Subgroups — #31820037

Order 17792.

1900 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984888	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581599	C₂	1	4	yes	maximal, complemented
4	#22581594	C₂	1	4	yes	maximal, complemented
5	D₄×D₅₅₆	C₂	1	4	yes	maximal, complemented
6	#22581587	C₂	1	4	yes	maximal, complemented
7	#22581522	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581517	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581514	C₂	1	4	yes	maximal, complemented
10	#22581464	C₂	1	1	yes	maximal, characteristic, complemented
11	#120×D₁₃₉	C₂	1	4	yes	maximal, complemented
12	#22581427	C₂	1	1	yes	maximal, characteristic, complemented
13	#525×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂⁴×D₁₃₉	C₂²	1	2	yes	complemented
15	#17984888	—	2	8	no	—
16	C₂×#14555786	C₂²	1	2	yes	complemented
17	C₂×#14555785	C₂²	1	1	yes	characteristic, complemented
18	C₂×#14555785	C₂²	1	2	yes	complemented
19	C₂²×D₅₅₆	—	2	8	no	—
20	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
21	C₂²×D₅₅₆	C₂²	1	2	yes	complemented
22	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₅₅₆	C₂²	1	4	yes	complemented
24	C₂²×D₅₅₆	C₂²	1	4	yes	complemented
25	C₂²×D₅₅₆	C₂²	1	2	yes	complemented
26	C₂×D₄×D₁₃₉	C₂²	1	4	yes	complemented
27	C₂×D₄×D₁₃₉	C₂²	1	8	yes	complemented
28	C₂×D₄×D₁₃₉	C₂²	1	2	yes	complemented
29	#17984839	C₂²	1	2	yes	complemented
30	#17984838	C₂²	1	2	yes	complemented
31	#17984831	C₂²	1	2	yes	complemented
32	#17984830	C₂²	1	2	yes	complemented
33	#17984827	C₂²	1	8	yes	complemented
34	#17984825	C₂²	1	2	yes	complemented
35	#17984821	C₂²	1	2	yes	complemented
36	#17984817	C₂²	1	8	yes	complemented
37	#17984817	C₂²	1	4	yes	complemented
38	#17984816	C₂²	1	2	yes	complemented
39	#17984816	C₂²	1	4	yes	complemented
40	#17984815	C₂²	1	8	yes	complemented
41	#17984812	C₂²	1	4	yes	complemented
42	#17984812	C₂²	1	8	yes	complemented
43	#17984812	C₂²	1	8	yes	complemented
44	#17984811	C₂²	1	2	yes	complemented
45	#17984810	C₂²	1	4	yes	complemented
46	#17984810	C₂²	1	8	yes	complemented
47	C₄×D₅₅₆	C₂²	1	4	yes	complemented
48	#17984804	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊D₁₃₉	C₂²	1	4	yes	complemented
50	#17984801	C₂²	1	1	yes	characteristic, complemented
51	C₂²⋊C₄×D₁₃₉	C₂²	1	4	yes	complemented
52	C₂²⋊C₄×D₁₃₉	C₂²	1	8	yes	complemented
53	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
