Subgroups — #30874297

Order 17152.

870 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863514	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863512	C₂	1	1	yes	maximal, characteristic
3	#22568980	C₂	1	1	yes	maximal, characteristic
4	#22568776	C₂	1	2	yes	maximal, complemented
5	#22560958	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560485	C₂	1	2	yes	maximal, complemented
7	#22560483	C₂	1	2	yes	maximal, complemented
8	#22560379	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560228	C₂	1	2	yes	maximal, complemented
10	C₈×Dic₆₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1945	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	1	yes	characteristic
13	C₂×#14524051	C₂²	1	2	yes	—
14	C₂×#14524049	C₂²	1	2	yes	—
15	#16863864	C₂²	1	2	yes	—
16	#16863514	—	2	4	no	—
17	#16863512	—	2	4	no	—
18	#16863490	C₄	1	1	yes	characteristic, complemented
19	#16863490	C₄	1	1	yes	characteristic, complemented
20	#16863483	C₄	1	2	yes	complemented
21	#16863483	C₄	1	2	yes	complemented
22	#16863478	C₄	1	1	yes	characteristic, complemented
23	#16863478	C₄	1	1	yes	characteristic, complemented
24	C₂×#14523979	C₂²	1	1	yes	characteristic
25	C₂×#14523978	C₂²	1	2	yes	—
26	#16863423	C₂²	1	2	yes	—
27	#16863421	—	2	2	no	—
28	#16863421	—	2	2	no	—
29	#16863419	C₂²	1	2	yes	—
30	#16863418	—	2	2	no	—
31	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×D₆₇	—	2	2	no	—
33	C₂×#14523971	C₂²	1	2	yes	—
34	C₂×#14523969	C₂²	1	2	yes	—
35	#16863390	C₂²	1	2	yes	—
36	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
38	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
39	#16863155	C₂²	1	2	yes	—
40	#16863151	C₂²	1	1	yes	characteristic
41	#16863150	C₂²	1	1	yes	characteristic
42	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
43	#427×C₆₇	C₂²	1	1	yes	characteristic
44	#424×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
46	#339×C₆₇	C₂²	1	2	yes	—
47	C₂×#11761628	C₂³	1	2	yes	—
48	#14524065	D₄	1	4	yes	—
49	#14524065	Q₈	1	4	yes	—
50	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
51	C₂×#11761617	C₂×C₄	1	2	yes	—
52	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
53	#14524051	—	2	4	no	—
54	#14524051	—	4	2	no	—
55	#14524049	—	2	4	no	—
56	#14524049	—	4	2	no	—
57	C₂×#11761612	C₂³	1	2	yes	—
58	C₂×#11761611	—	2	2	no	—
59	C₂×#11761611	—	2	2	no	—
60	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
61	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
62	C₂×C₈×D₆₇	—	2	2	no	—
63	C₂×C₈×D₆₇	—	2	2	no	—
64	#14524008	C₂×C₄	1	2	yes	—
65	#14524008	C₂×C₄	1	2	yes	—
66	#14524005	C₂×C₄	1	2	yes	—
67	#14524005	C₂×C₄	1	2	yes	—
68	(C₂²⋊D₆₇)⋊C₄	—	2	2	no	—
69	(C₂²⋊D₆₇)⋊C₄	—	2	2	no	—
70	#14524000	—	2	2	no	—
71	C₂×Dic₆₇⋊C₄	—	2	1	no	—
72	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
74	C₄²×D₆₇	—	2	2	no	—
75	#14523979	—	2	2	no	—
76	#14523978	—	4	2	no	—
77	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
78	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
79	Dic₆₇⋊C₈	—	2	2	no	—
80	Dic₆₇⋊C₈	—	2	2	no	—
