Subgroups — #28377801

Order 13952.

1424 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295862	C₂	1	2	yes	maximal, complemented
2	#20295840	C₂	1	2	yes	maximal, complemented
3	#20295839	C₂	1	2	yes	maximal, complemented
4	#20295833	C₂	1	4	yes	maximal, complemented
5	#20295832	C₂	1	4	yes	maximal, complemented
6	#20295828	C₂	1	2	yes	maximal, complemented
7	#20295827	C₂	1	2	yes	maximal, complemented
8	#20295824	C₂	1	2	yes	maximal, complemented
9	#20295823	C₂	1	4	yes	maximal, complemented
10	C₂×#16812721	C₂	1	1	yes	maximal, characteristic, complemented
11	#135×D₁₀₉	C₂	1	4	yes	maximal, complemented
12	C₂×#16812705	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#135×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#16812747	C₂²	1	4	yes	complemented
15	#16812746	—	2	2	no	—
16	#16812746	C₂²	1	4	yes	complemented
17	#16812745	—	2	2	no	—
18	#16812745	C₂²	1	4	yes	complemented
19	#16812744	—	2	4	no	—
20	#16812744	C₂²	1	4	yes	complemented
21	C₂×#13182029	—	2	4	no	—
22	C₂×#13182029	C₂²	1	1	yes	characteristic
23	C₂×#13182029	C₂²	1	2	yes	complemented
24	#41×D₁₀₉	—	2	4	no	—
25	#41×D₁₀₉	C₂²	1	8	yes	complemented
26	#16812731	C₂²	1	8	yes	complemented
27	#16812730	C₂²	1	4	yes	complemented
28	#16812729	C₂²	1	4	yes	complemented
29	#16812728	C₂²	1	8	yes	complemented
30	#16812726	C₂²	1	8	yes	complemented
31	#16812725	C₂²	1	4	yes	complemented
32	#16812724	C₂²	1	4	yes	complemented
33	#16812723	C₂²	1	8	yes	complemented
34	#16812722	C₂²	1	4	yes	complemented
35	#16812721	C₂²	1	2	yes	complemented
36	#16812721	C₂²	1	2	yes	complemented
37	#16812721	C₂²	1	2	yes	complemented
38	#16812721	C₂²	1	2	yes	complemented
39	#16812720	C₂²	1	8	yes	complemented
40	#16812719	C₂²	1	8	yes	complemented
41	#16812718	C₂²	1	4	yes	complemented
42	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic, complemented
43	Q₁₆×D₁₀₉	C₂²	1	4	yes	complemented
44	C₂×#13182018	C₂²	1	2	yes	complemented
45	SD₁₆×D₁₀₉	C₂²	1	8	yes	complemented
46	C₂×D₈₇₂	C₂²	1	1	yes	characteristic, complemented
47	D₈×D₁₀₉	C₂²	1	4	yes	complemented
48	#16812705	C₂²	1	2	yes	complemented
49	#16812705	C₂²	1	2	yes	complemented
50	#16812705	C₂²	1	2	yes	complemented
51	#16812705	C₂²	1	2	yes	complemented
52	C₂×#13182013	C₂²	1	1	yes	characteristic, complemented
53	C₂×#13182012	C₂²	1	2	yes	complemented
54	C₂×C₈×D₁₀₉	C₂²	1	2	yes	complemented
55	C₂×#41×C₁₀₉	C₂²	1	2	yes	complemented
56	#135×C₁₀₉	C₂²	1	2	yes	complemented
57	#135×C₁₀₉	C₂²	1	2	yes	complemented
58	#135×C₁₀₉	C₂²	1	2	yes	complemented
59	#135×C₁₀₉	C₂²	1	2	yes	complemented
60	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
61	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	complemented
