Subgroups — #30749673

Order 17152.

500 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×#16863448	C₂	1	1	yes	maximal, characteristic
2	#22560186	C₂	1	1	yes	maximal, characteristic
3	C₄×(C₂×D₆₇)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863146	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558043	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
9	#16863448	—	2	2	no	—
10	#16863448	—	2	2	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
14	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
15	C₂×(C₂×D₆₇)⋊C₈	C₄	1	1	yes	characteristic
16	C₂×(C₂×D₆₇)⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₆₇	—	2	1	no	—
19	#16863150	—	2	1	no	—
20	#16863150	C₄	1	1	yes	characteristic
21	#16863150	C₄	1	1	yes	characteristic
22	#16863149	—	2	2	no	—
23	#16863146	—	2	2	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₆₇	Q₈	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	—	2	1	no	—
31	C₂²×C₆₇⋊C₈	—	2	1	no	—
32	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
37	C₄²×D₆₇	—	2	1	no	—
38	C₄²×D₆₇	—	2	1	no	—
39	C₄²×D₆₇	—	2	1	no	—
40	C₄²×D₆₇	—	2	1	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
45	(C₂×D₆₇)⋊C₈	—	2	4	no	—
46	(C₂×D₆₇)⋊C₈	—	2	2	no	—
47	(C₂×D₆₇)⋊C₈	—	4	2	no	—
48	(C₂×D₆₇)⋊C₈	C₈	1	4	yes	complemented
49	C₄×C₆₇⋊C₈	—	2	1	no	—
50	C₄×C₆₇⋊C₈	—	2	1	no	—
51	C₈×Dic₆₇	—	2	4	no	—
52	C₂²×C₈×C₆₇	—	2	1	no	—
53	C₂²×C₈×C₆₇	—	2	1	no	—
54	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₆₇	—	2	1	no	—
58	C₄⋊C₈×C₆₇	—	2	1	no	—
59	C₄⋊C₈×C₆₇	—	2	1	no	—
60	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
61	C₄×C₈×C₆₇	—	2	1	no	—
62	C₄×C₈×C₆₇	—	2	1	no	—
63	C₂³×D₆₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₆₇	—	2	1	no	—
68	C₂×C₄×D₆₇	—	2	1	no	—
69	C₂×C₄×D₆₇	—	2	1	no	—
70	C₂×C₄×D₆₇	—	2	1	no	—
71	C₂×C₄×D₆₇	—	2	1	no	—
72	C₂×C₄×D₆₇	—	2	1	no	—
73	C₂×C₄×D₆₇	—	2	1	no	—
74	C₂×C₄×D₆₇	—	2	1	no	—
75	C₂×C₄×D₆₇	—	2	1	no	—
76	C₂×C₄×D₆₇	—	2	1	no	—
77	C₂×C₄×D₆₇	—	2	1	no	—
78	C₂×C₄×D₆₇	—	2	1	no	—
79	C₂×C₄×D₆₇	—	2	2	no	—
80	C₂×C₄×D₆₇	—	4	1	no	—
81	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
82	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
83	C₂×C₆₇⋊C₈	—	2	1	no	—
84	C₂×C₆₇⋊C₈	—	2	1	no	—
85	C₂×C₆₇⋊C₈	—	2	1	no	—
86	C₂×C₆₇⋊C₈	—	2	2	no	—
87	C₂×C₆₇⋊C₈	—	2	1	no	—
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	—	4	1	no	—
90	C₂×C₆₇⋊C₈	—	4	1	no	—
91	C₂×C₆₇⋊C₈	—	4	1	no	—
92	C₂×C₆₇⋊C₈	—	4	1	no	—
93	C₂×C₆₇⋊C₈	—	4	1	no	—
94	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
95	C₄×Dic₆₇	—	2	1	no	—
96	C₄×Dic₆₇	—	2	1	no	—
97	C₄×Dic₆₇	—	2	1	no	—
98	C₄×Dic₆₇	—	2	1	no	—
99	C₄×Dic₆₇	—	2	1	no	—
100	C₄×Dic₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₆₇	—	2	1	no	—
105	C₂×C₈×C₆₇	—	2	1	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	1	no	—
109	C₂×C₈×C₆₇	—	2	1	no	—
110	C₂×C₈×C₆₇	—	4	1	no	—
111	C₂×C₈×C₆₇	—	4	1	no	—
112	C₂×C₈×C₆₇	—	4	1	no	—
113	C₂×C₈×C₆₇	—	4	1	no	—