54	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
55	#123×C₁₃₉	C₂²	1	2	yes	complemented
56	C₂²⋊Q₈×C₁₃₉	C₂²	1	2	yes	complemented
57	#121×C₁₃₉	C₂²	1	2	yes	complemented
58	#120×C₁₃₉	C₂²	1	2	yes	complemented
59	C₄×D₄×C₁₃₉	C₂²	1	4	yes	complemented
60	#117×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
61	C₂³×D₁₃₉	—	2	8	no	—
62	C₂³×D₁₃₉	—	4	8	no	—
63	C₂³×D₁₃₉	C₂³	1	8	yes	—
64	C₂³×D₁₃₉	C₂³	1	2	yes	—
65	C₂³×D₁₃₉	C₂³	1	2	yes	—
66	C₂³×D₁₃₉	C₂³	1	4	yes	complemented
67	#14555786	—	2	8	no	—
68	#14555785	—	2	8	no	—
69	#14555785	—	2	4	no	—
70	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic, complemented
71	C₂×D₅₅₆	—	2	8	no	—
72	C₂×D₅₅₆	—	2	16	no	—
73	C₂×D₅₅₆	—	2	16	no	—
74	C₂×D₅₅₆	—	2	8	no	—
75	C₂×D₅₅₆	—	2	4	no	—
76	C₂×D₅₅₆	D₄	1	8	yes	complemented
77	C₂×D₅₅₆	C₂³	1	8	yes	—
78	C₂×D₅₅₆	C₂³	1	2	yes	—
79	C₂×D₅₅₆	C₂³	1	2	yes	—
80	C₂×D₅₅₆	C₂³	1	2	yes	—
81	C₂×D₅₅₆	C₂³	1	2	yes	—
82	C₂×D₅₅₆	C₂³	1	4	yes	complemented
83	C₂×D₅₅₆	C₂³	1	1	yes	characteristic, complemented
84	C₂×D₅₅₆	C₂³	1	8	yes	complemented
85	C₂×C₂²⋊D₁₃₉	C₂³	1	4	yes	—
86	C₂×C₂²⋊D₁₃₉	C₂³	1	2	yes	—
87	C₂×C₂²⋊D₁₃₉	C₂³	1	4	yes	complemented
88	C₂×C₂²⋊D₁₃₉	C₂³	1	8	yes	complemented
89	D₄×D₁₃₉	—	2	16	no	—
90	D₄×D₁₃₉	—	2	8	no	—
91	D₄×D₁₃₉	—	4	16	no	—
92	C₂×C₄×D₁₃₉	C₂³	1	4	yes	complemented
93	C₂×C₄×D₁₃₉	C₂³	1	2	yes	complemented
94	C₂×C₄×D₁₃₉	C₂³	1	8	yes	complemented
95	C₄⋊Dic₁₃₉	C₂³	1	2	yes	complemented
96	Dic₁₃₉⋊C₄	C₂³	1	4	yes	complemented
97	C₂²⋊Dic₁₃₉	C₂³	1	2	yes	—
98	C₂²⋊Dic₁₃₉	C₂³	1	4	yes	—
99	(C₂×D₁₃₉)⋊C₄	C₂³	1	8	yes	complemented
100	(C₂×D₁₃₉)⋊C₄	C₂³	1	8	yes	complemented
101	(C₂×D₁₃₉)⋊C₄	C₂³	1	8	yes	complemented
102	(C₂×D₁₃₉)⋊C₄	C₂³	1	4	yes	complemented
103	C₄×Dic₁₃₉	C₂³	1	2	yes	complemented
104	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
105	#41×C₁₃₉	—	2	4	no	—
106	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic, complemented
107	C₂×D₄×C₁₃₉	C₂³	1	4	yes	—
108	C₂×D₄×C₁₃₉	C₂³	1	2	yes	—
109	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic, complemented
110	C₂²×C₄×C₁₃₉	—	2	4	no	—
111	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
112	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
113	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
114	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
115	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	complemented
116	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	complemented
