Subgroups — C₄×#16863672 #30862724

Order 17152.

882 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863672	C₂	1	1	yes	maximal, characteristic
2	C₄×#14524076	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#14524049	C₂	1	1	yes	maximal, characteristic
5	#22561122	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560995	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560865	C₂	1	2	yes	maximal, complemented
8	#22560723	C₂	1	2	yes	maximal
9	#22560270	C₂	1	1	yes	maximal, characteristic
10	C₄×#14523961	C₂	1	2	yes	maximal, complemented
11	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524076	C₂²	1	2	yes	—
13	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
14	C₂×#14524049	C₂²	1	1	yes	characteristic
15	C₂×#14524049	C₂²	1	1	yes	characteristic
16	C₄×#11761628	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
18	#16863672	C₄	1	8	yes	direct product factor
19	#16863504	—	2	2	no	—
20	#16863498	—	2	2	no	—
21	C₄×Dic₂₆₈	—	2	1	no	—
22	C₄×Dic₂₆₈	—	2	1	no	—
23	C₄×Dic₂₆₈	—	2	1	no	—
24	C₄×Dic₂₆₈	—	2	1	no	—
25	C₄×Dic₂₆₈	—	2	2	no	—
26	C₄×#11761617	—	2	2	no	—
27	#16863438	C₂²	1	2	yes	—
28	C₂×#14523977	C₂²	1	1	yes	characteristic
29	#16863406	C₂²	1	2	yes	—
30	#16863398	C₂²	1	2	yes	—
31	C₂×#14523968	C₂²	1	2	yes	—
32	C₂×#14523961	C₂²	1	2	yes	—
33	C₂×#14523961	C₂²	1	2	yes	—
34	#16863351	C₂²	1	1	yes	characteristic
35	#16863351	C₂²	1	1	yes	characteristic
36	#16863350	—	2	1	no	—
37	#16863350	—	2	1	no	—
38	C₄×#11761612	C₂²	1	1	yes	characteristic
39	C₄×#11761611	—	2	1	no	—
40	C₄×C₈×D₆₇	—	2	1	no	—
41	#16863164	C₂²	1	2	yes	—
42	#16863164	C₂²	1	2	yes	—
43	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
46	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
47	#341×C₆₇	C₂²	1	2	yes	—
48	C₂×#11761628	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₃₄	C₂³	1	2	yes	—
50	#14524076	—	2	4	no	—
51	#14524076	C₂×C₄	1	8	yes	—
52	C₂×Dic₂₆₈	—	2	2	no	—
53	C₂×Dic₂₆₈	—	2	1	no	—
54	C₂×Dic₂₆₈	—	2	1	no	—
55	C₂×Dic₂₆₈	—	2	2	no	—
56	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	complemented
57	C₂×#11761617	—	2	2	no	—
58	#14524049	—	2	2	no	—
59	#14524049	—	2	2	no	—
60	#14524049	—	2	2	no	—
61	#14524049	C₂×C₄	1	4	yes	—
62	C₂×#11761612	C₂³	1	1	yes	characteristic
63	C₂×#11761612	C₂³	1	1	yes	characteristic
64	C₂×#11761611	—	2	1	no	—
65	C₂×#11761611	—	2	1	no	—
66	C₂×C₈×D₆₇	—	2	1	no	—
67	C₂×C₈×D₆₇	—	2	1	no	—
68	C₄×Dic₁₃₄	—	2	2	no	—
69	C₄×Dic₁₃₄	—	2	2	no	—
70	C₄×Dic₁₃₄	—	2	2	no	—
71	C₄×Dic₁₃₄	—	2	2	no	—
72	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
73	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
74	C₄×D₂₆₈	D₄	1	1	yes	characteristic