114	C₂×C₈×C₆₇	—	4	1	no	—
115	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₆₇	#41	1	1	yes	characteristic
119	C₄²×C₆₇	#41	1	1	yes	characteristic
120	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×D₆₇	—	2	1	no	—
122	C₂²×D₆₇	—	2	1	no	—
123	C₂²×D₆₇	—	2	1	no	—
124	C₂²×D₆₇	—	2	1	no	—
125	C₂²×D₆₇	—	2	1	no	—
126	C₂²×D₆₇	—	2	1	no	—
127	C₂²×D₆₇	C₄⋊C₈	1	2	yes	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
139	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
140	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
142	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
144	C₂×Dic₆₇	—	2	1	no	—
145	C₂×Dic₆₇	—	2	1	no	—
146	C₂×Dic₆₇	—	2	1	no	—
147	C₂×Dic₆₇	—	2	1	no	—
148	C₂×Dic₆₇	—	2	1	no	—
149	C₂×Dic₆₇	—	2	1	no	—
150	C₂×Dic₆₇	—	2	1	no	—
151	C₂×Dic₆₇	C₄⋊C₈	1	2	yes	—
152	C₂×Dic₆₇	#108	1	2	yes	—
153	C₄×D₆₇	—	2	2	no	—
154	C₄×D₆₇	—	2	2	no	—
155	C₄×D₆₇	—	2	2	no	—
156	C₄×D₆₇	—	2	2	no	—
157	C₄×D₆₇	—	4	2	no	—
158	C₄×D₆₇	—	4	2	no	—
159	C₄×D₆₇	—	4	1	no	—
160	C₄×D₆₇	—	4	1	no	—
161	C₄×D₆₇	—	4	1	no	—
162	C₄×D₆₇	—	4	1	no	—
163	C₈×C₆₇	—	2	2	no	—
164	C₈×C₆₇	—	4	1	no	—
165	C₈×C₆₇	—	4	2	no	—
166	C₈×C₆₇	—	8	1	no	—
167	C₆₇⋊C₈	—	2	2	no	—
168	C₆₇⋊C₈	—	4	1	no	—
169	C₆₇⋊C₈	—	4	2	no	—
170	C₆₇⋊C₈	—	8	1	no	—
171	C₂²×C₆₇	#385	1	1	yes	characteristic
172	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
173	C₂²×C₆₇	#424	1	1	yes	characteristic
174	C₂²×C₆₇	#427	1	1	yes	characteristic
175	C₂²×C₆₇	#428	1	1	yes	characteristic
176	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
177	C₂²×C₆₇	#430	1	1	yes	characteristic
178	C₂×D₆₇	—	2	2	no	—
179	C₂×D₆₇	—	2	2	no	—
180	C₂×D₆₇	—	2	2	no	—
181	C₂×D₆₇	—	4	1	no	—
182	C₂×D₆₇	—	4	1	no	—
183	C₂×D₆₇	—	4	1	no	—
184	C₂×D₆₇	—	4	1	no	—
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
188	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
189	C₄×C₆₇	#435	1	1	yes	characteristic
190	C₄×C₆₇	#435	1	1	yes	characteristic
191	C₄×C₆₇	#449	1	1	yes	characteristic
192	C₄×C₆₇	#452	1	1	yes	characteristic
193	C₄×C₆₇	#482	1	1	yes	characteristic
194	C₄×C₆₇	#485	1	1	yes	characteristic
195	Dic₆₇	—	2	2	no	—
196	Dic₆₇	—	2	2	no	—
197	Dic₆₇	—	4	1	no	—
198	#8831	—	67	1	no	maximal, Sylow
199	C₂×C₆₇	#1564	1	1	yes	characteristic
200	C₂×C₆₇	#1565	1	1	yes	characteristic
201	C₂×C₆₇	#1572	1	1	yes	characteristic
202	C₂×C₆₇	#1574	1	1	yes	characteristic
203	C₂×C₆₇	#1850	1	1	yes	characteristic
204	C₂×C₆₇	#1854	1	1	yes	characteristic
205	C₂×C₆₇	#1855	1	1	yes	characteristic
206	D₆₇	—	4	2	no	—
207	C₂×#431	—	67	1	no	—
208	#1797	—	67	1	no	—
209	C₄×C₂²⋊C₈	—	67	1	no	—
210	C₂×#335	—	67	1	no	—
211	C₂×#334	—	67	1	no	—
212	#1277	—	67	1	no	—
213	#1277	D₆₇	1	1	yes	characteristic, complemented
214	C₆₇	#8831	1	1	yes	characteristic, complemented, Sylow
215	C₂²×C₄²	—	67	1	no	—
216	#431	—	134	2	no	—
217	#431	—	134	2	no	—
218	C₂×C₄⋊C₈	—	67	1	no	—
219	C₂×C₄⋊C₈	—	67	1	no	—
220	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
