Subgroups — #30749682

Order 17152.

520 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863418	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×#16863147	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	C₂²×#11761611	—	2	1	no	—
9	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
10	#16863418	—	2	2	no	—
11	#16863418	—	2	2	no	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×Dic₆₇⋊C₈	—	2	1	no	—
15	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
16	C₂×(C₂×D₆₇)⋊C₈	C₄	1	1	yes	characteristic
17	C₂×(C₂×D₆₇)⋊C₈	C₄	1	1	yes	characteristic
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	#16863150	—	2	1	no	—
20	#16863150	C₄	1	1	yes	characteristic
21	#16863150	C₄	1	1	yes	characteristic
22	#16863147	—	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	C₂×#11761611	—	2	4	no	—
31	C₂×#11761611	—	4	2	no	—
32	C₂×#11761611	—	4	2	no	—
33	C₂²×C₆₇⋊C₈	—	2	1	no	—
34	C₂²×C₆₇⋊C₈	—	2	1	no	—
35	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
40	C₄²×D₆₇	—	2	1	no	—
41	C₄²×D₆₇	—	2	1	no	—
42	C₄²×D₆₇	—	2	1	no	—
43	C₄²×D₆₇	—	2	1	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
47	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
48	Dic₆₇⋊C₈	—	2	4	no	—
49	(C₂×D₆₇)⋊C₈	—	2	2	no	—
50	(C₂×D₆₇)⋊C₈	—	2	2	no	—
51	(C₂×D₆₇)⋊C₈	—	4	2	no	—
52	C₄×C₆₇⋊C₈	—	2	1	no	—
53	C₄×C₆₇⋊C₈	—	2	1	no	—
54	C₂²×C₈×C₆₇	—	2	1	no	—
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₆₇	—	2	1	no	—
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₄⋊C₈×C₆₇	—	2	1	no	—
62	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
63	C₄×C₈×C₆₇	—	2	1	no	—
64	C₄×C₈×C₆₇	—	2	1	no	—
65	C₂³×D₆₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
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	1	no	—
80	C₂×C₄×D₆₇	—	2	1	no	—
81	C₂×C₄×D₆₇	—	2	2	no	—
82	C₂×C₄×D₆₇	—	4	1	no	—
83	C₂×C₄×D₆₇	SD₁₆	1	2	yes	—
84	C₂×C₄×D₆₇	SD₁₆	1	2	yes	—
85	#11761611	—	4	8	no	—
86	C₂×C₆₇⋊C₈	—	2	1	no	—
87	C₂×C₆₇⋊C₈	—	2	1	no	—
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	—	2	1	no	—
91	C₂×C₆₇⋊C₈	—	2	1	no	—
92	C₂×C₆₇⋊C₈	—	4	1	no	—
93	C₂×C₆₇⋊C₈	—	4	1	no	—
94	C₂×C₆₇⋊C₈	—	4	1	no	—
95	C₂×C₆₇⋊C₈	—	4	1	no	—
96	C₂×C₆₇⋊C₈	—	4	1	no	—
97	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	complemented
98	C₄×Dic₆₇	—	2	1	no	—
99	C₄×Dic₆₇	—	2	1	no	—
100	C₄×Dic₆₇	—	2	1	no	—
101	C₄×Dic₆₇	—	2	1	no	—
102	C₄×Dic₆₇	—	2	1	no	—
103	C₄×Dic₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	—	2	1	no	—
109	C₂×C₈×C₆₇	—	2	1	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	—	4	1	no	—
114	C₂×C₈×C₆₇	—	4	1	no	—
115	C₂×C₈×C₆₇	—	4	1	no	—
116	C₂×C₈×C₆₇	—	4	1	no	—
117	C₂×C₈×C₆₇	—	4	1	no	—
118	C₂×C₈×C₆₇	M₁₆	1	2	yes	complemented
119	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₆₇	#41	1	1	yes	characteristic
122	C₄²×C₆₇	#41	1	1	yes	characteristic