75	C₄×D₂₆₈	D₄	1	1	yes	characteristic
76	C₄×C₂²⋊D₆₇	—	2	1	no	—
77	C₄×C₂²⋊D₆₇	—	2	1	no	—
78	#14524002	C₂³	1	1	yes	characteristic
79	#14524000	—	2	1	no	—
80	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
81	C₂×Dic₆₇⋊C₄	—	2	2	no	—
82	C₂×C₄×Dic₆₇	—	2	2	no	—
83	C₄²×D₆₇	—	2	1	no	—
84	#14523977	—	2	2	no	—
85	#14523977	—	2	1	no	—
86	#14523977	—	2	1	no	—
87	#14523975	—	2	2	no	—
88	#14523968	—	2	2	no	—
89	#14523968	—	2	2	no	—
90	#14523968	—	2	2	no	—
91	#14523968	—	2	2	no	—
92	#14523968	—	2	2	no	—
93	#14523965	—	2	2	no	—
94	#14523961	—	2	4	no	—
95	#14523961	C₂×C₄	1	8	yes	complemented
96	#14523954	—	2	1	no	—
97	#14523953	—	2	1	no	—
98	#14523953	—	2	1	no	—
99	C₄×C₆₇⋊C₈	—	2	1	no	—
100	C₈×Dic₆₇	—	2	1	no	—
101	C₈×Dic₆₇	—	2	1	no	—
102	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
103	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
104	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
105	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	#108×C₆₇	—	2	2	no	—
108	#108×C₆₇	C₂×C₄	1	4	yes	complemented
109	#97×C₆₇	—	2	2	no	—
110	C₄×C₈×C₆₇	—	2	2	no	—
111	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
112	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
113	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
114	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
115	#11761628	—	2	1	no	—
116	#11761628	—	2	1	no	—
117	#11761628	C₂²×C₄	1	2	yes	—
118	#11761628	#41	1	2	yes	—
119	C₂×Dic₁₃₄	—	2	2	no	—
120	C₂×Dic₁₃₄	—	2	2	no	—
121	C₂×Dic₁₃₄	—	2	2	no	—
122	C₂×Dic₁₃₄	—	2	2	no	—
123	C₂×Dic₁₃₄	—	2	2	no	—
124	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
125	C₂×Dic₁₃₄	C₂×D₄	1	1	yes	characteristic
126	C₂×D₂₆₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₂²⋊D₆₇	—	2	1	no	—
128	C₂²×Dic₆₇	—	2	2	no	—
129	C₂²×Dic₆₇	—	2	2	no	—
130	C₂×C₄×D₆₇	—	2	1	no	—
131	C₂×C₄×D₆₇	—	2	1	no	—
132	C₂×C₄×D₆₇	—	2	1	no	—
133	Dic₂₆₈	—	2	8	no	—
134	Dic₂₆₈	—	2	4	no	—
135	Dic₂₆₈	—	2	4	no	—
136	#11761617	—	2	8	no	—
137	#11761612	—	2	1	no	—
138	#11761612	—	2	1	no	—
139	#11761612	C₂²×C₄	1	2	yes	—
140	#11761612	#41	1	2	yes	—
141	#11761611	—	2	4	no	—
142	#11761611	—	2	4	no	—
143	C₂×C₆₇⋊C₈	—	2	1	no	—
144	C₂×C₆₇⋊C₈	—	2	1	no	—
145	C₈×D₆₇	—	2	4	no	—
146	C₈×D₆₇	—	2	4	no	—
147	C₄⋊Dic₆₇	—	2	2	no	—
148	C₄⋊Dic₆₇	—	2	2	no	—
149	C₄⋊Dic₆₇	C₂×D₄	1	1	yes	characteristic
150	C₄⋊Dic₆₇	C₂×D₄	1	1	yes	characteristic
151	Dic₆₇⋊C₄	—	2	1	no	—
152	Dic₆₇⋊C₄	—	2	1	no	—
153	Dic₆₇⋊C₄	—	4	2	no	—
154	Dic₆₇⋊C₄	—	4	2	no	—
155	C₂²⋊Dic₆₇	—	2	1	no	—
156	(C₂×D₆₇)⋊C₄	—	2	1	no	—
157	(C₂×D₆₇)⋊C₄	—	2	1	no	—
158	C₄×Dic₆₇	—	2	1	no	—
159	C₄×Dic₆₇	—	2	1	no	—
160	C₄×Dic₆₇	—	2	1	no	—
