	semidirect
185. C₂²×D₄	C₂×D₁₀₉	semidirect
186. C₂³×C₄	C₂×D₁₀₉	semidirect
216. #41	C₂²⋊D₁₀₉	semidirect
267. Q₈	C₂×C₂²⋊D₁₀₉	semidirect
276. D₄	C₂×C₂²⋊D₁₀₉	semidirect
287. C₂×C₄	C₂×C₂²⋊D₁₀₉	semidirect
305. C₂²	#16812745	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812745	2
#20295695	2
#20295687	2
#20295664	2
D₄×C₂²⋊D₁₀₉	2
#20295604	2
#20295590	2
#20295537	2
C₂×#16812673	2
#20295509	2
C₂²×#41×C₁₀₉	2
#3488	109