123	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×D₆₇	—	2	1	no	—
125	C₂²×D₆₇	—	2	1	no	—
126	C₂²×D₆₇	—	2	1	no	—
127	C₂²×D₆₇	—	2	1	no	—
128	C₂²×D₆₇	—	2	1	no	—
129	C₂²×D₆₇	—	2	1	no	—
130	C₂²×D₆₇	#106	1	2	yes	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
142	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
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	—	2	1	no	—
152	C₂×Dic₆₇	—	2	1	no	—
153	C₂×Dic₆₇	—	2	1	no	—
154	C₂×Dic₆₇	#106	1	2	yes	—
155	C₂×Dic₆₇	#108	1	2	yes	—
156	C₄×D₆₇	—	2	2	no	—
157	C₄×D₆₇	—	2	2	no	—
158	C₄×D₆₇	—	2	2	no	—
159	C₄×D₆₇	—	2	2	no	—
160	C₄×D₆₇	—	4	2	no	—
161	C₄×D₆₇	—	4	2	no	—
162	C₄×D₆₇	—	4	1	no	—
163	C₄×D₆₇	—	4	1	no	—
164	C₄×D₆₇	—	4	1	no	—
165	C₄×D₆₇	—	4	1	no	—
166	C₈×C₆₇	—	2	2	no	—
167	C₈×C₆₇	—	4	1	no	—
168	C₈×C₆₇	—	4	2	no	—
169	C₈×C₆₇	—	8	1	no	—
170	C₆₇⋊C₈	—	2	2	no	—
171	C₆₇⋊C₈	—	4	1	no	—
172	C₆₇⋊C₈	—	4	2	no	—
173	C₆₇⋊C₈	—	8	1	no	—
174	C₂²×C₆₇	#385	1	1	yes	characteristic
175	C₂²×C₆₇	#423	1	1	yes	characteristic
176	C₂²×C₆₇	#424	1	1	yes	characteristic
177	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
178	C₂²×C₆₇	#428	1	1	yes	characteristic
179	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
180	C₂²×C₆₇	#430	1	1	yes	characteristic
181	C₂×D₆₇	—	2	2	no	—
182	C₂×D₆₇	—	2	2	no	—
183	C₂×D₆₇	—	2	2	no	—
184	C₂×D₆₇	—	4	1	no	—
185	C₂×D₆₇	—	4	1	no	—
186	C₂×D₆₇	—	4	1	no	—
187	C₂×D₆₇	—	4	1	no	—
188	C₄×C₆₇	—	2	1	no	—
189	C₄×C₆₇	—	2	1	no	—
190	C₄×C₆₇	#435	1	1	yes	characteristic
191	C₄×C₆₇	#435	1	1	yes	characteristic
192	C₄×C₆₇	#436	1	1	yes	characteristic
193	C₄×C₆₇	#436	1	1	yes	characteristic
194	C₄×C₆₇	#448	1	1	yes	characteristic
195	C₄×C₆₇	#450	1	1	yes	characteristic
196	C₄×C₆₇	#481	1	1	yes	characteristic
197	C₄×C₆₇	#483	1	1	yes	characteristic
198	Dic₆₇	—	2	2	no	—
199	Dic₆₇	—	2	2	no	—
200	Dic₆₇	—	4	1	no	—
201	#8833	—	67	1	no	maximal, Sylow
202	C₂×C₆₇	#1566	1	1	yes	characteristic
203	C₂×C₆₇	#1569	1	1	yes	characteristic
204	C₂×C₆₇	#1570	1	1	yes	characteristic
205	C₂×C₆₇	#1574	1	1	yes	characteristic
206	C₂×C₆₇	#1851	1	1	yes	characteristic
207	C₂×C₆₇	#1853	1	1	yes	characteristic
208	C₂×C₆₇	#1855	1	1	yes	characteristic
209	D₆₇	—	4	2	no	—
210	C₂×#423	—	67	1	no	—
211	#1797	—	67	1	no	—
212	C₄×C₂²⋊C₈	—	67	1	no	—
213	C₂×#334	—	67	1	no	—
214	C₂×#334	—	67	1	no	—
215	#1277	—	67	1	no	—
216	#1277	D₆₇	1	1	yes	characteristic, complemented
217	C₆₇	#8833	1	1	yes	characteristic, complemented, Sylow
218	C₂²×M₁₆	—	134	1	no	—
219	C₂²×C₄²	—	67	1	no	—
220	#423	—	134	2	no	—
221	#423	—	134	2	no	—
222	C₂×C₄⋊C₈	—	67	1	no	—
223	C₂×C₄⋊C₈	—	67	1	no	—
224	C₂×C₄⋊C₈	—	134	1	no	—
225	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
226	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
227	C₂×C₂²⋊C₈	—	67	1	no	—
228	C₂×C₂²⋊C₈	—	67	1	no	—
229	C₂×C₂²⋊C₈	—	134	1	no	—
230	C₂×C₄×C₈	—	67	1	no	—