161	C₄×Dic₆₇	—	4	2	no	—
162	C₄×Dic₆₇	—	4	2	no	—
163	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
166	M₁₆×C₆₇	—	2	2	no	—
167	M₁₆×C₆₇	—	2	2	no	—
168	M₁₆×C₆₇	—	2	2	no	—
169	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	—	2	1	no	—
172	C₂×C₈×C₆₇	—	2	1	no	—
173	C₂×C₈×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
175	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₆₇	#41	1	2	yes	—
180	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
185	C₂²×D₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	4	1	no	—
188	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
189	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
190	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
191	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
198	C₂²⋊D₆₇	—	2	2	no	—
199	C₂²⋊D₆₇	—	2	2	no	—
200	C₂×Dic₆₇	—	2	1	no	—
201	C₂×Dic₆₇	—	2	4	no	—
202	C₂×Dic₆₇	—	2	1	no	—
203	C₂×Dic₆₇	—	2	1	no	—
204	C₂×Dic₆₇	—	4	2	no	—
205	C₂×Dic₆₇	—	4	2	no	—
206	C₂×Dic₆₇	—	4	2	no	—
207	C₂×Dic₆₇	—	8	2	no	—
208	D₂₆₈	—	2	1	no	—
209	D₂₆₈	C₄×D₄	1	2	yes	—
210	C₄×D₆₇	—	2	2	no	—
211	C₄×D₆₇	—	2	2	no	—
212	C₄×D₆₇	—	4	2	no	—
213	C₄×D₆₇	—	4	2	no	—
214	Dic₁₃₄	—	2	1	no	—
215	Dic₁₃₄	—	2	4	no	—
216	Dic₁₃₄	—	2	4	no	—
217	Dic₁₃₄	—	4	2	no	—
218	Dic₁₃₄	—	4	2	no	—
219	Dic₁₃₄	C₄×D₄	1	2	yes	—
220	C₈×C₆₇	—	2	4	no	—
221	C₈×C₆₇	—	2	1	no	—
222	C₈×C₆₇	—	2	1	no	—
223	C₈×C₆₇	—	4	2	no	—
224	C₈×C₆₇	C₄×D₄	1	2	yes	—
225	C₈×C₆₇	C₄×D₄	1	2	yes	—
226	C₆₇⋊C₈	—	2	2	no	—
227	C₆₇⋊C₈	—	2	2	no	—
228	C₂²×C₆₇	—	2	1	no	—
229	C₂²×C₆₇	—	2	1	no	—
230	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
231	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
232	C₂²×C₆₇	#529	1	1	yes	characteristic
233	C₂×D₆₇	—	2	2	no	—
234	C₂×D₆₇	—	4	1	no	—
235	C₂×D₆₇	—	4	1	no	—
236	C₄×C₆₇	—	2	1	no	—
237	C₄×C₆₇	—	2	1	no	—
238	C₄×C₆₇	—	2	1	no	—
239	C₄×C₆₇	#473	1	2	yes	complemented
240	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
241	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
242	C₄×C₆₇	#525	1	1	yes	characteristic
243	C₄×C₆₇	#532	1	1	yes	characteristic
244	Dic₆₇	—	2	2	no	—
245	Dic₆₇	—	4	1	no	—
246	Dic₆₇	—	4	1	no	—
247	Dic₆₇	—	4	4	no	—
248	Dic₆₇	—	8	2	no	—
249	C₄×#473	—	67	1	no	maximal, Sylow
250	C₂×C₆₇	—	2	1	no	—
251	C₂×C₆₇	—	2	1	no	—
252	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
253	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
254	C₂×C₆₇	#3067	1	1	yes	characteristic
255	D₆₇	—	4	2	no	—
256	C₂×#473	—	67	1	no	—
257	C₄×#137	—	67	2	no	—
258	C₂×C₄×Q₁₆	—	67	1	no	—
259	C₄×#131	—	67	1	no	—
260	#1979	—	67	1	no	—
261	#1952	—	67	1	no	—
262	#1930	—	67	2	no	—
263	#1908	—	67	2	no	—