117	C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	complemented
118	C₂²⋊C₄×C₁₃₉	C₂³	1	2	yes	—
119	C₂²⋊C₄×C₁₃₉	C₂³	1	4	yes	—
120	C₂²⋊C₄×C₁₃₉	C₂³	1	4	yes	complemented
121	C₄²×C₁₃₉	C₂³	1	2	yes	complemented
122	C₂²×D₁₃₉	—	2	4	no	—
123	C₂²×D₁₃₉	—	2	16	no	—
124	C₂²×D₁₃₉	—	2	16	no	—
125	C₂²×D₁₃₉	—	2	8	no	—
126	C₂²×D₁₃₉	—	4	4	no	—
127	C₂²×D₁₃₉	—	4	8	no	—
128	C₂²×D₁₃₉	—	4	4	no	—
129	C₂²×D₁₃₉	—	4	8	no	—
130	C₂²×D₁₃₉	—	4	32	no	—
131	C₂²×D₁₃₉	C₂×D₄	1	8	yes	complemented
132	C₂²×D₁₃₉	C₂⁴	1	4	yes	—
133	C₂²×D₁₃₉	C₂⁴	1	2	yes	—
134	C₂²×D₁₃₉	C₂⁴	1	4	yes	—
135	Dic₂₇₈	—	2	2	no	—
136	D₅₅₆	—	2	8	no	—
137	D₅₅₆	—	2	16	no	—
138	D₅₅₆	—	2	4	no	—
139	D₅₅₆	—	2	4	no	—
140	D₅₅₆	—	2	2	no	—
141	D₅₅₆	—	4	8	no	—
142	D₅₅₆	C₂×D₄	1	8	yes	complemented
143	D₅₅₆	C₂×D₄	1	8	yes	complemented
144	C₂²⋊D₁₃₉	—	2	4	no	—
145	C₂²⋊D₁₃₉	—	2	8	no	—
146	C₂²⋊D₁₃₉	—	2	8	no	—
147	C₂²⋊D₁₃₉	—	4	8	no	—
148	C₂×Dic₁₃₉	C₂⁴	1	4	yes	—
149	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
150	C₄×D₁₃₉	—	2	8	no	—
151	C₄×D₁₃₉	—	2	4	no	—
152	C₄×D₁₃₉	—	4	8	no	—
153	C₂³×C₁₃₉	—	2	2	no	—
154	C₂³×C₁₃₉	—	2	4	no	—
155	C₂³×C₁₃₉	C₂⁴	1	2	yes	—
156	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
157	C₂³×C₁₃₉	C₂⁴	1	2	yes	—
158	Q₈×C₁₃₉	—	2	2	no	—
159	D₄×C₁₃₉	—	2	4	no	—
160	D₄×C₁₃₉	—	2	8	no	—
161	D₄×C₁₃₉	—	2	2	no	—
162	C₂×C₄×C₁₃₉	—	2	2	no	—
163	C₂×C₄×C₁₃₉	—	2	4	no	—
164	C₂×C₄×C₁₃₉	—	2	4	no	—
165	C₂×C₄×C₁₃₉	—	2	4	no	—
166	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	complemented
167	C₂×C₄×C₁₃₉	C₂⁴	1	4	yes	—
168	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
169	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
170	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
171	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
172	C₂²×C₁₃₉	—	2	8	no	—
173	C₂²×C₁₃₉	—	2	2	no	—
174	C₂²×C₁₃₉	—	2	2	no	—
175	C₂²×C₁₃₉	—	2	2	no	—
176	C₂²×C₁₃₉	—	2	2	no	—
177	C₂²×C₁₃₉	—	2	4	no	—
178	C₂²×C₁₃₉	C₂²×D₄	1	2	yes	—
179	C₂²×C₁₃₉	H(5,𝔽₂)	1	4	yes	complemented
180	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
181	C₂×D₁₃₉	—	2	8	no	—
182	C₂×D₁₃₉	—	2	2	no	—
183	C₂×D₁₃₉	—	2	4	no	—
184	C₂×D₁₃₉	—	2	4	no	—
185	C₂×D₁₃₉	—	2	8	no	—
186	C₂×D₁₃₉	—	4	8	no	—
187	C₂×D₁₃₉	—	4	32	no	—
188	C₂×D₁₃₉	—	4	8	no	—
189	C₂×D₁₃₉	—	8	8	no	—
190	C₂×D₁₃₉	C₂²×D₄	1	8	yes	—
191	C₄×C₁₃₉	—	2	2	no	—
192	C₄×C₁₃₉	—	2	2	no	—
193	C₄×C₁₃₉	—	2	4	no	—
194	C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
