Subgroups — Q₈×#14273654 #28605388

Order 15104.

1096 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×Q₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#21425074	C₂	1	3	yes	maximal, complemented
4	#21425041	C₂	1	3	yes	maximal, complemented
5	#21424116	C₂	1	3	yes	maximal
6	C₄×#14273654	C₂	1	3	yes	maximal
7	Q₈×#41×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₅₉	—	2	1	no	—
9	#16837533	—	2	3	no	—
10	Q₈×C₅₉⋊C₈	—	2	1	no	—
11	#16837426	C₂²	1	3	yes	—
12	C₂×#14273654	C₂²	1	3	yes	—
13	C₂×#14273654	C₂²	1	1	yes	characteristic
14	C₄×#11745664	C₂²	1	3	yes	—
15	C₄⋊C₄×Dic₅₉	—	2	3	no	—
16	C₄×C₄⋊Dic₅₉	C₂²	1	3	yes	—
17	C₄²×Dic₅₉	C₂²	1	3	yes	—
18	#16837161	C₂²	1	3	yes	—
19	#16837161	C₂²	1	3	yes	—
20	#16837161	C₂²	1	6	yes	—
21	Q₈²×C₅₉	C₄	1	1	yes	characteristic, complemented
22	C₅₉×#557	C₄	1	3	yes	complemented
23	C₅₉×#554	—	2	3	no	—
24	C₅₉×#554	C₄	1	3	yes	complemented
25	D₄×Q₈×C₅₉	—	2	1	no	—
26	D₄×Q₈×C₅₉	C₄	1	1	yes	characteristic, complemented
27	C₅₉×#533	—	2	3	no	—
28	C₅₉×#533	C₂²	1	3	yes	—
29	C₄×#41×C₅₉	C₂²	1	3	yes	—
30	C₂×C₄×Q₈×C₅₉	—	2	1	no	—
31	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×#11745664	C₂³	1	3	yes	—
33	C₂×#11745664	C₂³	1	1	yes	characteristic
34	Q₈×Dic₅₉	—	2	8	no	—
35	C₂×C₄⋊Dic₅₉	—	2	3	no	—
36	C₂×C₄×Dic₅₉	—	2	3	no	—
37	C₂×C₄×Dic₅₉	—	2	3	no	—
38	C₂×C₄×Dic₅₉	C₂³	1	3	yes	—
39	C₂×C₄×Dic₅₉	C₂³	1	1	yes	characteristic
40	C₄⋊(C₅₉⋊C₈)	—	2	3	no	—
41	C₄⋊(C₅₉⋊C₈)	—	2	3	no	—
42	#14273654	—	2	6	no	—
43	#14273654	Q₈	1	4	yes	direct product factor
44	#14273638	—	2	3	no	—
45	C₄×C₅₉⋊C₈	—	2	3	no	—
46	C₅₉⋊#95	—	2	3	no	—
47	C₂×#41×C₅₉	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₅₉	—	2	1	no	—
49	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₅₉	—	2	3	no	—
51	C₄⋊Q₈×C₅₉	—	2	3	no	—
52	C₄⋊Q₈×C₅₉	C₂×C₄	1	3	yes	—
53	C₄⋊Q₈×C₅₉	C₂×C₄	1	3	yes	—
54	#125×C₅₉	—	2	6	no	—
55	#125×C₅₉	C₂×C₄	1	6	yes	—
56	C₂²⋊Q₈×C₅₉	—	2	3	no	—
57	C₂²⋊Q₈×C₅₉	—	2	3	no	—
58	C₂²⋊Q₈×C₅₉	—	2	3	no	—
59	C₂²⋊Q₈×C₅₉	—	2	3	no	—
60	C₂²⋊Q₈×C₅₉	—	2	6	no	—
61	C₄×Q₈×C₅₉	—	2	1	no	—
62	C₄×Q₈×C₅₉	—	2	3	no	—
63	C₄×Q₈×C₅₉	—	2	3	no	—
64	C₄×Q₈×C₅₉	—	2	3	no	—
65	C₄×Q₈×C₅₉	C₂×C₄	1	3	yes	—
66	C₄×Q₈×C₅₉	C₂×C₄	1	3	yes	—
67	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₅₉	C₂×C₄	1	3	yes	—
69	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
70	C₄×Q₈×C₅₉	D₄	1	3	yes	—
71	C₄×D₄×C₅₉	—	2	3	no	—
72	C₄×D₄×C₅₉	—	2	3	no	—
73	C₄×D₄×C₅₉	C₂×C₄	1	3	yes	—
74	C₄×D₄×C₅₉	C₂×C₄	1	3	yes	—
75	#117×C₅₉	—	2	3	no	—
76	#117×C₅₉	C₂³	1	3	yes	—
77	C₂×C₄⋊C₄×C₅₉	—	2	3	no	—
78	C₂×C₄⋊C₄×C₅₉	D₄	1	3	yes	—
79	C₂×C₄²×C₅₉	—	2	3	no	—
80	C₂×C₄²×C₅₉	C₂³	1	3	yes	—
81	#41×C₅₉	—	2	1	no	—
82	#41×C₅₉	—	2	1	no	—
83	#41×C₅₉	C₂×Q₈	1	2	yes	—
84	#41×C₅₉	#41	1	2	yes	—
85	C₂×Q₈×C₅₉	—	2	4	no	—