264	#1816	—	67	1	no	—
265	C₄×#108	D₆₇	1	1	yes	characteristic, complemented
266	C₄×#101	—	67	2	no	—
267	C₆₇	C₄×#473	1	1	yes	characteristic, complemented, Sylow
268	C₂×#137	—	67	2	no	—
269	C₂²×Q₁₆	—	67	1	no	—
270	C₂×#131	—	67	1	no	—
271	C₂×#131	—	67	1	no	—
272	C₄×#41	—	67	1	no	—
273	C₂×C₄×Q₈	—	67	2	no	—
274	#473	—	67	8	no	—
275	#441	—	134	2	no	—
276	#440	—	134	2	no	—
277	C₄×Q₁₆	—	134	2	no	—
278	C₄×Q₁₆	—	134	1	no	—
279	C₄×Q₁₆	—	134	1	no	—
280	C₄×Q₁₆	—	134	1	no	—
281	C₄×Q₁₆	—	134	1	no	—
282	C₄×SD₁₆	—	134	2	no	—
283	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
284	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
285	#428	—	67	2	no	—
286	C₂×#107	—	67	1	no	—
287	#419	—	67	2	no	—
288	#417	—	67	2	no	—
289	C₂×#103	—	67	2	no	—
290	C₂×#101	—	67	2	no	—
291	C₂×#101	—	67	2	no	—
292	#405	—	67	1	no	—
293	#405	—	67	1	no	—
294	#405	—	134	1	no	—
295	#405	—	134	1	no	—
296	C₄×M₁₆	—	67	1	no	—
297	C₄×M₁₆	—	134	1	no	—
298	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
299	C₂×C₄×C₈	—	134	1	no	—
300	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
301	#343	—	67	2	no	—
302	#343	—	67	2	no	—
303	#341	C₂×D₆₇	1	2	yes	—
304	C₂×#41	—	67	1	no	—
305	C₂²×Q₈	—	67	2	no	—
306	#137	—	67	8	no	—
307	#137	—	134	4	no	—
308	C₂×Q₁₆	—	67	4	no	—
309	C₂×Q₁₆	—	134	2	no	—
310	C₂×Q₁₆	—	134	1	no	—
311	C₂×Q₁₆	—	134	1	no	—
312	C₂×Q₁₆	—	134	2	no	—
313	C₂×SD₁₆	—	134	2	no	—
314	#131	—	67	4	no	—
315	#131	—	134	2	no	—
316	#131	—	134	2	no	—
317	#131	—	134	2	no	—
318	C₂×M₁₆	—	67	1	no	—
319	C₂×M₁₆	—	67	1	no	—
320	C₂×M₁₆	—	134	1	no	—
321	C₂×M₁₆	—	134	1	no	—
322	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
323	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
324	C₂²×C₈	—	134	1	no	—
325	C₂²×C₈	—	134	1	no	—
326	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
327	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
328	C₄×Q₈	—	67	1	no	—
329	C₄×Q₈	—	67	1	no	—
330	C₄×Q₈	—	134	2	no	—
331	C₄×Q₈	—	134	2	no	—
332	C₄×Q₈	—	134	2	no	—
333	C₄×Q₈	—	134	2	no	—
334	C₄×D₄	—	67	1	no	—
335	C₄×D₄	—	67	1	no	—
336	C₄×D₄	—	134	1	no	—
337	C₄×D₄	—	134	1	no	—
338	#117	—	67	1	no	—
339	#117	—	134	1	no	—
340	C₂×C₄⋊C₄	—	67	2	no	—
341	C₂×C₄⋊C₄	—	134	2	no	—
342	C₂×C₄²	—	134	2	no	—
343	C₂×C₄²	—	134	1	no	—
344	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
345	#108	—	2	2	no	—
346	#108	C₄×D₆₇	1	4	yes	complemented
347	#107	—	134	2	no	—
348	#107	—	134	1	no	—
349	#107	—	134	1	no	—
350	#106	—	134	2	no	—
351	#103	—	134	2	no	—
352	#103	—	134	2	no	—
353	#103	—	134	2	no	—
354	#103	—	134	2	no	—
355	#103	—	134	2	no	—
356	#102	—	134	2	no	—
357	#101	—	67	8	no	—
358	#101	—	134	4	no	—
