Subgroups — #30874290

Order 17152.

858 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863519	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863514	C₂	1	1	yes	maximal, characteristic
3	#22569011	C₂	1	1	yes	maximal, characteristic
4	#22568774	C₂	1	2	yes	maximal, complemented
5	#22560962	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560957	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560487	C₂	1	2	yes	maximal, complemented
8	#22560487	C₂	1	2	yes	maximal, complemented
9	#22560384	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560230	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1771	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	2	yes	—
14	C₂×#14524049	C₂²	1	2	yes	—
15	C₄×#11761628	C₂²	1	2	yes	—
16	#16863519	—	2	4	no	—
17	#16863514	—	2	4	no	—
18	#16863509	C₄	1	2	yes	complemented
19	#16863504	C₄	1	2	yes	complemented
20	#16863498	C₄	1	2	yes	complemented
21	#16863498	C₄	1	2	yes	complemented
22	C₂×#14523979	C₂²	1	1	yes	characteristic
23	C₂×#14523979	C₂²	1	1	yes	characteristic
24	C₂×#14523979	C₂²	1	2	yes	—
25	#16863434	C₂²	1	1	yes	characteristic
26	#16863422	C₂²	1	2	yes	—
27	#16863422	C₂²	1	2	yes	—
28	#16863421	—	2	2	no	—
29	#16863421	—	2	2	no	—
30	#16863420	C₂²	1	2	yes	—
31	#16863418	—	2	2	no	—
32	C₄⋊C₈×D₆₇	—	2	2	no	—
33	C₂×#14523971	C₂²	1	2	yes	—
34	C₂×#14523971	C₂²	1	2	yes	—
35	#16863393	C₂²	1	2	yes	—
36	#16863156	C₂²	1	2	yes	—
37	#16863156	C₂²	1	2	yes	—
38	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
43	#336×C₆₇	C₂²	1	1	yes	characteristic
44	#336×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#11761628	C₂³	1	2	yes	—
46	#14524065	D₄	1	4	yes	—
47	#14524065	Q₈	1	4	yes	—
48	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
49	C₂×#11761617	C₂×C₄	1	2	yes	—
50	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
51	#14524049	—	2	4	no	—
52	#14524049	—	2	4	no	—
53	#14524049	—	4	2	no	—
54	#14524049	—	4	2	no	—
55	C₂×#11761612	C₂³	1	2	yes	—
56	C₂×#11761612	C₂³	1	2	yes	—
57	C₂×#11761611	—	2	2	no	—
58	C₂×#11761611	—	2	2	no	—
59	C₂×C₈×D₆₇	—	2	2	no	—
60	C₂×C₈×D₆₇	—	2	2	no	—
61	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
63	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
64	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
65	C₄×C₂²⋊D₆₇	—	2	2	no	—
66	C₄×C₂²⋊D₆₇	—	2	2	no	—
67	#14524002	C₂³	1	2	yes	—
68	#14524000	—	2	2	no	—
69	C₄²×D₆₇	—	2	2	no	—
70	#14523979	—	2	2	no	—
71	#14523979	—	2	2	no	—
72	#14523979	—	4	2	no	—
73	#14523977	C₂×C₄	1	1	yes	characteristic
74	#14523977	C₂×C₄	1	1	yes	characteristic
75	#14523975	C₂×C₄	1	2	yes	—
76	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
77	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
78	Dic₆₇⋊C₈	—	2	2	no	—
79	Dic₆₇⋊C₈	—	2	2	no	—
80	Dic₆₇⋊C₈	—	2	2	no	—
81	Dic₆₇⋊C₈	—	2	2	no	—
82	#14523971	—	2	4	no	—
83	#14523971	—	2	4	no	—
84	#14523968	C₂×C₄	1	2	yes	—
85	#14523968	C₂×C₄	1	2	yes	—
86	#14523965	C₂×C₄	1	2	yes	—
87	#14523965	C₂×C₄	1	2	yes	—
88	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
97	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
98	#97×C₆₇	—	2	2	no	—
99	#97×C₆₇	C₂×C₄	1	2	yes	—
100	#97×C₆₇	C₂×C₄	1	2	yes	—
101	C₄×C₈×C₆₇	—	2	2	no	—
102	#95×C₆₇	—	2	1	no	—
103	#11761628	—	2	2	no	—
104	#11761628	—	2	2	no	—
105	#11761628	C₂×D₄	1	4	yes	—
106	#11761628	C₂×Q₈	1	4	yes	—
107	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