62	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
63	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
64	#13182030	—	2	4	no	—
65	#13182030	C₂³	1	8	yes	—
66	#13182029	—	2	2	no	—
67	#13182029	—	2	2	no	—
68	#13182029	—	2	8	no	—
69	#13182029	—	2	4	no	—
70	#13182029	—	2	8	no	—
71	#13182029	—	2	4	no	—
72	#13182029	D₄	1	2	yes	—
73	#13182029	D₄	1	2	yes	—
74	#13182029	C₂³	1	2	yes	—
75	#13182029	C₂³	1	2	yes	—
76	#13182029	C₂³	1	4	yes	—
77	#13182029	C₂³	1	4	yes	—
78	#13182028	—	2	8	no	—
79	#13182028	—	2	8	no	—
80	#13182028	—	2	4	no	—
81	#13182028	C₂³	1	8	yes	—
82	C₂×Dic₂₁₈	—	2	4	no	—
83	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
84	C₂×Dic₂₁₈	C₂³	1	2	yes	—
85	Q₈×D₁₀₉	—	2	4	no	—
86	Q₈×D₁₀₉	C₂³	1	8	yes	—
87	C₂×D₄₃₆	—	2	4	no	—
88	C₂×D₄₃₆	D₄	1	1	yes	characteristic
89	C₂×D₄₃₆	C₂³	1	2	yes	—
90	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
91	C₂×C₂²⋊D₁₀₉	—	2	8	no	—
92	C₂×C₂²⋊D₁₀₉	D₄	1	2	yes	—
93	D₄×D₁₀₉	—	2	8	no	—
94	D₄×D₁₀₉	—	2	8	no	—
95	D₄×D₁₀₉	—	2	4	no	—
96	D₄×D₁₀₉	C₂³	1	8	yes	—
97	C₂×C₄×D₁₀₉	—	2	2	no	—
98	C₂×C₄×D₁₀₉	—	2	8	no	—
99	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
100	Dic₄₃₆	C₂³	1	2	yes	—
101	Dic₄₃₆	C₂³	1	2	yes	—
102	#13182019	C₂³	1	8	yes	—
103	#13182018	C₂³	1	4	yes	—
104	#13182018	C₂³	1	4	yes	—
105	#13182017	C₂³	1	8	yes	—
106	#13182016	C₂³	1	8	yes	—
107	D₈₇₂	C₂³	1	2	yes	—
108	D₈₇₂	C₂³	1	2	yes	—
109	#13182014	C₂³	1	8	yes	—
110	#13182013	C₂³	1	2	yes	—
111	#13182013	C₂³	1	2	yes	—
112	#13182012	C₂³	1	4	yes	—
113	#13182012	C₂³	1	4	yes	—
114	C₂×#11724784	C₂³	1	2	yes	—
115	C₈×D₁₀₉	C₂³	1	4	yes	—
116	C₈×D₁₀₉	C₂³	1	4	yes	—
117	#41×C₁₀₉	—	2	2	no	—
118	#41×C₁₀₉	—	2	2	no	—
119	#41×C₁₀₉	C₂³	1	4	yes	—
120	#41×C₁₀₉	C₂³	1	4	yes	—
121	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
122	C₂×D₄×C₁₀₉	—	2	2	no	—
123	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
124	C₂²×C₄×C₁₀₉	—	2	2	no	—
125	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
126	Q₁₆×C₁₀₉	C₂³	1	2	yes	—
127	Q₁₆×C₁₀₉	C₂³	1	2	yes	—
128	SD₁₆×C₁₀₉	C₂³	1	4	yes	—
129	SD₁₆×C₁₀₉	C₂³	1	4	yes	—
130	D₈×C₁₀₉	C₂³	1	2	yes	—
131	D₈×C₁₀₉	C₂³	1	2	yes	—
132	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
133	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
134	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
135	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
136	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
137	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
138	C₂³×C₁₀₉	—	2	2	no	—
139	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
140	C₂²×D₁₀₉	—	2	2	no	—
141	C₂²×D₁₀₉	—	2	8	no	—
142	C₂²×D₁₀₉	C₂×D₄	1	2	yes	—
143	Q₈×C₁₀₉	—	2	2	no	—