195	C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
196	Dic₁₃₉	—	2	2	no	—
197	Dic₁₃₉	—	2	4	no	—
198	C₂×C₁₃₉	—	2	2	no	—
199	C₂×C₁₃₉	—	2	4	no	—
200	C₂×C₁₃₉	—	4	2	no	—
201	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
202	C₂×C₁₃₉	C₂×H(5,𝔽₂)	1	2	yes	—
203	D₁₃₉	—	2	8	no	—
204	D₁₃₉	—	4	4	no	—
205	D₁₃₉	—	4	2	no	—
206	C₁₃₉	#3484	1	1	yes	characteristic, complemented, Sylow
207	#3484	—	139	1	no	maximal, Sylow
208	C₂×H(5,𝔽₂)	—	139	1	no	—
209	C₂³×D₄	—	139	1	no	—
210	#546	—	139	4	no	—
211	#546	—	139	4	no	—
212	D₄²	—	139	4	no	—
213	D₄²	—	139	4	no	—
214	#535	—	139	1	no	—
215	#535	—	139	1	no	—
216	#534	—	139	4	no	—
217	#525	—	139	1	no	—
218	#525	D₁₃₉	1	1	yes	characteristic, complemented
219	C₂×#120	—	139	4	no	—
220	#518	—	139	1	no	—
221	C₂⁵	—	139	2	no	—
222	H(5,𝔽₂)	—	278	8	no	—
223	C₂×#41	—	139	2	no	—
224	C₂×#41	—	139	1	no	—
225	C₂×#41	—	139	2	no	—
226	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic, complemented
227	C₂²×D₄	—	278	8	no	—
228	C₂²×D₄	—	139	4	no	—
229	C₂²×D₄	—	139	2	no	—
230	C₂²×D₄	—	139	8	no	—
231	C₂²×D₄	—	139	2	no	—
232	C₂²×D₄	—	139	1	no	—
233	C₂²×D₄	—	139	4	no	—
234	C₂²×D₄	—	139	4	no	—
235	C₂²×D₄	—	139	1	no	—
236	C₂²×D₄	—	139	2	no	—
237	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic, complemented
238	#127	—	139	2	no	—
239	#127	—	139	2	no	—
240	#124	—	139	2	no	—
241	#124	—	139	2	no	—
242	#123	—	139	2	no	—
243	#123	—	139	8	no	—
244	#123	C₂×D₁₃₉	1	2	yes	complemented
245	C₂²⋊Q₈	—	139	2	no	—
246	C₂²⋊Q₈	C₂×D₁₃₉	1	2	yes	complemented
247	#121	—	139	2	no	—
248	#121	—	139	4	no	—
249	#121	—	139	8	no	—
250	#121	—	139	4	no	—
251	#121	—	139	8	no	—
252	#121	C₂×D₁₃₉	1	2	yes	complemented
253	#120	—	139	2	no	—
254	#120	—	139	4	no	—
255	#120	—	139	8	no	—
256	#120	—	139	4	no	—
257	#120	—	139	8	no	—
258	#120	—	139	8	no	—
259	#120	C₂×D₁₃₉	1	2	yes	complemented
260	C₄×D₄	—	139	4	no	—
261	C₄×D₄	—	139	4	no	—
262	C₄×D₄	—	139	4	no	—
263	C₄×D₄	C₂×D₁₃₉	1	4	yes	complemented
264	#117	—	139	1	no	—
265	#117	C₂×D₁₃₉	1	1	yes	characteristic, complemented
266	C₂×C₂²⋊C₄	—	139	4	no	—
267	C₂×C₂²⋊C₄	—	139	8	no	—
268	C₂⁴	—	278	8	no	—
269	C₂⁴	—	556	8	no	—
270	C₂⁴	—	139	8	no	—
271	C₂⁴	—	139	2	no	—
272	C₂⁴	—	139	4	no	—
273	C₂⁴	—	139	2	no	—
274	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
275	#41	—	2	4	no	—
276	#41	—	278	8	no	—
277	#41	—	278	8	no	—
278	#41	—	278	4	no	—
279	C₂×Q₈	—	139	1	no	—