86	C₂×Q₈×C₅₉	—	2	3	no	—
87	C₂×Q₈×C₅₉	—	2	3	no	—
88	C₂×Q₈×C₅₉	—	4	4	no	—
89	C₂×Q₈×C₅₉	C₂²⋊C₄	1	3	yes	—
90	C₂×Q₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×Q₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×D₄×C₅₉	—	2	1	no	—
93	C₂×D₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₉	—	2	1	no	—
95	C₂²×C₄×C₅₉	—	2	3	no	—
96	C₂²×C₄×C₅₉	—	2	3	no	—
97	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
98	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
99	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
100	C₂²×Dic₅₉	—	2	1	no	—
101	C₂²×Dic₅₉	—	2	3	no	—
102	C₄⋊C₄×C₅₉	—	2	6	no	—
103	C₄⋊C₄×C₅₉	—	2	3	no	—
104	C₄⋊C₄×C₅₉	—	2	3	no	—
105	C₄⋊C₄×C₅₉	—	2	3	no	—
106	C₄⋊C₄×C₅₉	—	2	3	no	—
107	C₄⋊C₄×C₅₉	—	2	6	no	—
108	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	3	yes	—
109	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	6	yes	—
110	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	3	yes	—
111	C₄⋊C₄×C₅₉	C₂²×C₄	1	3	yes	—
112	C₄⋊C₄×C₅₉	C₂²×C₄	1	3	yes	—
113	C₂²⋊C₄×C₅₉	—	2	3	no	—
114	C₂²⋊C₄×C₅₉	—	2	3	no	—
115	C₂²⋊C₄×C₅₉	—	2	3	no	—
116	C₄²×C₅₉	—	2	3	no	—
117	C₄²×C₅₉	—	2	3	no	—
118	C₄²×C₅₉	—	2	3	no	—
119	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
120	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
121	C₄²×C₅₉	C₂×D₄	1	3	yes	—
122	C₄²×C₅₉	C₂×D₄	1	3	yes	—
123	C₄⋊Dic₅₉	—	2	12	no	—
124	C₄×Dic₅₉	—	2	12	no	—
125	C₄×Dic₅₉	—	2	1	no	—
126	C₄×Dic₅₉	—	2	3	no	—
127	C₄×Dic₅₉	—	2	3	no	—
128	C₄×Dic₅₉	—	4	6	no	—
129	C₄×Dic₅₉	C₂×Q₈	1	2	yes	complemented
130	#11745664	—	2	1	no	—
131	#11745664	—	2	3	no	—
132	#11745664	—	2	3	no	—
133	#11745664	C₂×Q₈	1	2	yes	complemented
134	C₂×C₅₉⋊C₈	—	2	1	no	—
135	C₂×C₅₉⋊C₈	—	2	3	no	—
136	C₂³×C₅₉	—	2	1	no	—
137	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
138	Q₈×C₅₉	—	2	1	no	—
139	Q₈×C₅₉	—	2	6	no	—
140	Q₈×C₅₉	—	4	6	no	—
141	Q₈×C₅₉	—	8	3	no	—
142	Q₈×C₅₉	#104	1	4	yes	complemented
143	Q₈×C₅₉	C₄×Q₈	1	2	yes	complemented
144	D₄×C₅₉	—	2	2	no	—
145	D₄×C₅₉	—	2	2	no	—
146	D₄×C₅₉	—	2	1	no	—
147	D₄×C₅₉	C₄×Q₈	1	2	yes	complemented
148	C₂×C₄×C₅₉	—	2	1	no	—
149	C₂×C₄×C₅₉	—	2	2	no	—
150	C₂×C₄×C₅₉	—	2	2	no	—
151	C₂×C₄×C₅₉	—	2	6	no	—
152	C₂×C₄×C₅₉	—	2	3	no	—
153	C₂×C₄×C₅₉	—	2	3	no	—
154	C₂×C₄×C₅₉	—	4	3	no	—
155	C₂×C₄×C₅₉	—	4	6	no	—
156	C₂×C₄×C₅₉	—	8	3	no	—
157	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
158	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
159	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
160	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
161	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
166	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂×Dic₅₉	—	2	2	no	—
168	C₂×Dic₅₉	—	2	2	no	—