222	C₂×C₂²⋊C₈	—	67	1	no	—
223	C₂×C₂²⋊C₈	—	67	1	no	—
224	C₂×C₂²⋊C₈	—	134	1	no	—
225	C₂×C₂²⋊C₈	—	134	1	no	—
226	C₂×C₄×C₈	—	67	1	no	—
227	C₂×C₄×C₈	—	134	1	no	—
228	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
229	#336	—	67	1	no	—
230	#336	—	67	1	no	—
231	#336	—	134	1	no	—
232	#335	—	134	2	no	—
233	#334	—	134	2	no	—
234	C₂³×C₄	—	67	1	no	—
235	C₂³×C₄	—	67	1	no	—
236	C₂³×C₄	—	67	1	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	67	1	no	—
240	C₂²×C₈	—	67	1	no	—
241	C₂²×C₈	—	134	1	no	—
242	C₂²×C₈	—	134	1	no	—
243	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
244	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
245	C₂×C₄²	—	67	1	no	—
246	C₂×C₄²	—	67	1	no	—
247	C₂×C₄²	—	67	1	no	—
248	C₂×C₄²	—	134	1	no	—
249	C₂×C₄²	—	134	1	no	—
250	C₂×C₄²	—	134	1	no	—
251	C₂×C₄²	—	134	1	no	—
252	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	67	2	no	—
257	C₄⋊C₈	—	134	1	no	—
258	C₄⋊C₈	—	134	1	no	—
259	C₄⋊C₈	—	134	1	no	—
260	C₄⋊C₈	D₂₆₈	1	2	yes	—
261	C₂²⋊C₈	—	268	2	no	—
262	C₂²⋊C₈	—	67	4	no	—
263	C₂²⋊C₈	—	134	2	no	—
264	C₂²⋊C₈	—	134	4	no	—
265	C₄×C₈	—	2	1	no	—
266	C₄×C₈	—	2	1	no	—
267	C₄×C₈	—	134	4	no	—
268	C₄×C₈	—	134	1	no	—
269	C₄×C₈	—	134	1	no	—
270	C₂⁴	—	67	1	no	—
271	C₂²×C₄	—	268	1	no	—
272	C₂²×C₄	—	67	1	no	—
273	C₂²×C₄	—	67	1	no	—
274	C₂²×C₄	—	67	2	no	—
275	C₂²×C₄	—	67	2	no	—
276	C₂²×C₄	—	67	1	no	—
277	C₂²×C₄	—	134	2	no	—
278	C₂²×C₄	—	134	1	no	—
279	C₂²×C₄	—	134	1	no	—
280	C₂²×C₄	—	134	1	no	—
281	C₂²×C₄	—	134	1	no	—
282	C₂²×C₄	—	134	1	no	—
283	C₂²×C₄	—	134	1	no	—
284	C₂²×C₄	—	134	1	no	—
285	C₂²×C₄	—	134	1	no	—
286	C₂²×C₄	—	134	1	no	—
287	C₂²×C₄	—	134	1	no	—
288	C₂²×C₄	—	134	1	no	—
289	C₂²×C₄	—	134	1	no	—
290	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
292	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	4	1	no	—
300	C₂×C₈	—	4	1	no	—
301	C₂×C₈	—	4	1	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	—	268	1	no	—
305	C₂×C₈	—	268	1	no	—
306	C₂×C₈	—	268	1	no	—
307	C₂×C₈	—	268	1	no	—
308	C₂×C₈	—	268	1	no	—
309	C₂×C₈	—	67	2	no	—
310	C₂×C₈	—	134	1	no	—
311	C₂×C₈	—	134	2	no	—
312	C₂×C₈	—	134	1	no	—
313	C₂×C₈	—	134	1	no	—
314	C₂×C₈	—	134	1	no	—
315	C₂×C₈	—	134	1	no	—
316	C₂×C₈	C₈×D₆₇	1	2	yes	—
317	C₄²	—	134	1	no	—
318	C₄²	—	134	1	no	—
319	C₄²	—	134	1	no	—
320	C₄²	—	134	1	no	—
321	C₄²	—	134	1	no	—
322	C₄²	—	134	1	no	—
323	C₄²	D₄×D₆₇	1	1	yes	characteristic
324	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
325	C₄²	#11761627	1	1	yes	characteristic
326	C₄²	#11761628	1	1	yes	characteristic
327	C₂³	—	67	2	no	—
328	C₂³	—	134	1	no	—
329	C₂³	—	134	1	no	—
330	C₂³	—	134	1	no	—
331	C₂³	—	134	1	no	—
332	C₂³	—	134	1	no	—
333	C₂³	—	134	1	no	—
334	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	268	2	no	—
338	C₂×C₄	—	268	2	no	—
339	C₂×C₄	—	268	1	no	—
340	C₂×C₄	—	268	1	no	—
341	C₂×C₄	—	268	1	no	—
342	C₂×C₄	—	268	1	no	—
343	C₂×C₄	—	67	2	no	—