280	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic, complemented
281	C₂×D₄	—	278	8	no	—
282	C₂×D₄	—	278	16	no	—
283	C₂×D₄	—	278	16	no	—
284	C₂×D₄	—	278	8	no	—
285	C₂×D₄	—	278	16	no	—
286	C₂×D₄	—	278	8	no	—
287	C₂×D₄	—	278	4	no	—
288	C₂×D₄	—	556	16	no	—
289	C₂×D₄	—	139	4	no	—
290	C₂×D₄	—	139	2	no	—
291	C₂×D₄	—	139	4	no	—
292	C₂×D₄	—	139	8	no	—
293	C₂×D₄	—	139	8	no	—
294	C₂×D₄	—	139	4	no	—
295	C₂×D₄	—	139	8	no	—
296	C₂×D₄	—	139	2	no	—
297	C₂×D₄	—	139	2	no	—
298	C₂×D₄	—	139	1	no	—
299	C₂×D₄	—	139	8	no	—
300	C₂×D₄	—	139	2	no	—
301	C₂×D₄	—	139	2	no	—
302	C₂×D₄	C₂²×D₁₃₉	1	4	yes	—
303	C₂×D₄	C₂²×D₁₃₉	1	2	yes	—
304	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic, complemented
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	139	4	no	—
307	C₂²×C₄	—	139	2	no	—
308	C₂²×C₄	—	139	8	no	—
309	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
310	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
311	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
312	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
313	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	complemented
314	C₄⋊C₄	—	139	4	no	—
315	C₄⋊C₄	—	139	2	no	—
316	C₄⋊C₄	C₂²×D₁₃₉	1	2	yes	complemented
317	C₄⋊C₄	C₂²×D₁₃₉	1	4	yes	complemented
318	C₂²⋊C₄	—	139	2	no	—
319	C₂²⋊C₄	—	139	4	no	—
320	C₂²⋊C₄	—	139	8	no	—
321	C₂²⋊C₄	—	139	8	no	—
322	C₂²⋊C₄	—	139	8	no	—
323	C₂²⋊C₄	—	139	4	no	—
324	C₂²⋊C₄	C₂²×D₁₃₉	1	2	yes	—
325	C₂²⋊C₄	C₂²×D₁₃₉	1	4	yes	—
326	C₂²⋊C₄	C₂²×D₁₃₉	1	4	yes	complemented
327	C₄²	—	139	2	no	—
328	C₄²	C₂²×D₁₃₉	1	2	yes	complemented
329	C₂³	—	2	2	no	—
330	C₂³	—	2	4	no	—
331	C₂³	—	278	16	no	—
332	C₂³	—	278	8	no	—
333	C₂³	—	278	4	no	—
334	C₂³	—	278	16	no	—
335	C₂³	—	556	4	no	—
336	C₂³	—	556	8	no	—
337	C₂³	—	556	4	no	—
338	C₂³	—	556	8	no	—
339	C₂³	—	556	32	no	—
340	C₂³	—	139	4	no	—
341	C₂³	—	139	2	no	—
342	C₂³	—	139	4	no	—
343	C₂³	—	139	8	no	—
344	C₂³	C₂³×D₁₃₉	1	2	yes	—
345	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
346	C₂³	C₂³×D₁₃₉	1	2	yes	—
347	Q₈	—	2	2	no	—
348	Q₈	—	278	2	no	—
349	D₄	—	2	4	no	—
350	D₄	—	2	8	no	—
351	D₄	—	2	2	no	—
352	D₄	—	278	16	no	—
353	D₄	—	278	8	no	—
354	D₄	—	278	8	no	—
355	D₄	—	278	4	no	—
356	D₄	—	278	8	no	—
357	D₄	—	278	4	no	—
358	D₄	—	278	4	no	—
359	D₄	—	278	2	no	—
360	D₄	—	556	8	no	—
361	D₄	—	556	8	no	—
362	D₄	—	139	8	no	—
363	D₄	—	139	8	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	278	8	no	—