359	#97	—	2	2	no	—
360	#97	—	134	1	no	—
361	#97	—	134	1	no	—
362	#97	—	134	1	no	—
363	C₄×C₈	—	2	2	no	—
364	C₄×C₈	—	134	1	no	—
365	C₄×C₈	—	134	1	no	—
366	C₄×C₈	—	134	1	no	—
367	C₄×C₈	D₂₆₈	1	1	yes	characteristic
368	C₄×C₈	D₂₆₈	1	1	yes	characteristic
369	C₄×C₈	D₂₆₈	1	1	yes	characteristic
370	C₄×C₈	D₂₆₈	1	1	yes	characteristic
371	#41	—	67	2	no	—
372	#41	—	67	2	no	—
373	#41	—	134	1	no	—
374	#41	—	134	1	no	—
375	C₂×Q₈	—	67	4	no	—
376	C₂×Q₈	—	67	1	no	—
377	C₂×Q₈	—	134	2	no	—
378	C₂×Q₈	—	134	2	no	—
379	C₂×Q₈	—	134	2	no	—
380	C₂×Q₈	—	134	2	no	—
381	C₂×Q₈	—	134	2	no	—
382	C₂×D₄	—	67	1	no	—
383	C₂×D₄	—	134	1	no	—
384	C₂²×C₄	—	134	2	no	—
385	C₂²×C₄	—	134	1	no	—
386	C₂²×C₄	—	134	1	no	—
387	C₂²×C₄	—	134	2	no	—
388	C₂²×C₄	—	134	1	no	—
389	C₂²×C₄	#11761629	1	1	yes	characteristic
390	C₂²×C₄	#11761629	1	1	yes	characteristic
391	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
392	Q₁₆	—	134	8	no	—
393	Q₁₆	—	134	4	no	—
394	Q₁₆	—	134	4	no	—
395	SD₁₆	—	134	8	no	—
396	M₁₆	—	2	2	no	—
397	M₁₆	—	2	2	no	—
398	M₁₆	—	2	2	no	—
399	M₁₆	—	67	2	no	—
400	M₁₆	—	67	2	no	—
401	M₁₆	—	134	4	no	—
402	M₁₆	—	134	1	no	—
403	M₁₆	—	134	1	no	—
404	M₁₆	—	134	4	no	—
405	M₁₆	C₂×C₄×D₆₇	1	4	yes	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	134	4	no	—
411	C₂×C₈	—	134	1	no	—
412	C₂×C₈	—	134	1	no	—
413	C₂×C₈	—	134	4	no	—
414	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
415	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
416	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
417	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
418	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
419	C₂×C₈	#11761628	1	2	yes	—
420	C₄⋊C₄	—	268	2	no	—
421	C₄⋊C₄	—	268	2	no	—
422	C₄⋊C₄	—	67	1	no	—
423	C₄⋊C₄	—	67	1	no	—
424	C₄⋊C₄	—	134	2	no	—
425	C₄⋊C₄	—	134	2	no	—
426	C₄⋊C₄	—	134	1	no	—
427	C₄⋊C₄	—	134	1	no	—
428	C₂²⋊C₄	—	134	1	no	—
429	C₂²⋊C₄	—	134	1	no	—
430	C₂²⋊C₄	—	134	1	no	—
431	C₄²	—	268	2	no	—
432	C₄²	—	268	2	no	—
433	C₄²	—	134	1	no	—
434	C₄²	—	134	1	no	—
435	C₄²	—	134	1	no	—
436	C₄²	D₄×D₆₇	1	1	yes	characteristic
437	C₄²	D₄×D₆₇	1	1	yes	characteristic
438	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
439	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
440	C₂³	—	134	1	no	—
441	C₂³	C₂×#11761629	1	1	yes	characteristic
442	Q₈	—	268	2	no	—
443	Q₈	—	268	2	no	—
444	Q₈	—	67	2	no	—
445	Q₈	—	134	4	no	—
446	Q₈	—	134	4	no	—
447	Q₈	—	134	1	no	—
448	D₄	—	67	2	no	—
449	D₄	—	134	2	no	—
450	D₄	—	134	2	no	—
451	D₄	—	134	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	268	2	no	—
455	C₂×C₄	—	268	2	no	—
456	C₂×C₄	—	268	2	no	—
457	C₂×C₄	—	268	2	no	—