231	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
232	#336	—	67	1	no	—
233	#336	—	67	1	no	—
234	#336	—	134	1	no	—
235	#334	—	134	2	no	—
236	#334	—	134	2	no	—
237	C₂³×C₄	—	67	1	no	—
238	C₂³×C₄	—	67	1	no	—
239	C₂³×C₄	—	67	1	no	—
240	C₂×M₁₆	—	268	2	no	—
241	C₂×M₁₆	—	268	2	no	—
242	C₂×M₁₆	—	134	4	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	2	1	no	—
245	C₂²×C₈	—	67	1	no	—
246	C₂²×C₈	—	67	1	no	—
247	C₂²×C₈	—	134	1	no	—
248	C₂²×C₈	—	134	1	no	—
249	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
250	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
251	C₂×C₄²	—	67	1	no	—
252	C₂×C₄²	—	67	1	no	—
253	C₂×C₄²	—	67	1	no	—
254	C₂×C₄²	—	134	1	no	—
255	C₂×C₄²	—	134	1	no	—
256	C₂×C₄²	—	134	1	no	—
257	C₂×C₄²	—	134	1	no	—
258	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	67	2	no	—
263	C₄⋊C₈	—	134	4	no	—
264	C₄⋊C₈	—	134	1	no	—
265	C₄⋊C₈	—	134	1	no	—
266	C₄⋊C₈	—	134	1	no	—
267	C₄⋊C₈	D₂₆₈	1	2	yes	—
268	C₂²⋊C₈	—	268	2	no	—
269	C₂²⋊C₈	—	134	2	no	—
270	C₂²⋊C₈	—	134	2	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	2	1	no	—
273	C₄×C₈	—	134	1	no	—
274	C₄×C₈	—	134	1	no	—
275	C₂⁴	—	67	1	no	—
276	C₂²×C₄	—	268	1	no	—
277	C₂²×C₄	—	67	1	no	—
278	C₂²×C₄	—	67	1	no	—
279	C₂²×C₄	—	67	2	no	—
280	C₂²×C₄	—	67	2	no	—
281	C₂²×C₄	—	67	1	no	—
282	C₂²×C₄	—	134	2	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	—	134	1	no	—
291	C₂²×C₄	—	134	1	no	—
292	C₂²×C₄	—	134	1	no	—
293	C₂²×C₄	—	134	1	no	—
294	C₂²×C₄	—	134	1	no	—
295	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
296	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
297	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
298	M₁₆	—	268	8	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	4	1	no	—
309	C₂×C₈	—	4	1	no	—
310	C₂×C₈	—	268	1	no	—
311	C₂×C₈	—	268	1	no	—
312	C₂×C₈	—	268	1	no	—
313	C₂×C₈	—	268	1	no	—
314	C₂×C₈	—	268	1	no	—
315	C₂×C₈	—	67	2	no	—
316	C₂×C₈	—	134	1	no	—
317	C₂×C₈	—	134	2	no	—
318	C₂×C₈	—	134	1	no	—
319	C₂×C₈	—	134	1	no	—
320	C₂×C₈	—	134	1	no	—
321	C₂×C₈	—	134	1	no	—
322	C₂×C₈	#11761611	1	2	yes	complemented
323	C₄²	—	134	1	no	—
324	C₄²	—	134	1	no	—
325	C₄²	—	134	1	no	—
326	C₄²	—	134	1	no	—
327	C₄²	—	134	1	no	—
328	C₄²	—	134	1	no	—
329	C₄²	D₄×D₆₇	1	1	yes	characteristic
330	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
331	C₄²	#11761627	1	1	yes	characteristic
332	C₄²	#11761628	1	1	yes	characteristic
333	C₂³	—	67	2	no	—
334	C₂³	—	134	1	no	—
335	C₂³	—	134	1	no	—
336	C₂³	—	134	1	no	—
337	C₂³	—	134	1	no	—
338	C₂³	—	134	1	no	—
339	C₂³	—	134	1	no	—
340	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	268	2	no	—
344	C₂×C₄	—	268	2	no	—
345	C₂×C₄	—	268	1	no	—
346	C₂×C₄	—	268	1	no	—
347	C₂×C₄	—	268	1	no	—
348	C₂×C₄	—	268	1	no	—
349	C₂×C₄	—	67	2	no	—
350	C₂×C₄	—	67	2	no	—
351	C₂×C₄	—	134	1	no	—
352	C₂×C₄	—	134	1	no	—