144	Q₈×C₁₀₉	C₂⁴	1	4	yes	—
145	D₄×C₁₀₉	—	2	4	no	—
146	D₄×C₁₀₉	—	2	4	no	—
147	D₄×C₁₀₉	—	2	2	no	—
148	D₄×C₁₀₉	C₂⁴	1	4	yes	—
149	C₂×C₄×C₁₀₉	—	2	2	no	—
150	C₂×C₄×C₁₀₉	—	2	4	no	—
151	C₂×C₄×C₁₀₉	—	2	4	no	—
152	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
156	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
157	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
158	C₂²⋊D₁₀₉	—	2	4	no	—
159	C₂²⋊D₁₀₉	—	2	4	no	—
160	C₂²⋊D₁₀₉	—	2	8	no	—
161	C₂²⋊D₁₀₉	—	2	8	no	—
162	C₂²⋊D₁₀₉	C₂×D₄	1	4	yes	—
163	C₂²⋊D₁₀₉	C₂×D₄	1	4	yes	—
164	C₂×Dic₁₀₉	—	2	2	no	—
165	C₂×Dic₁₀₉	—	2	8	no	—
166	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
167	D₄₃₆	—	2	8	no	—
168	D₄₃₆	—	2	2	no	—
169	D₄₃₆	C₂×D₄	1	2	yes	—
170	D₄₃₆	C₂×D₄	1	2	yes	—
171	D₄₃₆	C₂⁴	1	4	yes	—
172	C₄×D₁₀₉	—	2	8	no	—
173	C₄×D₁₀₉	—	2	4	no	—
174	C₄×D₁₀₉	—	2	4	no	—
175	C₄×D₁₀₉	—	2	8	no	—
176	C₄×D₁₀₉	C₂×D₄	1	4	yes	—
177	C₄×D₁₀₉	C₂⁴	1	4	yes	—
178	Dic₂₁₈	—	2	2	no	—
179	Dic₂₁₈	—	2	8	no	—
180	Dic₂₁₈	C₂×D₄	1	2	yes	—
181	Dic₂₁₈	C₂×D₄	1	2	yes	—
182	Dic₂₁₈	C₂⁴	1	4	yes	—
183	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
184	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
185	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
186	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
187	#11724784	C₂⁴	1	4	yes	—
188	C₂²×C₁₀₉	—	2	4	no	—
189	C₂²×C₁₀₉	—	4	1	no	—
190	C₂²×C₁₀₉	—	8	2	no	—
191	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
192	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
193	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
194	C₂×D₁₀₉	—	2	4	no	—
195	C₂×D₁₀₉	—	4	2	no	—
196	C₂×D₁₀₉	—	8	8	no	—
197	C₂×D₁₀₉	—	8	2	no	—
198	C₂×D₁₀₉	C₂²×D₄	1	4	yes	—
199	C₄×C₁₀₉	—	2	4	no	—
200	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
201	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
202	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
203	C₄×C₁₀₉	C₂⁵	1	1	yes	characteristic
204	Dic₁₀₉	—	2	4	no	—
205	Dic₁₀₉	C₂²×D₄	1	4	yes	—
206	C₂×C₁₀₉	—	2	1	no	—
207	C₂×C₁₀₉	—	2	1	no	—
208	C₂×C₁₀₉	—	2	1	no	—
209	C₂×C₁₀₉	—	4	4	no	—
210	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
211	D₁₀₉	—	2	4	no	—
212	D₁₀₉	—	4	4	no	—
213	#3584	—	109	1	no	maximal, Sylow
214	C₁₀₉	#3584	1	1	yes	characteristic, complemented, Sylow
215	#585	—	109	2	no	—
216	#578	—	109	2	no	—
217	#578	—	109	2	no	—
218	#577	—	109	4	no	—
219	#577	—	109	4	no	—
220	#576	—	109	2	no	—
221	#576	—	109	2	no	—
222	#575	—	109	2	no	—
223	#575	—	109	4	no	—
224	C₂×#135	—	109	4	no	—
225	C₂×#135	—	109	1	no	—
226	C₂×#135	D₁₀₉	1	1	yes	characteristic, complemented
227	C₂×#131	—	109	1	no	—
228	2₋¹⁺⁴	—	109	4	no	—
229	2₋¹⁺⁴	—	109	4	no	—
230	2₋¹⁺⁴	—	218	2	no	—