108	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
109	C₂×C₂²⋊D₆₇	—	2	2	no	—
110	C₂×C₄×D₆₇	—	2	2	no	—
111	C₂×C₄×D₆₇	—	2	2	no	—
112	C₂×C₄×D₆₇	—	2	2	no	—
113	Dic₂₆₈	C₄⋊C₄	1	4	yes	complemented
114	#11761617	C₄⋊C₄	1	8	yes	complemented
115	D₅₃₆	C₄⋊C₄	1	4	yes	complemented
116	#11761612	—	2	2	no	—
117	#11761612	—	2	2	no	—
118	#11761612	—	2	2	no	—
119	#11761612	—	2	2	no	—
120	#11761611	—	4	4	no	—
121	#11761611	—	4	4	no	—
122	C₂×C₆₇⋊C₈	—	2	2	no	—
123	C₂×C₆₇⋊C₈	—	2	2	no	—
124	C₈×D₆₇	—	4	4	no	—
125	C₈×D₆₇	—	4	4	no	—
126	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
127	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
128	Dic₆₇⋊C₄	—	2	2	no	—
129	Dic₆₇⋊C₄	—	2	2	no	—
130	C₂²⋊Dic₆₇	—	2	2	no	—
131	(C₂×D₆₇)⋊C₄	—	2	2	no	—
132	(C₂×D₆₇)⋊C₄	—	2	2	no	—
133	C₄×Dic₆₇	—	2	2	no	—
134	C₄×Dic₆₇	—	2	2	no	—
135	C₄×Dic₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	C₂×C₈×C₆₇	—	2	1	no	—
142	C₂×C₈×C₆₇	—	2	1	no	—
143	C₂×C₈×C₆₇	—	2	1	no	—
144	C₂×C₈×C₆₇	—	2	1	no	—
145	C₂×C₈×C₆₇	—	2	1	no	—
146	C₂×C₈×C₆₇	—	2	1	no	—
147	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
148	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
151	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
153	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
155	C₄⋊C₄×C₆₇	—	4	2	no	—
156	C₄²×C₆₇	—	2	1	no	—
157	C₄²×C₆₇	—	2	1	no	—
158	C₄²×C₆₇	—	2	1	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
161	C₄²×C₆₇	C₂×D₄	1	2	yes	—
162	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
163	C₂²×D₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
173	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
174	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
180	C₂²⋊D₆₇	—	2	4	no	—
181	C₂²⋊D₆₇	—	2	4	no	—
182	C₂×Dic₆₇	—	2	2	no	—
183	C₂×Dic₆₇	—	2	2	no	—
184	C₂×Dic₆₇	—	2	2	no	—
185	D₂₆₈	—	2	2	no	—
186	D₂₆₈	C₂×C₄⋊C₄	1	4	yes	—
187	C₄×D₆₇	—	2	4	no	—
188	C₄×D₆₇	—	2	4	no	—
189	C₄×D₆₇	—	4	4	no	—
190	C₄×D₆₇	—	8	2	no	—
191	Dic₁₃₄	—	2	2	no	—
192	Dic₁₃₄	C₂×C₄⋊C₄	1	4	yes	—
193	C₈×C₆₇	—	2	2	no	—
194	C₈×C₆₇	—	2	1	no	—
195	C₈×C₆₇	—	2	2	no	—
196	C₈×C₆₇	—	2	1	no	—
197	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
198	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
199	C₆₇⋊C₈	—	4	2	no	—
200	C₆₇⋊C₈	—	4	2	no	—
201	C₂²×C₆₇	—	2	1	no	—
202	C₂²×C₆₇	—	2	1	no	—
203	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂²×C₆₇	#547	1	1	yes	characteristic
205	C₂²×C₆₇	#554	1	1	yes	characteristic
206	C₂×D₆₇	—	2	4	no	—
207	C₂×D₆₇	—	4	2	no	—
208	C₂×D₆₇	—	4	2	no	—
209	C₄×C₆₇	—	2	2	no	—
210	C₄×C₆₇	—	2	2	no	—
211	C₄×C₆₇	—	4	2	no	—
212	C₄×C₆₇	—	8	1	no	—
213	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
214	C₄×C₆₇	#520	1	1	yes	characteristic
215	C₄×C₆₇	D₄²	1	1	yes	characteristic
216	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
217	Dic₆₇	—	2	4	no	—
218	Dic₆₇	—	4	2	no	—
219	Dic₆₇	—	4	2	no	—
220	#22687	—	67	1	no	maximal, Sylow
221	C₂×C₆₇	—	2	1	no	—
222	C₂×C₆₇	—	2	1	no	—
223	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
224	C₂×C₆₇	#2955	1	1	yes	characteristic
225	C₂×C₆₇	#2956	1	1	yes	characteristic
226	D₆₇	—	4	4	no	—
227	C₂×#444	—	67	1	no	—
228	C₂×#443	—	67	1	no	—
229	#2943	—	67	1	no	—
230	#2906	—	67	2	no	—
231	#1946	—	67	1	no	—
232	#1945	—	67	1	no	—
233	#1866	—	67	2	no	—
234	#1866	—	67	2	no	—
235	#1849	—	67	1	no	—
236	#1806	—	67	2	no	—