169	C₂×Dic₅₉	—	2	2	no	—
170	C₂×Dic₅₉	—	2	6	no	—
171	C₂×Dic₅₉	—	4	3	no	—
172	C₂×Dic₅₉	—	4	3	no	—
173	C₅₉⋊C₈	—	2	2	no	—
174	C₅₉⋊C₈	—	4	3	no	—
175	Q₈×#104	—	59	1	no	maximal, Sylow
176	C₂²×C₅₉	—	2	1	no	—
177	C₂²×C₅₉	—	2	1	no	—
178	C₂²×C₅₉	—	2	2	no	—
179	C₂²×C₅₉	—	4	1	no	—
180	C₂²×C₅₉	—	4	1	no	—
181	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₅₉	#548	1	1	yes	characteristic
183	C₂²×C₅₉	D₄×Q₈	1	1	yes	characteristic
184	C₄×C₅₉	—	2	2	no	—
185	C₄×C₅₉	—	2	3	no	—
186	C₄×C₅₉	—	2	3	no	—
187	C₄×C₅₉	—	2	3	no	—
188	C₄×C₅₉	—	4	3	no	—
189	C₄×C₅₉	—	4	3	no	—
190	C₄×C₅₉	C₂×#104	1	6	yes	—
191	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
192	C₄×C₅₉	#519	1	1	yes	characteristic
193	C₄×C₅₉	#548	1	1	yes	characteristic
194	C₄×C₅₉	D₄×Q₈	1	1	yes	characteristic
195	Dic₅₉	—	2	4	no	—
196	Dic₅₉	—	4	6	no	—
197	Q₈×#41	D₅₉	1	1	yes	characteristic
198	Q₈×M₁₆	—	59	1	no	—
199	C₄²×Q₈	—	59	1	no	—
200	#1994	—	59	3	no	—
201	#1986	—	59	3	no	—
202	#1806	—	59	3	no	—
203	C₄×#104	—	59	3	no	—
204	C₂×C₅₉	—	2	1	no	—
205	C₂×C₅₉	—	2	1	no	—
206	C₂×C₅₉	—	4	2	no	—
207	C₂×C₅₉	Q₈×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₅₉	C₂²×#104	1	1	yes	characteristic
209	C₂×C₅₉	#2901	1	1	yes	characteristic
210	Q₈²	Dic₅₉	1	1	yes	characteristic, complemented
211	#557	Dic₅₉	1	3	yes	complemented
212	#554	—	2	3	no	—
213	#554	Dic₅₉	1	3	yes	complemented
214	D₄×Q₈	—	2	1	no	—
215	D₄×Q₈	Dic₅₉	1	1	yes	characteristic, complemented
216	#533	—	2	3	no	—
217	#533	C₂×D₅₉	1	3	yes	—
218	C₄×#41	C₂×D₅₉	1	3	yes	—
219	C₂×C₄×Q₈	—	2	1	no	—
220	C₂×C₄×Q₈	—	118	1	no	—
221	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
222	#446	—	118	3	no	—
223	C₈×Q₈	—	118	1	no	—
224	#423	—	59	3	no	—
225	C₂×#104	—	59	1	no	—
226	C₂×#104	—	59	3	no	—
227	C₄×M₁₆	—	59	3	no	—
228	C₄×C₄⋊C₄	—	59	3	no	—
229	C₄×C₄⋊C₄	—	118	3	no	—
230	C₄³	—	59	3	no	—
231	#339	—	59	6	no	—
232	#339	—	59	3	no	—
233	#339	—	59	3	no	—
234	C₅₉	Q₈×#104	1	1	yes	characteristic, complemented, Sylow
235	C₂×#41	C₂²×D₅₉	1	1	yes	characteristic
236	C₂²×Q₈	—	2	1	no	—
237	C₂²×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
238	C₂×M₁₆	—	59	1	no	—
239	C₂×M₁₆	—	59	3	no	—
240	C₄⋊Q₈	—	2	3	no	—
241	C₄⋊Q₈	—	2	3	no	—
242	C₄⋊Q₈	C₂×Dic₅₉	1	3	yes	—
243	C₄⋊Q₈	C₂×Dic₅₉	1	3	yes	—
244	#125	—	2	6	no	—
245	#125	C₂×Dic₅₉	1	6	yes	—
246	C₂²⋊Q₈	—	2	3	no	—
247	C₂²⋊Q₈	—	2	3	no	—
248	C₂²⋊Q₈	—	2	3	no	—
249	C₂²⋊Q₈	—	2	3	no	—
250	C₂²⋊Q₈	—	2	6	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	3	no	—
253	C₄×Q₈	—	2	3	no	—
254	C₄×Q₈	—	2	3	no	—
255	C₄×Q₈	—	118	8	no	—
256	C₄×Q₈	C₂×Dic₅₉	1	3	yes	—
257	C₄×Q₈	C₂×Dic₅₉	1	3	yes	—
258	C₄×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
259	C₄×Q₈	C₂×Dic₅₉	1	3	yes	—
260	C₄×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