231	H(5,𝔽₂)	—	109	4	no	—
232	H(5,𝔽₂)	—	109	4	no	—
233	H(5,𝔽₂)	—	218	4	no	—
234	H(5,𝔽₂)	—	218	2	no	—
235	C₂×#41	—	109	1	no	—
236	C₂×#41	—	109	8	no	—
237	C₂×#41	—	109	2	no	—
238	C₂×#41	—	218	4	no	—
239	C₂×#41	—	218	4	no	—
240	C₂×#41	C₂×D₁₀₉	1	2	yes	complemented
241	#137	—	109	4	no	—
242	#137	—	109	4	no	—
243	#137	—	109	8	no	—
244	#137	—	109	8	no	—
245	AGL(1,ℤ/8)	—	109	4	no	—
246	AGL(1,ℤ/8)	—	109	4	no	—
247	AGL(1,ℤ/8)	—	109	8	no	—
248	AGL(1,ℤ/8)	—	109	8	no	—
249	#135	—	109	8	no	—
250	#135	—	109	4	no	—
251	#135	—	109	8	no	—
252	#135	—	109	4	no	—
253	#135	—	109	2	no	—
254	#135	—	109	2	no	—
255	#135	—	109	2	no	—
256	#135	—	109	2	no	—
257	#135	C₂×D₁₀₉	1	2	yes	complemented
258	#135	C₂×D₁₀₉	1	2	yes	complemented
259	#135	C₂×D₁₀₉	1	2	yes	complemented
260	#135	C₂×D₁₀₉	1	2	yes	complemented
261	C₂×Q₁₆	—	109	4	no	—
262	C₂×Q₁₆	—	109	1	no	—
263	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic, complemented
264	C₂×SD₁₆	—	109	8	no	—
265	C₂×SD₁₆	—	109	2	no	—
266	C₂×SD₁₆	C₂×D₁₀₉	1	2	yes	complemented
267	C₂×D₈	—	109	4	no	—
268	C₂×D₈	—	109	1	no	—
269	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
270	#131	—	109	2	no	—
271	#131	—	109	2	no	—
272	#131	—	109	2	no	—
273	#131	—	109	2	no	—
274	C₂×M₁₆	—	109	1	no	—
275	C₂×M₁₆	—	109	2	no	—
276	C₂²×C₈	—	109	2	no	—
277	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
278	#41	—	2	2	no	—
279	#41	—	2	2	no	—
280	#41	—	109	2	no	—
281	#41	—	109	2	no	—
282	#41	—	109	2	no	—
283	#41	—	109	2	no	—
284	#41	—	109	8	no	—
285	#41	—	109	8	no	—
286	#41	—	109	4	no	—
287	#41	—	109	4	no	—
288	#41	—	218	8	no	—
289	#41	—	218	8	no	—
290	#41	—	218	4	no	—
291	#41	—	218	4	no	—
292	#41	—	218	8	no	—
293	#41	—	218	8	no	—
294	#41	—	218	4	no	—
295	#41	—	218	4	no	—
296	#41	—	218	2	no	—
297	#41	—	218	2	no	—
298	#41	C₂²×D₁₀₉	1	4	yes	—
299	#41	C₂²×D₁₀₉	1	4	yes	—
300	C₂×Q₈	—	109	1	no	—
301	C₂×Q₈	—	109	8	no	—
302	C₂×Q₈	—	109	2	no	—
303	C₂×Q₈	—	218	4	no	—
304	C₂×Q₈	—	218	4	no	—
305	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	109	2	no	—
308	C₂×D₄	—	109	8	no	—
309	C₂×D₄	—	109	1	no	—
310	C₂×D₄	—	109	2	no	—
311	C₂×D₄	—	218	4	no	—
312	C₂×D₄	—	218	8	no	—
313	C₂×D₄	—	218	4	no	—
314	C₂×D₄	—	218	8	no	—
315	C₂×D₄	—	218	8	no	—
316	C₂×D₄	—	218	2	no	—
317	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	109	2	no	—
320	C₂²×C₄	—	218	8	no	—
321	C₂²×C₄	—	218	2	no	—
322	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
323	Q₁₆	—	109	8	no	—
324	Q₁₆	—	109	2	no	—
325	Q₁₆	—	109	2	no	—
326	Q₁₆	C₂²×D₁₀₉	1	2	yes	—
327	Q₁₆	C₂²×D₁₀₉	1	2	yes	—
328	SD₁₆	—	109	8	no	—
329	SD₁₆	—	109	8	no	—
330	SD₁₆	—	109	4	no	—
331	SD₁₆	—	109	4	no	—
332	SD₁₆	C₂²×D₁₀₉	1	4	yes	—
333	SD₁₆	C₂²×D₁₀₉	1	4	yes	—