344	C₂×C₄	—	67	2	no	—
345	C₂×C₄	—	134	1	no	—
346	C₂×C₄	—	134	1	no	—
347	C₂×C₄	—	134	1	no	—
348	C₂×C₄	—	134	1	no	—
349	C₂×C₄	—	134	1	no	—
350	C₂×C₄	—	134	1	no	—
351	C₂×C₄	—	134	1	no	—
352	C₂×C₄	—	134	2	no	—
353	C₂×C₄	—	134	2	no	—
354	C₂×C₄	—	134	2	no	—
355	C₂×C₄	—	134	2	no	—
356	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
357	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
358	C₂×C₄	#14524013	1	1	yes	characteristic
359	C₂×C₄	#14524021	1	1	yes	characteristic
360	C₂×C₄	#14524023	1	1	yes	characteristic
361	C₂×C₄	#14524025	1	1	yes	characteristic
362	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
363	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
364	C₂×C₄	#14524049	1	1	yes	characteristic
365	C₂×C₄	#14524051	1	1	yes	characteristic
366	C₂×C₄	#14524062	1	1	yes	characteristic
367	C₂×C₄	#14524066	1	1	yes	characteristic
368	C₂×C₄	#14524070	1	1	yes	characteristic
369	C₂×C₄	#14524072	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	1	no	—
373	C₈	—	8	1	no	—
374	C₈	—	268	2	no	—
375	C₈	—	268	1	no	—
376	C₈	—	536	1	no	—
377	C₈	—	134	2	no	—
378	C₂²	—	268	1	no	—
379	C₂²	—	268	1	no	—
380	C₂²	—	268	1	no	—
381	C₂²	—	268	1	no	—
382	C₂²	—	134	2	no	—
383	C₂²	—	134	2	no	—
384	C₂²	—	134	2	no	—
385	C₂²	#16863270	1	1	yes	characteristic
386	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
387	C₂²	#16863421	1	1	yes	characteristic
388	C₂²	#16863433	1	1	yes	characteristic
389	C₂²	#16863435	1	1	yes	characteristic
390	C₂²	#108×D₆₇	1	1	yes	characteristic
391	C₂²	#16863443	1	1	yes	characteristic
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	268	1	no	—
395	C₄	—	134	2	no	—
396	C₄	—	134	2	no	—
397	C₄	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
398	C₄	C₈×D₂₆₈	1	1	yes	characteristic
399	C₄	#16863469	1	1	yes	characteristic
400	C₄	#16863470	1	1	yes	characteristic
401	C₄	#16863544	1	1	yes	characteristic
402	C₄	#16863559	1	1	yes	characteristic
403	C₄	#16863718	1	1	yes	characteristic
404	C₄	#16863736	1	1	yes	characteristic
405	C₂	—	268	2	no	—
406	C₂	#22558956	1	1	yes	characteristic
407	C₂	#22558958	1	1	yes	characteristic
408	C₂	#22558988	1	1	yes	characteristic
409	C₂	#22558996	1	1	yes	characteristic
410	C₂	#22560388	1	1	yes	characteristic
411	C₂	#22560411	1	1	yes	characteristic
412	C₂	#22560415	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1277	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂³×D₆₇	C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1277	D₆₇
hypocenter	C₆₇	#8831
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
4. C₂×#16863149	C₂	semidirect
5. C₂×#16863146	C₂	semidirect
6. #22558043	C₂	semidirect
7. C₆₇×#1277	C₂	semidirect
48. (C₂×D₆₇)⋊C₈	C₈	semidirect
213. #1277	D₆₇	semidirect
214. C₆₇	#8831	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863448	2
#22560186	2
C₄×(C₂×D₆₇)⋊C₈	2
C₂×#16863149	2
C₂×#16863146	2
#22558043	2
C₆₇×#1277	2
#8831	67