458	C₂×C₄	—	268	2	no	—
459	C₂×C₄	—	536	2	no	—
460	C₂×C₄	—	134	4	no	—
461	C₂×C₄	—	134	1	no	—
462	C₂×C₄	—	134	1	no	—
463	C₂×C₄	—	134	1	no	—
464	C₂×C₄	—	134	2	no	—
465	C₂×C₄	—	134	2	no	—
466	C₂×C₄	#14524018	1	1	yes	characteristic
467	C₂×C₄	#14524018	1	1	yes	characteristic
468	C₂×C₄	#14524018	1	1	yes	characteristic
469	C₂×C₄	#14524018	1	1	yes	characteristic
470	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
471	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
473	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
474	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
475	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
476	C₈	—	2	4	no	—
477	C₈	—	2	1	no	—
478	C₈	—	2	1	no	—
479	C₈	—	4	2	no	—
480	C₈	—	134	2	no	—
481	C₈	—	134	2	no	—
482	C₈	C₄×D₂₆₈	1	2	yes	—
483	C₈	C₄×D₂₆₈	1	2	yes	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	268	1	no	—
487	C₂²	—	268	1	no	—
488	C₂²	—	134	2	no	—
489	C₂²	C₄×#11761629	1	1	yes	characteristic
490	C₂²	C₂×#14524018	1	1	yes	characteristic
491	C₂²	#16863933	1	1	yes	characteristic
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	268	4	no	—
496	C₄	—	268	1	no	—
497	C₄	—	268	1	no	—
498	C₄	—	536	2	no	—
499	C₄	—	134	2	no	—
500	C₄	#16863672	1	2	yes	direct product factor
501	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
502	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
503	C₄	#16863908	1	1	yes	characteristic
504	C₄	#16863943	1	1	yes	characteristic
505	C₂	—	2	1	no	—
506	C₂	—	2	1	no	—
507	C₂	—	268	2	no	—
508	C₂	C₄×#14524018	1	1	yes	characteristic
509	C₂	C₂×#16863672	1	1	yes	characteristic
510	C₂	#22569930	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524018
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂×#11761628	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#108	D₆₇
hypocenter	C₆₇	C₄×#473
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₂₆₈	C₂	semidirect
5. #22561122	C₂	semidirect
6. #22560995	C₂	semidirect
7. #22560865	C₂	semidirect
10. C₄×#14523961	C₂	semidirect
11. C₄×#108×C₆₇	C₂	semidirect
18. #16863672	C₄	direct (N × H)
56. C₂×Dic₂₆₈	C₂×C₄	semidirect
95. #14523961	C₂×C₄	semidirect
108. #108×C₆₇	C₂×C₄	semidirect
239. C₄×C₆₇	#473	semidirect
265. C₄×#108	D₆₇	semidirect
267. C₆₇	C₄×#473	semidirect
346. #108	C₄×D₆₇	semidirect
500. C₄	#16863672	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863672	2
C₄×#14524076	2
C₂×C₄×Dic₂₆₈	2
C₄×#14524049	2
#22561122	2
#22560995	2
#22560865	2
#22560723	2
#22560270	2
C₄×#14523961	2
C₄×#108×C₆₇	2
C₄×#473	67

Subgroups — C4×#16863672 #30862724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863672 #30862724