334	D₈	—	109	8	no	—
335	D₈	—	109	2	no	—
336	D₈	—	109	2	no	—
337	D₈	C₂²×D₁₀₉	1	2	yes	—
338	D₈	C₂²×D₁₀₉	1	2	yes	—
339	M₁₆	—	109	2	no	—
340	M₁₆	—	109	2	no	—
341	M₁₆	—	109	4	no	—
342	M₁₆	—	109	4	no	—
343	C₂×C₈	—	109	2	no	—
344	C₂×C₈	—	109	4	no	—
345	C₂×C₈	—	109	4	no	—
346	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
347	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
348	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
349	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
350	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
351	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
352	C₂³	—	2	2	no	—
353	C₂³	—	109	2	no	—
354	C₂³	—	218	8	no	—
355	C₂³	—	218	2	no	—
356	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
357	Q₈	—	2	2	no	—
358	Q₈	—	109	2	no	—
359	Q₈	—	109	2	no	—
360	Q₈	—	109	4	no	—
361	Q₈	—	218	8	no	—
362	Q₈	—	218	2	no	—
363	Q₈	C₂³×D₁₀₉	1	4	yes	—
364	D₄	—	2	4	no	—
365	D₄	—	2	4	no	—
366	D₄	—	2	2	no	—
367	D₄	—	109	4	no	—
368	D₄	—	109	4	no	—
369	D₄	—	109	2	no	—
370	D₄	—	109	2	no	—
371	D₄	—	109	4	no	—
372	D₄	—	218	8	no	—
373	D₄	—	218	8	no	—
374	D₄	—	218	8	no	—
375	D₄	—	218	4	no	—
376	D₄	—	218	4	no	—
377	D₄	—	218	2	no	—
378	D₄	C₂³×D₁₀₉	1	4	yes	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	109	2	no	—
383	C₂×C₄	—	109	4	no	—
384	C₂×C₄	—	109	4	no	—
385	C₂×C₄	—	218	8	no	—
386	C₂×C₄	—	218	8	no	—
387	C₂×C₄	—	218	8	no	—
388	C₂×C₄	—	218	2	no	—
389	C₂×C₄	—	218	4	no	—
390	C₂×C₄	—	218	4	no	—
391	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
392	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
393	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
394	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
395	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
396	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
397	C₈	—	109	4	no	—
398	C₈	C₂³×D₁₀₉	1	1	yes	characteristic
399	C₈	C₂³×D₁₀₉	1	1	yes	characteristic
400	C₈	C₂³×D₁₀₉	1	1	yes	characteristic
401	C₈	C₂³×D₁₀₉	1	1	yes	characteristic
402	C₂²	—	2	4	no	—
403	C₂²	—	4	1	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	872	2	no	—
406	C₂²	—	872	8	no	—
407	C₂²	—	109	4	no	—
408	C₂²	—	436	2	no	—
409	C₂²	—	218	4	no	—
410	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
411	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
412	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
413	C₄	—	2	4	no	—
414	C₄	—	109	4	no	—
415	C₄	—	218	4	no	—
416	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
417	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
418	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
419	C₄	C₂⁴×D₁₀₉	1	1	yes	characteristic