261	C₄×Q₈	C₂²⋊D₅₉	1	3	yes	—
262	C₄×D₄	—	2	3	no	—
263	C₄×D₄	—	2	3	no	—
264	C₄×D₄	C₂×Dic₅₉	1	3	yes	—
265	C₄×D₄	C₂×Dic₅₉	1	3	yes	—
266	#117	—	2	3	no	—
267	#117	C₂²×D₅₉	1	3	yes	—
268	C₂×C₄⋊C₄	—	2	3	no	—
269	C₂×C₄⋊C₄	—	118	3	no	—
270	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	3	yes	—
271	C₂×C₄²	—	2	3	no	—
272	C₂×C₄²	—	59	1	no	—
273	C₂×C₄²	—	59	3	no	—
274	C₂×C₄²	—	118	3	no	—
275	C₂×C₄²	—	118	3	no	—
276	C₂×C₄²	C₂²×D₅₉	1	3	yes	—
277	C₄⋊C₈	—	118	3	no	—
278	C₄⋊C₈	—	118	3	no	—
279	#104	—	59	4	no	—
280	#104	—	118	6	no	—
281	#97	—	118	3	no	—
282	C₄×C₈	—	118	3	no	—
283	#95	—	118	3	no	—
284	#41	—	2	1	no	—
285	#41	—	2	1	no	—
286	#41	Q₈×D₅₉	1	2	yes	—
287	#41	#11745689	1	2	yes	—
288	C₂×Q₈	—	2	4	no	—
289	C₂×Q₈	—	2	3	no	—
290	C₂×Q₈	—	2	3	no	—
291	C₂×Q₈	—	4	4	no	—
292	C₂×Q₈	C₂²⋊Dic₅₉	1	3	yes	—
293	C₂×Q₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
294	C₂×Q₈	C₂²×Dic₅₉	1	1	yes	characteristic
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	C₂²×Dic₅₉	1	1	yes	characteristic
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	2	3	no	—
299	C₂²×C₄	—	2	3	no	—
300	C₂²×C₄	—	118	1	no	—
301	C₂²×C₄	—	118	3	no	—
302	C₂²×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
303	C₂²×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
304	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
305	M₁₆	—	59	2	no	—
306	M₁₆	—	118	1	no	—
307	M₁₆	—	118	3	no	—
308	M₁₆	—	118	3	no	—
309	C₂×C₈	—	118	1	no	—
310	C₂×C₈	—	118	3	no	—
311	C₄⋊C₄	—	2	6	no	—
312	C₄⋊C₄	—	2	3	no	—
313	C₄⋊C₄	—	2	3	no	—
314	C₄⋊C₄	—	2	3	no	—
315	C₄⋊C₄	—	2	3	no	—
316	C₄⋊C₄	—	2	6	no	—
317	C₄⋊C₄	—	118	12	no	—
318	C₄⋊C₄	C₂²⋊Dic₅₉	1	3	yes	—
319	C₄⋊C₄	C₂²⋊Dic₅₉	1	6	yes	—
320	C₄⋊C₄	C₂²⋊Dic₅₉	1	3	yes	—
321	C₄⋊C₄	C₂²×Dic₅₉	1	3	yes	—
322	C₄⋊C₄	C₂²×Dic₅₉	1	3	yes	—
323	C₂²⋊C₄	—	2	3	no	—
324	C₂²⋊C₄	—	2	3	no	—
325	C₂²⋊C₄	—	2	3	no	—
326	C₄²	—	2	3	no	—
327	C₄²	—	2	3	no	—
328	C₄²	—	2	3	no	—
329	C₄²	—	59	2	no	—
330	C₄²	—	118	12	no	—
331	C₄²	—	118	1	no	—
332	C₄²	—	118	3	no	—
333	C₄²	—	118	3	no	—
334	C₄²	—	236	6	no	—
335	C₄²	C₂²×Dic₅₉	1	3	yes	—
336	C₄²	C₂²×Dic₅₉	1	3	yes	—
337	C₄²	C₂×C₂²⋊D₅₉	1	3	yes	—
338	C₄²	C₂×C₂²⋊D₅₉	1	3	yes	—
339	C₂³	—	2	1	no	—
340	C₂³	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
341	Q₈	—	2	6	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	4	6	no	—
344	Q₈	—	8	3	no	—
345	Q₈	#14273654	1	4	yes	direct product factor
346	Q₈	Q₈×Dic₅₉	1	2	yes	complemented
347	D₄	—	2	2	no	—
348	D₄	—	2	2	no	—
349	D₄	—	2	1	no	—
350	D₄	Q₈×Dic₅₉	1	2	yes	complemented
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	6	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	3	no	—
356	C₂×C₄	—	2	3	no	—
357	C₂×C₄	—	4	3	no	—
358	C₂×C₄	—	4	6	no	—