81	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
82	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
83	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
84	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
85	#14523971	—	2	4	no	—
86	#14523969	—	2	4	no	—
87	#14523966	C₂×C₄	1	2	yes	—
88	#14523966	C₂×C₄	1	2	yes	—
89	#14523963	C₂×C₄	1	2	yes	—
90	#14523963	C₂×C₄	1	2	yes	—
91	C₈×Dic₆₇	—	2	2	no	—
92	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
93	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
94	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
95	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
96	C₆₇⋊#95	—	2	1	no	—
97	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
98	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	#117×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	#108×C₆₇	—	2	2	no	—
103	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
104	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
105	#106×C₆₇	C₂×C₄	1	2	yes	—
106	C₄⋊C₈×C₆₇	—	2	2	no	—
107	C₄⋊C₈×C₆₇	—	2	2	no	—
108	C₄×C₈×C₆₇	—	2	2	no	—
109	#11761628	—	2	2	no	—
110	#11761628	—	2	2	no	—
111	#11761628	C₂×D₄	1	4	yes	—
112	#11761628	C₂×Q₈	1	4	yes	—
113	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
114	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
115	C₂×C₂²⋊D₆₇	—	2	2	no	—
116	C₂²×Dic₆₇	—	2	1	no	—
117	C₂²×Dic₆₇	#41	1	1	yes	characteristic
118	C₂²×Dic₆₇	#41	1	1	yes	characteristic
119	C₂×C₄×D₆₇	—	2	2	no	—
120	C₂×C₄×D₆₇	—	2	2	no	—
121	C₂×C₄×D₆₇	—	2	2	no	—
122	Dic₂₆₈	C₄⋊C₄	1	4	yes	complemented
123	#11761617	C₄⋊C₄	1	8	yes	complemented
124	D₅₃₆	C₄⋊C₄	1	4	yes	complemented
125	#11761612	—	2	2	no	—
126	#11761612	—	2	2	no	—
127	#11761611	—	4	4	no	—
128	#11761611	—	4	4	no	—
129	C₂×C₆₇⋊C₈	—	2	1	no	—
130	C₂×C₆₇⋊C₈	—	2	1	no	—
131	C₂×C₆₇⋊C₈	—	2	1	no	—
132	C₂×C₆₇⋊C₈	—	2	1	no	—
133	C₂×C₆₇⋊C₈	—	2	2	no	—
134	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
138	C₈×D₆₇	—	4	4	no	—
139	C₈×D₆₇	—	4	4	no	—
140	Dic₆₇⋊C₄	—	2	4	no	—
141	Dic₆₇⋊C₄	—	2	2	no	—
142	Dic₆₇⋊C₄	—	2	2	no	—
143	(C₂×D₆₇)⋊C₄	—	2	2	no	—
144	(C₂×D₆₇)⋊C₄	—	2	2	no	—
145	C₄×Dic₆₇	—	2	2	no	—
146	C₄×Dic₆₇	—	2	2	no	—
147	C₄×Dic₆₇	—	2	1	no	—
148	C₄×Dic₆₇	—	2	1	no	—
149	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
152	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
155	M₁₆×C₆₇	—	2	2	no	—
156	M₁₆×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
166	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
168	C₂²⋊C₄×C₆₇	—	2	2	no	—
169	C₄²×C₆₇	—	2	2	no	—
170	C₄²×C₆₇	—	2	1	no	—
171	C₄²×C₆₇	—	2	1	no	—
172	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
173	C₂²×D₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	4	1	no	—
178	C₂×C₄×C₆₇	—	4	1	no	—
179	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
183	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
185	C₂²⋊D₆₇	—	2	4	no	—
186	C₂²⋊D₆₇	—	2	4	no	—
187	C₂×Dic₆₇	—	2	2	no	—
188	C₂×Dic₆₇	—	2	2	no	—
189	C₂×Dic₆₇	—	2	2	no	—
190	C₂×Dic₆₇	—	4	1	no	—