420	C₂	—	2	1	no	—
421	C₂	—	2	1	no	—
422	C₂	—	2	1	no	—
423	C₂	—	4	4	no	—
424	C₂	—	436	4	no	—
425	C₂	—	218	4	no	—
426	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₀₉
commutator, cosocle	C₄×C₁₀₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₉
Fitting	C₂×#135×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂³×D₄
hypercenter	C₂×#135	D₁₀₉
hypocenter	C₁₀₉	#3584
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. #20295862	C₂	semidirect
2. #20295840	C₂	semidirect
3. #20295839	C₂	semidirect
4. #20295833	C₂	semidirect
5. #20295832	C₂	semidirect
6. #20295828	C₂	semidirect
7. #20295827	C₂	semidirect
8. #20295824	C₂	semidirect
9. #20295823	C₂	semidirect
10. C₂×#16812721	C₂	semidirect
11. #135×D₁₀₉	C₂	semidirect
12. C₂×#16812705	C₂	semidirect
13. C₂×#135×C₁₀₉	C₂	semidirect
14. #16812747	C₂²	semidirect
16. #16812746	C₂²	semidirect
18. #16812745	C₂²	semidirect
20. #16812744	C₂²	semidirect
23. C₂×#13182029	C₂²	semidirect
25. #41×D₁₀₉	C₂²	semidirect
26. #16812731	C₂²	semidirect
27. #16812730	C₂²	semidirect
28. #16812729	C₂²	semidirect
29. #16812728	C₂²	semidirect
30. #16812726	C₂²	semidirect
31. #16812725	C₂²	semidirect
32. #16812724	C₂²	semidirect
33. #16812723	C₂²	semidirect
34. #16812722	C₂²	semidirect
35. #16812721	C₂²	semidirect
36. #16812721	C₂²	semidirect
37. #16812721	C₂²	semidirect
38. #16812721	C₂²	semidirect
39. #16812720	C₂²	semidirect
40. #16812719	C₂²	semidirect
41. #16812718	C₂²	semidirect
42. C₂×Dic₄₃₆	C₂²	semidirect
43. Q₁₆×D₁₀₉	C₂²	semidirect
44. C₂×#13182018	C₂²	semidirect
45. SD₁₆×D₁₀₉	C₂²	semidirect
46. C₂×D₈₇₂	C₂²	semidirect
47. D₈×D₁₀₉	C₂²	semidirect
48. #16812705	C₂²	semidirect
49. #16812705	C₂²	semidirect
50. #16812705	C₂²	semidirect
51. #16812705	C₂²	semidirect
52. C₂×#13182013	C₂²	semidirect
53. C₂×#13182012	C₂²	semidirect
54. C₂×C₈×D₁₀₉	C₂²	semidirect
55. C₂×#41×C₁₀₉	C₂²	semidirect
56. #135×C₁₀₉	C₂²	semidirect
57. #135×C₁₀₉	C₂²	semidirect
58. #135×C₁₀₉	C₂²	semidirect
59. #135×C₁₀₉	C₂²	semidirect
60. C₂×Q₁₆×C₁₀₉	C₂²	semidirect
61. C₂×SD₁₆×C₁₀₉	C₂²	semidirect
62. C₂×D₈×C₁₀₉	C₂²	semidirect
63. C₂²×C₈×C₁₀₉	C₂²	semidirect
214. C₁₀₉	#3584	semidirect
226. C₂×#135	D₁₀₉	semidirect
240. C₂×#41	C₂×D₁₀₉	semidirect
257. #135	C₂×D₁₀₉	semidirect
258. #135	C₂×D₁₀₉	semidirect
259. #135	C₂×D₁₀₉	semidirect
260. #135	C₂×D₁₀₉	semidirect
263. C₂×Q₁₆	C₂×D₁₀₉	semidirect
266. C₂×SD₁₆	C₂×D₁₀₉	semidirect
269. C₂×D₈	C₂×D₁₀₉	semidirect
277. C₂²×C₈	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295862	2
#20295840	2
#20295839	2
#20295833	2
#20295832	2
#20295828	2
#20295827	2
#20295824	2
#20295823	2
C₂×#16812721	2
#135×D₁₀₉	2
C₂×#16812705	2
C₂×#135×C₁₀₉	2
#3584	109