369	C₂×C₄	—	278	4	no	—
370	C₂×C₄	—	556	8	no	—
371	C₂×C₄	—	139	4	no	—
372	C₂×C₄	—	139	2	no	—
373	C₂×C₄	D₄×D₁₃₉	1	4	yes	complemented
374	C₂×C₄	C₂³×D₁₃₉	1	4	yes	—
375	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
376	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
377	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
378	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
379	C₂²	—	2	8	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	4	no	—
385	C₂²	—	278	8	no	—
386	C₂²	—	278	2	no	—
387	C₂²	—	278	4	no	—
388	C₂²	—	278	4	no	—
389	C₂²	—	278	8	no	—
390	C₂²	—	556	8	no	—
391	C₂²	—	556	32	no	—
392	C₂²	—	556	8	no	—
393	C₂²	—	1112	8	no	—
394	C₂²	—	139	8	no	—
395	C₂²	C₂×D₄×D₁₃₉	1	2	yes	—
396	C₂²	#17984888	1	4	yes	complemented
397	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
398	C₄	—	2	2	no	—
399	C₄	—	2	2	no	—
400	C₄	—	2	4	no	—
401	C₄	—	278	2	no	—
402	C₄	—	278	4	no	—
403	C₄	C₂×D₄×D₁₃₉	1	2	yes	—
404	C₄	C₂×D₄×D₁₃₉	1	2	yes	—
405	C₂	—	2	2	no	—
406	C₂	—	2	4	no	—
407	C₂	—	4	2	no	—
408	C₂	—	278	8	no	—
409	C₂	—	556	4	no	—
410	C₂	—	556	2	no	—
411	C₂	C₂²×D₄×D₁₃₉	1	1	yes	characteristic
412	C₂	C₂×#17984888	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₉
commutator, socle, cosocle	C₂²×C₁₃₉	C₂⁵
Fitting	#525×C₁₃₉	C₂
hypercenter	#525	D₁₃₉
hypocenter	C₁₃₉	#3484
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#17984888	C₂	semidirect
2. C₂³×D₅₅₆	C₂	semidirect
3. #22581599	C₂	semidirect
4. #22581594	C₂	semidirect
5. D₄×D₅₅₆	C₂	semidirect
6. #22581587	C₂	semidirect
7. #22581522	C₂	semidirect
8. #22581517	C₂	semidirect
9. #22581514	C₂	semidirect
10. #22581464	C₂	semidirect
11. #120×D₁₃₉	C₂	semidirect
12. #22581427	C₂	semidirect
13. #525×C₁₃₉	C₂	semidirect
14. C₂⁴×D₁₃₉	C₂²	semidirect
16. C₂×#14555786	C₂²	semidirect
17. C₂×#14555785	C₂²	semidirect
18. C₂×#14555785	C₂²	semidirect
21. C₂²×D₅₅₆	C₂²	semidirect
22. C₂²×D₅₅₆	C₂²	semidirect
23. C₂²×D₅₅₆	C₂²	semidirect
24. C₂²×D₅₅₆	C₂²	semidirect
25. C₂²×D₅₅₆	C₂²	semidirect
26. C₂×D₄×D₁₃₉	C₂²	semidirect
27. C₂×D₄×D₁₃₉	C₂²	semidirect
28. C₂×D₄×D₁₃₉	C₂²	semidirect
29. #17984839	C₂²	semidirect
30. #17984838	C₂²	semidirect
31. #17984831	C₂²	semidirect
32. #17984830	C₂²	semidirect
33. #17984827	C₂²	semidirect
34. #17984825	C₂²	semidirect
35. #17984821	C₂²	semidirect
36. #17984817	C₂²	semidirect
37. #17984817	C₂²	semidirect