191	C₂×Dic₆₇	—	4	1	no	—
192	C₂×Dic₆₇	—	4	1	no	—
193	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
194	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
195	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
196	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
197	D₂₆₈	—	2	2	no	—
198	D₂₆₈	C₂×C₄⋊C₄	1	4	yes	—
199	C₄×D₆₇	—	2	4	no	—
200	C₄×D₆₇	—	2	4	no	—
201	C₄×D₆₇	—	4	4	no	—
202	C₄×D₆₇	—	8	2	no	—
203	Dic₁₃₄	—	2	2	no	—
204	Dic₁₃₄	C₂×C₄⋊C₄	1	4	yes	—
205	C₈×C₆₇	—	2	1	no	—
206	C₈×C₆₇	—	2	1	no	—
207	C₈×C₆₇	—	4	2	no	—
208	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
209	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
210	C₆₇⋊C₈	—	2	1	no	—
211	C₆₇⋊C₈	—	2	1	no	—
212	C₆₇⋊C₈	—	2	1	no	—
213	C₆₇⋊C₈	—	2	1	no	—
214	C₆₇⋊C₈	—	4	2	no	—
215	C₂²×C₆₇	—	2	1	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂²×C₆₇	#547	1	1	yes	characteristic
219	C₂²×C₆₇	#554	1	1	yes	characteristic
220	C₂×D₆₇	—	2	4	no	—
221	C₂×D₆₇	—	4	2	no	—
222	C₂×D₆₇	—	4	2	no	—
223	C₄×C₆₇	—	4	2	no	—
224	C₄×C₆₇	—	4	2	no	—
225	C₄×C₆₇	—	4	2	no	—
226	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
227	C₄×C₆₇	#520	1	1	yes	characteristic
228	C₄×C₆₇	D₄²	1	1	yes	characteristic
229	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
230	Dic₆₇	—	2	4	no	—
231	Dic₆₇	—	2	1	no	—
232	Dic₆₇	—	2	1	no	—
233	Dic₆₇	—	2	1	no	—
234	Dic₆₇	—	4	2	no	—
235	Dic₆₇	#443	1	2	yes	complemented
236	#22688	—	67	1	no	maximal, Sylow
237	C₂×C₆₇	—	2	1	no	—
238	C₂×C₆₇	—	2	1	no	—
239	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
241	C₂×C₆₇	#2957	1	1	yes	characteristic
242	D₆₇	—	4	4	no	—
243	C₂×#443	—	67	1	no	—
244	C₂×#443	—	67	1	no	—
245	C₄×#135	—	67	1	no	—
246	#2906	—	67	2	no	—
247	#1945	—	67	1	no	—
248	#1945	D₆₇	1	1	yes	characteristic, complemented
249	#1866	—	67	2	no	—
250	#1866	—	67	2	no	—
251	#1848	—	67	1	no	—
252	#1806	—	67	2	no	—
253	C₈×C₄⋊C₄	—	67	1	no	—
254	C₆₇	#22688	1	1	yes	characteristic, complemented, Sylow
255	C₂×#135	—	67	1	no	—
256	C₂×#131	—	67	2	no	—
257	C₂×#131	—	67	2	no	—
258	C₄×#41	—	67	2	no	—
259	#443	—	134	4	no	—
260	#443	—	134	4	no	—
261	C₄×Q₁₆	—	67	1	no	—
262	C₄×Q₁₆	—	67	1	no	—
263	C₄×SD₁₆	—	67	2	no	—
264	C₄×SD₁₆	—	67	2	no	—
265	C₄×D₈	—	67	1	no	—
266	C₄×D₈	—	67	1	no	—
267	C₂×#108	—	67	2	no	—
268	C₂×#108	—	67	1	no	—
269	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
270	#427	C₂×D₆₇	1	1	yes	characteristic
271	#424	—	67	2	no	—
272	#424	—	134	2	no	—
273	#424	—	134	2	no	—
274	#424	C₂×D₆₇	1	2	yes	—
275	#423	—	67	2	no	—
276	#423	—	134	2	no	—
277	C₂×C₄⋊C₈	—	67	1	no	—
278	C₂×C₄⋊C₈	—	134	2	no	—
279	C₂×#104	—	67	2	no	—
280	C₂×#104	—	67	2	no	—
281	#416	—	67	2	no	—
282	C₂×C₄×C₈	—	67	1	no	—
283	C₂×C₄×C₈	—	67	1	no	—
284	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
285	C₄×C₄⋊C₄	—	67	1	no	—
286	#339	—	67	2	no	—
287	#339	C₂×D₆₇	1	2	yes	—
288	#336	—	67	1	no	—
289	#336	—	67	1	no	—
290	C₂×#41	—	67	2	no	—
291	#135	—	67	4	no	—
292	#135	—	67	4	no	—
293	C₂×Q₁₆	—	67	1	no	—
294	C₂×SD₁₆	—	67	2	no	—
295	C₂×D₈	—	67	1	no	—
296	#131	—	268	2	no	—