237	#1771	D₆₇	1	1	yes	characteristic, complemented
238	C₆₇	#22687	1	1	yes	characteristic, complemented, Sylow
239	C₂×#135	—	67	1	no	—
240	C₂×#131	—	67	2	no	—
241	C₂×#131	—	67	2	no	—
242	C₄×#41	—	67	2	no	—
243	#444	—	134	4	no	—
244	#443	—	134	4	no	—
245	#442	—	67	2	no	—
246	#441	—	67	2	no	—
247	#440	—	67	2	no	—
248	#440	—	67	2	no	—
249	C₂×#108	—	67	2	no	—
250	C₂×#108	—	67	1	no	—
251	C₂×#108	—	67	1	no	—
252	#427	—	67	1	no	—
253	#424	—	67	2	no	—
254	#424	—	67	2	no	—
255	#424	—	134	2	no	—
256	#424	—	134	2	no	—
257	#423	—	67	2	no	—
258	#423	—	134	2	no	—
259	C₂×C₄⋊C₈	—	134	2	no	—
260	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
261	C₂×#104	—	67	2	no	—
262	C₂×#104	—	67	2	no	—
263	#416	—	67	2	no	—
264	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
265	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
266	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
267	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
268	#339	—	67	2	no	—
269	#339	—	67	2	no	—
270	#336	C₂×D₆₇	1	1	yes	characteristic
271	#336	C₂×D₆₇	1	1	yes	characteristic
272	C₂×#41	—	67	2	no	—
273	#135	—	67	4	no	—
274	#135	—	67	4	no	—
275	C₂×Q₁₆	—	67	1	no	—
276	C₂×SD₁₆	—	67	2	no	—
277	C₂×D₈	—	67	1	no	—
278	#131	—	268	2	no	—
279	#131	—	268	2	no	—
280	#131	—	134	4	no	—
281	#131	—	134	4	no	—
282	C₂×M₁₆	—	67	2	no	—
283	C₂×M₁₆	—	67	2	no	—
284	C₂×M₁₆	—	134	2	no	—
285	C₂×M₁₆	—	134	2	no	—
286	C₂²×C₈	—	134	2	no	—
287	C₂²×C₈	—	134	2	no	—
288	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
289	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
290	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
292	C₄×Q₈	—	67	2	no	—
293	C₄×Q₈	—	67	2	no	—
294	C₄×D₄	—	67	2	no	—
295	C₄×D₄	—	67	2	no	—
296	C₄×D₄	—	134	2	no	—
297	C₄×D₄	—	134	2	no	—
298	#117	—	67	2	no	—
299	#117	—	134	2	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄²	—	134	2	no	—
302	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
303	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
304	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
305	#108	—	268	2	no	—
306	#108	—	134	2	no	—
307	#108	—	134	2	no	—
308	#107	—	67	1	no	—
309	#107	—	67	1	no	—
310	#106	—	67	2	no	—
311	C₄⋊C₈	—	134	2	no	—
312	C₄⋊C₈	—	134	2	no	—
313	C₄⋊C₈	—	134	2	no	—
314	C₄⋊C₈	—	134	2	no	—
315	C₄⋊C₈	—	134	2	no	—
316	C₄⋊C₈	—	134	2	no	—
317	C₄⋊C₈	D₂₆₈	1	2	yes	—
318	C₄⋊C₈	D₂₆₈	1	2	yes	—
319	#104	—	134	4	no	—
320	#104	—	134	4	no	—
321	#103	—	67	2	no	—
322	#103	—	67	2	no	—
323	#102	—	67	2	no	—
324	#102	—	67	2	no	—
325	#97	—	2	2	no	—
326	#97	C₄×D₆₇	1	2	yes	—
327	#97	C₄×D₆₇	1	2	yes	—
328	C₄×C₈	—	2	2	no	—
329	#95	—	2	1	no	—
330	#41	—	67	4	no	—
331	#41	—	67	4	no	—
332	#41	—	134	2	no	—
333	#41	—	134	2	no	—
334	C₂×Q₈	—	67	2	no	—
335	C₂×D₄	—	67	2	no	—
336	C₂×D₄	—	134	2	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	134	2	no	—
340	C₂²×C₄	—	134	2	no	—
341	C₂²×C₄	—	134	2	no	—
342	C₂²×C₄	#11761628	1	1	yes	characteristic
343	C₂²×C₄	#11761628	1	1	yes	characteristic
344	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
345	Q₁₆	—	67	4	no	—
346	SD₁₆	—	67	8	no	—
347	D₈	—	67	4	no	—
348	M₁₆	—	268	4	no	—
349	M₁₆	—	268	4	no	—
350	M₁₆	—	134	2	no	—
351	M₁₆	—	134	2	no	—
352	M₁₆	—	134	2	no	—
353	M₁₆	—	134	2	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	268	4	no	—
361	C₂×C₈	—	268	4	no	—
362	C₂×C₈	—	134	2	no	—
363	C₂×C₈	—	134	2	no	—