359	C₂×C₄	—	8	3	no	—
360	C₂×C₄	—	118	2	no	—
361	C₂×C₄	—	118	2	no	—
362	C₂×C₄	—	118	2	no	—
363	C₂×C₄	—	118	6	no	—
364	C₂×C₄	—	236	3	no	—
365	C₂×C₄	—	236	3	no	—
366	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	3	yes	—
367	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	3	yes	—
368	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	3	yes	—
369	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	3	yes	—
370	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
371	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
372	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
373	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
374	C₂×C₄	#14273775	1	1	yes	characteristic
375	C₂×C₄	#14273775	1	1	yes	characteristic
376	C₈	—	118	2	no	—
377	C₈	—	236	3	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	4	1	no	—
382	C₂²	—	4	1	no	—
383	C₂²	C₂²×C₂²⋊Dic₅₉	1	1	yes	characteristic
384	C₂²	#16838063	1	1	yes	characteristic
385	C₂²	Q₈×C₂²⋊D₅₉	1	1	yes	characteristic
386	C₄	—	2	3	no	—
387	C₄	—	2	3	no	—
388	C₄	—	2	3	no	—
389	C₄	—	2	2	no	—
390	C₄	—	4	3	no	—
391	C₄	—	4	3	no	—
392	C₄	—	118	4	no	—
393	C₄	—	236	6	no	—
394	C₄	C₂×#14273654	1	6	yes	—
395	C₄	C₂×Q₈×Dic₅₉	1	1	yes	characteristic
396	C₄	#16837880	1	1	yes	characteristic
397	C₄	#16838063	1	1	yes	characteristic
398	C₄	Q₈×C₂²⋊D₅₉	1	1	yes	characteristic
399	C₂	—	2	1	no	—
400	C₂	—	2	1	no	—
401	C₂	—	4	2	no	—
402	C₂	Q₈×C₂²⋊Dic₅₉	1	1	yes	characteristic
403	C₂	C₂²×#14273654	1	1	yes	characteristic
404	C₂	#21432643	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	Q₈×#41×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×C₂²⋊C₄
prime radical	C₂×#41×C₅₉	C₂³
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	Q₈×#41	D₅₉
hypocenter	C₅₉	Q₈×#104
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#11745664	C₂	semidirect
2. C₄×Q₈×Dic₅₉	C₂	semidirect
3. #21425074	C₂	semidirect
4. #21425041	C₂	semidirect
21. Q₈²×C₅₉	C₄	semidirect
22. C₅₉×#557	C₄	semidirect
24. C₅₉×#554	C₄	semidirect
26. D₄×Q₈×C₅₉	C₄	semidirect
43. #14273654	Q₈	direct (N × H)
129. C₄×Dic₅₉	C₂×Q₈	semidirect
133. #11745664	C₂×Q₈	semidirect
142. Q₈×C₅₉	#104	semidirect
143. Q₈×C₅₉	C₄×Q₈	semidirect
147. D₄×C₅₉	C₄×Q₈	semidirect
210. Q₈²	Dic₅₉	semidirect
211. #557	Dic₅₉	semidirect
213. #554	Dic₅₉	semidirect
215. D₄×Q₈	Dic₅₉	semidirect
234. C₅₉	Q₈×#104	semidirect
345. Q₈	#14273654	direct (N × H)
346. Q₈	Q₈×Dic₅₉	semidirect
350. D₄	Q₈×Dic₅₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11745664	2
C₄×Q₈×Dic₅₉	2
#21425074	2
#21425041	2
#21424116	2
C₄×#14273654	2
Q₈×#41×C₅₉	2
Q₈×#104	59

Subgroups — Q8×#14273654 #28605388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14273654 #28605388