297	#131	—	268	2	no	—
298	#131	—	134	4	no	—
299	#131	—	134	4	no	—
300	C₂×M₁₆	—	67	2	no	—
301	C₂×M₁₆	—	134	2	no	—
302	C₂×M₁₆	—	134	2	no	—
303	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
304	C₂²×C₈	—	67	1	no	—
305	C₂²×C₈	—	67	1	no	—
306	C₂²×C₈	—	134	2	no	—
307	C₂²×C₈	—	134	2	no	—
308	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
309	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
310	C₄×Q₈	—	67	2	no	—
311	C₄×Q₈	—	67	2	no	—
312	C₄×D₄	—	67	2	no	—
313	C₄×D₄	—	67	2	no	—
314	C₄×D₄	—	134	2	no	—
315	C₄×D₄	—	134	2	no	—
316	#117	—	134	2	no	—
317	#117	C₂²×D₆₇	1	2	yes	—
318	C₂×C₄⋊C₄	—	134	1	no	—
319	C₂×C₄²	—	67	1	no	—
320	C₂×C₄²	—	67	1	no	—
321	C₂×C₄²	—	134	2	no	—
322	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
323	#108	—	2	2	no	—
324	#108	—	268	2	no	—
325	#108	—	134	2	no	—
326	#107	C₄×D₆₇	1	1	yes	characteristic
327	#107	C₄×D₆₇	1	1	yes	characteristic
328	#106	C₄×D₆₇	1	2	yes	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	67	1	no	—
332	C₄⋊C₈	—	67	1	no	—
333	C₄⋊C₈	—	67	1	no	—
334	C₄⋊C₈	—	67	1	no	—
335	C₄⋊C₈	—	134	2	no	—
336	C₄⋊C₈	—	134	2	no	—
337	C₄⋊C₈	—	134	2	no	—
338	C₄⋊C₈	—	134	2	no	—
339	#104	—	134	4	no	—
340	#104	—	134	4	no	—
341	#103	—	67	2	no	—
342	#103	—	67	2	no	—
343	#102	—	67	2	no	—
344	#102	—	67	2	no	—
345	C₄×C₈	—	2	2	no	—
346	C₄×C₈	—	67	1	no	—
347	C₄×C₈	—	67	1	no	—
348	C₄×C₈	—	67	1	no	—
349	C₄×C₈	—	67	1	no	—
350	C₄×C₈	—	134	2	no	—
351	#95	—	134	1	no	—
352	#41	—	67	4	no	—
353	#41	—	67	4	no	—
354	#41	—	134	2	no	—
355	#41	—	134	2	no	—
356	C₂×Q₈	—	67	2	no	—
357	C₂×D₄	—	67	2	no	—
358	C₂×D₄	—	134	2	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	67	1	no	—
361	C₂²×C₄	—	67	1	no	—
362	C₂²×C₄	—	134	2	no	—
363	C₂²×C₄	—	134	2	no	—
364	C₂²×C₄	—	134	1	no	—
365	C₂²×C₄	—	134	2	no	—
366	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
367	Q₁₆	—	67	4	no	—
368	SD₁₆	—	67	8	no	—
369	D₈	—	67	4	no	—
370	M₁₆	—	2	2	no	—
371	M₁₆	—	2	2	no	—
372	M₁₆	—	268	4	no	—
373	M₁₆	—	268	4	no	—
374	M₁₆	—	134	2	no	—
375	M₁₆	—	134	2	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	268	4	no	—
380	C₂×C₈	—	268	4	no	—
381	C₂×C₈	—	67	1	no	—
382	C₂×C₈	—	67	1	no	—
383	C₂×C₈	—	67	1	no	—
384	C₂×C₈	—	67	1	no	—
385	C₂×C₈	—	134	2	no	—
386	C₂×C₈	—	134	1	no	—
387	C₂×C₈	—	134	1	no	—
388	C₂×C₈	—	134	1	no	—
389	C₂×C₈	—	134	1	no	—
390	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
391	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
392	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
393	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
394	C₂×C₈	D₄×D₆₇	1	2	yes	—
395	C₂×C₈	Q₈×D₆₇	1	2	yes	—
396	C₄⋊C₄	—	134	2	no	—
397	C₄⋊C₄	—	134	2	no	—
398	C₄⋊C₄	—	134	4	no	—
399	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
400	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
401	C₂²⋊C₄	—	2	2	no	—
402	C₂²⋊C₄	—	134	2	no	—
403	C₂²⋊C₄	—	134	2	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	67	1	no	—
408	C₄²	—	67	1	no	—
409	C₄²	—	67	1	no	—
410	C₄²	—	67	1	no	—
411	C₄²	—	134	2	no	—