38. #17984816	C₂²	semidirect
39. #17984816	C₂²	semidirect
40. #17984815	C₂²	semidirect
41. #17984812	C₂²	semidirect
42. #17984812	C₂²	semidirect
43. #17984812	C₂²	semidirect
44. #17984811	C₂²	semidirect
45. #17984810	C₂²	semidirect
46. #17984810	C₂²	semidirect
47. C₄×D₅₅₆	C₂²	semidirect
48. #17984804	C₂²	semidirect
49. C₄×C₂²⋊D₁₃₉	C₂²	semidirect
50. #17984801	C₂²	semidirect
51. C₂²⋊C₄×D₁₃₉	C₂²	semidirect
52. C₂²⋊C₄×D₁₃₉	C₂²	semidirect
53. C₂×#41×C₁₃₉	C₂²	semidirect
54. C₂³×C₄×C₁₃₉	C₂²	semidirect
55. #123×C₁₃₉	C₂²	semidirect
56. C₂²⋊Q₈×C₁₃₉	C₂²	semidirect
57. #121×C₁₃₉	C₂²	semidirect
58. #120×C₁₃₉	C₂²	semidirect
59. C₄×D₄×C₁₃₉	C₂²	semidirect
60. #117×C₁₃₉	C₂²	semidirect
66. C₂³×D₁₃₉	C₂³	semidirect
70. C₂×Dic₂₇₈	C₂³	semidirect
76. C₂×D₅₅₆	D₄	semidirect
82. C₂×D₅₅₆	C₂³	semidirect
83. C₂×D₅₅₆	C₂³	semidirect
84. C₂×D₅₅₆	C₂³	semidirect
87. C₂×C₂²⋊D₁₃₉	C₂³	semidirect
88. C₂×C₂²⋊D₁₃₉	C₂³	semidirect
92. C₂×C₄×D₁₃₉	C₂³	semidirect
93. C₂×C₄×D₁₃₉	C₂³	semidirect
94. C₂×C₄×D₁₃₉	C₂³	semidirect
95. C₄⋊Dic₁₃₉	C₂³	semidirect
96. Dic₁₃₉⋊C₄	C₂³	semidirect
99. (C₂×D₁₃₉)⋊C₄	C₂³	semidirect
100. (C₂×D₁₃₉)⋊C₄	C₂³	semidirect
101. (C₂×D₁₃₉)⋊C₄	C₂³	semidirect
102. (C₂×D₁₃₉)⋊C₄	C₂³	semidirect
103. C₄×Dic₁₃₉	C₂³	semidirect
106. C₂×Q₈×C₁₃₉	C₂³	semidirect
109. C₂×D₄×C₁₃₉	C₂³	semidirect
115. C₂²×C₄×C₁₃₉	C₂³	semidirect
116. C₄⋊C₄×C₁₃₉	C₂³	semidirect
117. C₄⋊C₄×C₁₃₉	C₂³	semidirect
120. C₂²⋊C₄×C₁₃₉	C₂³	semidirect
121. C₄²×C₁₃₉	C₂³	semidirect
131. C₂²×D₁₃₉	C₂×D₄	semidirect
142. D₅₅₆	C₂×D₄	semidirect
143. D₅₅₆	C₂×D₄	semidirect
166. C₂×C₄×C₁₃₉	C₂×D₄	semidirect
179. C₂²×C₁₃₉	H(5,𝔽₂)	semidirect
206. C₁₃₉	#3484	semidirect
218. #525	D₁₃₉	semidirect
226. C₂×#41	C₂×D₁₃₉	semidirect
237. C₂³×C₄	C₂×D₁₃₉	semidirect
244. #123	C₂×D₁₃₉	semidirect
246. C₂²⋊Q₈	C₂×D₁₃₉	semidirect
252. #121	C₂×D₁₃₉	semidirect
259. #120	C₂×D₁₃₉	semidirect
263. C₄×D₄	C₂×D₁₃₉	semidirect
265. #117	C₂×D₁₃₉	semidirect
280. C₂×Q₈	C₂²×D₁₃₉	semidirect
304. C₂×D₄	C₂²×D₁₃₉	semidirect
313. C₂²×C₄	C₂²×D₁₃₉	semidirect
316. C₄⋊C₄	C₂²×D₁₃₉	semidirect
317. C₄⋊C₄	C₂²×D₁₃₉	semidirect
326. C₂²⋊C₄	C₂²×D₁₃₉	semidirect
328. C₄²	C₂²×D₁₃₉	semidirect
373. C₂×C₄	D₄×D₁₃₉	semidirect
396. C₂²	#17984888	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984888	2
C₂³×D₅₅₆	2
#22581599	2
#22581594	2
D₄×D₅₅₆	2
#22581587	2
#22581522	2
#22581517	2
#22581514	2
#22581464	2
#120×D₁₃₉	2
#22581427	2
#525×C₁₃₉	2
#3484	139