353	C₂×C₄	—	134	1	no	—
354	C₂×C₄	—	134	1	no	—
355	C₂×C₄	—	134	1	no	—
356	C₂×C₄	—	134	1	no	—
357	C₂×C₄	—	134	1	no	—
358	C₂×C₄	—	134	2	no	—
359	C₂×C₄	—	134	2	no	—
360	C₂×C₄	—	134	2	no	—
361	C₂×C₄	—	134	2	no	—
362	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
363	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
364	C₂×C₄	#14524013	1	1	yes	characteristic
365	C₂×C₄	#14524021	1	1	yes	characteristic
366	C₂×C₄	#14524023	1	1	yes	characteristic
367	C₂×C₄	#14524025	1	1	yes	characteristic
368	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
369	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
370	C₂×C₄	#14524049	1	1	yes	characteristic
371	C₂×C₄	#14524051	1	1	yes	characteristic
372	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
373	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
374	C₂×C₄	#14524067	1	1	yes	characteristic
375	C₂×C₄	#14524075	1	1	yes	characteristic
376	C₈	—	2	2	no	—
377	C₈	—	4	2	no	—
378	C₈	—	4	1	no	—
379	C₈	—	8	1	no	—
380	C₈	—	268	2	no	—
381	C₈	—	268	1	no	—
382	C₈	—	536	1	no	—
383	C₈	—	134	2	no	—
384	C₂²	—	268	1	no	—
385	C₂²	—	268	1	no	—
386	C₂²	—	268	1	no	—
387	C₂²	—	268	1	no	—
388	C₂²	—	134	2	no	—
389	C₂²	—	134	2	no	—
390	C₂²	—	134	2	no	—
391	C₂²	#16863270	1	1	yes	characteristic
392	C₂²	#16863418	1	1	yes	characteristic
393	C₂²	#16863421	1	1	yes	characteristic
394	C₂²	#106×D₆₇	1	1	yes	characteristic
395	C₂²	#16863437	1	1	yes	characteristic
396	C₂²	#108×D₆₇	1	1	yes	characteristic
397	C₂²	#16863443	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	268	1	no	—
401	C₄	—	134	2	no	—
402	C₄	—	134	2	no	—
403	C₄	#16863464	1	1	yes	characteristic
404	C₄	#16863469	1	1	yes	characteristic
405	C₄	#16863471	1	1	yes	characteristic
406	C₄	#16863475	1	1	yes	characteristic
407	C₄	#16863536	1	1	yes	characteristic
408	C₄	#16863548	1	1	yes	characteristic
409	C₄	#16863713	1	1	yes	characteristic
410	C₄	#16863725	1	1	yes	characteristic
411	C₂	—	268	2	no	—
412	C₂	#22558963	1	1	yes	characteristic
413	C₂	#22558975	1	1	yes	characteristic
414	C₂	#22558978	1	1	yes	characteristic
415	C₂	#22558999	1	1	yes	characteristic
416	C₂	#22560393	1	1	yes	characteristic
417	C₂	#22560403	1	1	yes	characteristic
418	C₂	#22560415	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
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	#8833
LCS	G ▷ C₂²×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×#16863147	C₂	semidirect
5. C₂×#16863146	C₂	semidirect
6. #22558043	C₂	semidirect
7. C₆₇×#1277	C₂	semidirect
97. C₂×C₆₇⋊C₈	M₁₆	semidirect
118. C₂×C₈×C₆₇	M₁₆	semidirect
216. #1277	D₆₇	semidirect
217. C₆₇	#8833	semidirect
322. C₂×C₈	#11761611	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863418	2
#22560186	2
C₄×(C₂×D₆₇)⋊C₈	2
C₂×#16863147	2
C₂×#16863146	2
#22558043	2
C₆₇×#1277	2
#8833	67