412	C₄²	—	134	2	no	—
413	C₄²	—	134	1	no	—
414	C₄²	—	134	1	no	—
415	C₂³	—	134	2	no	—
416	C₂³	#41×D₆₇	1	1	yes	characteristic
417	Q₈	—	67	4	no	—
418	Q₈	—	134	2	no	—
419	D₄	—	67	4	no	—
420	D₄	—	134	4	no	—
421	D₄	—	134	4	no	—
422	D₄	—	134	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	268	4	no	—
429	C₂×C₄	—	268	1	no	—
430	C₂×C₄	—	268	1	no	—
431	C₂×C₄	—	268	1	no	—
432	C₂×C₄	—	536	2	no	—
433	C₂×C₄	—	67	1	no	—
434	C₂×C₄	—	67	1	no	—
435	C₂×C₄	—	67	1	no	—
436	C₂×C₄	—	67	1	no	—
437	C₂×C₄	—	134	2	no	—
438	C₂×C₄	—	134	2	no	—
439	C₂×C₄	—	134	2	no	—
440	C₂×C₄	—	134	4	no	—
441	C₂×C₄	—	134	4	no	—
442	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
443	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
444	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
445	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
446	C₂×C₄	#14524089	1	1	yes	characteristic
447	C₂×C₄	#14524091	1	1	yes	characteristic
448	C₈	—	2	1	no	—
449	C₈	—	2	1	no	—
450	C₈	—	4	2	no	—
451	C₈	—	268	2	no	—
452	C₈	—	134	1	no	—
453	C₈	—	134	1	no	—
454	C₈	—	134	1	no	—
455	C₈	—	134	1	no	—
456	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
457	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
458	C₂²	—	2	1	no	—
459	C₂²	—	2	1	no	—
460	C₂²	—	268	2	no	—
461	C₂²	—	268	2	no	—
462	C₂²	—	134	4	no	—
463	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
464	C₂²	#16864048	1	1	yes	characteristic
465	C₂²	#16864102	1	1	yes	characteristic
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	268	2	no	—
470	C₄	—	67	2	no	—
471	C₄	—	134	4	no	—
472	C₄	—	134	1	no	—
473	C₄	—	134	1	no	—
474	C₄	—	134	1	no	—
475	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
476	C₄	#16863878	1	1	yes	characteristic
477	C₄	#16864030	1	1	yes	characteristic
478	C₄	#16864061	1	1	yes	characteristic
479	C₂	—	2	1	no	—
480	C₂	—	2	1	no	—
481	C₂	—	268	4	no	—
482	C₂	#22563110	1	1	yes	characteristic
483	C₂	#443×D₆₇	1	1	yes	characteristic
484	C₂	#22569121	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1945	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1945	D₆₇
hypocenter	C₆₇	#22688
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1945

Complements

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

N	H ≅ G/N	type
4. #22568776	C₂	semidirect
5. #22560958	C₂	semidirect
6. #22560485	C₂	semidirect
7. #22560483	C₂	semidirect
8. #22560379	C₂	semidirect
9. #22560228	C₂	semidirect
10. C₈×Dic₆₇⋊C₄	C₂	semidirect
11. C₆₇×#1945	C₂	semidirect
18. #16863490	C₄	semidirect
19. #16863490	C₄	semidirect
20. #16863483	C₄	semidirect
21. #16863483	C₄	semidirect
22. #16863478	C₄	semidirect
23. #16863478	C₄	semidirect
122. Dic₂₆₈	C₄⋊C₄	semidirect
123. #11761617	C₄⋊C₄	semidirect
124. D₅₃₆	C₄⋊C₄	semidirect
235. Dic₆₇	#443	semidirect
248. #1945	D₆₇	semidirect
254. C₆₇	#22688	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863514	2
C₂×#16863512	2
#22568980	2
#22568776	2
#22560958	2
#22560485	2
#22560483	2
#22560379	2
#22560228	2
C₈×Dic₆₇⋊C₄	2
C₆₇×#1945	2
#22688	67