364	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
365	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
366	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
367	C₂×C₈	D₄×D₆₇	1	2	yes	—
368	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
369	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
370	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
371	C₂×C₈	Q₈×D₆₇	1	2	yes	—
372	C₄⋊C₄	—	4	2	no	—
373	C₄⋊C₄	—	67	2	no	—
374	C₄⋊C₄	—	67	2	no	—
375	C₄⋊C₄	—	134	2	no	—
376	C₄⋊C₄	—	134	2	no	—
377	C₂²⋊C₄	—	134	2	no	—
378	C₂²⋊C₄	—	134	2	no	—
379	C₂²⋊C₄	—	134	2	no	—
380	C₄²	—	2	1	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	2	1	no	—
383	C₄²	—	2	1	no	—
384	C₄²	—	134	2	no	—
385	C₄²	—	134	2	no	—
386	C₄²	—	134	2	no	—
387	C₄²	C₂×C₄×D₆₇	1	2	yes	—
388	C₄²	C₂×D₂₆₈	1	2	yes	—
389	C₂³	—	134	2	no	—
390	C₂³	C₂×#11761628	1	1	yes	characteristic
391	Q₈	—	67	4	no	—
392	Q₈	—	134	2	no	—
393	D₄	—	67	4	no	—
394	D₄	—	134	4	no	—
395	D₄	—	134	4	no	—
396	D₄	—	134	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	268	4	no	—
406	C₂×C₄	—	536	2	no	—
407	C₂×C₄	—	134	2	no	—
408	C₂×C₄	—	134	2	no	—
409	C₂×C₄	—	134	2	no	—
410	C₂×C₄	—	134	4	no	—
411	C₂×C₄	—	134	4	no	—
412	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
413	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
414	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
417	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
418	C₂×C₄	#14524089	1	1	yes	characteristic
419	C₂×C₄	#14524090	1	1	yes	characteristic
420	C₈	—	2	1	no	—
421	C₈	—	2	2	no	—
422	C₈	—	2	1	no	—
423	C₈	—	2	2	no	—
424	C₈	—	268	2	no	—
425	C₈	—	268	2	no	—
426	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
427	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	268	2	no	—
431	C₂²	—	268	2	no	—
432	C₂²	—	134	4	no	—
433	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
434	C₂²	#16864051	1	1	yes	characteristic
435	C₂²	#16864106	1	1	yes	characteristic
436	C₄	—	2	2	no	—
437	C₄	—	2	2	no	—
438	C₄	—	4	2	no	—
439	C₄	—	8	1	no	—
440	C₄	—	268	2	no	—
441	C₄	—	268	2	no	—
442	C₄	—	134	4	no	—
443	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
444	C₄	#16863876	1	1	yes	characteristic
445	C₄	D₄×D₂₆₈	1	1	yes	characteristic
446	C₄	Q₈×D₂₆₈	1	1	yes	characteristic
447	C₂	—	2	1	no	—
448	C₂	—	2	1	no	—
449	C₂	—	268	4	no	—
450	C₂	C₄⋊C₄×D₂₆₈	1	1	yes	characteristic
451	C₂	#22569113	1	1	yes	characteristic
452	C₂	#22569120	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1771	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1771	D₆₇
hypocenter	C₆₇	#22687
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #1771

Complements

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

N	H ≅ G/N	type
4. #22568774	C₂	semidirect
5. #22560962	C₂	semidirect
6. #22560957	C₂	semidirect
7. #22560487	C₂	semidirect
8. #22560487	C₂	semidirect
9. #22560384	C₂	semidirect
10. #22560230	C₂	semidirect
11. C₆₇×#1771	C₂	semidirect
18. #16863509	C₄	semidirect
19. #16863504	C₄	semidirect
20. #16863498	C₄	semidirect
21. #16863498	C₄	semidirect
113. Dic₂₆₈	C₄⋊C₄	semidirect
114. #11761617	C₄⋊C₄	semidirect
115. D₅₃₆	C₄⋊C₄	semidirect
237. #1771	D₆₇	semidirect
238. C₆₇	#22687	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863519	2
C₂×#16863514	2
#22569011	2
#22568774	2
#22560962	2
#22560957	2
#22560487	2
#22560384	2
#22560230	2
C₆₇×#1771	2
#22687	67