Subgroups — #30748673

Order 17152.

546 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11761611	C₂	1	1	yes	maximal, characteristic
2	#22560186	C₂	1	1	yes	maximal, characteristic
3	#22560186	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863148	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863146	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558044	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761611	C₄	1	1	yes	characteristic
9	C₂²×#11761611	C₄	1	1	yes	characteristic
10	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
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	C₄×#11761611	—	2	2	no	—
16	C₄×#11761611	—	2	2	no	—
17	C₂×#14523953	C₄	1	1	yes	characteristic
18	C₂×#14523953	C₄	1	1	yes	characteristic
19	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	#16863150	—	2	1	no	—
21	#16863150	—	2	1	no	—
22	#16863148	—	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	2	no	—
31	C₂×#11761611	—	2	2	no	—
32	C₂×#11761611	—	2	2	no	—
33	C₂×#11761611	—	2	2	no	—
34	C₂×#11761611	—	2	2	no	—
35	C₂×#11761611	C₈	1	4	yes	complemented
36	C₂²×C₆₇⋊C₈	—	2	1	no	—
37	C₂²×C₆₇⋊C₈	—	2	1	no	—
38	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
43	C₄²×D₆₇	—	2	1	no	—
44	C₄²×D₆₇	—	2	1	no	—
45	C₄²×D₆₇	—	2	1	no	—
46	C₄²×D₆₇	—	2	1	no	—
47	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
48	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
49	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
50	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
51	(C₂×D₆₇)⋊C₈	—	4	2	no	—
52	(C₂×D₆₇)⋊C₈	—	4	2	no	—
53	#14523953	—	2	2	no	—
54	#14523953	—	2	2	no	—
55	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
56	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
57	C₂²×C₈×C₆₇	—	2	1	no	—
58	C₂²×C₈×C₆₇	—	2	1	no	—
59	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₆₇	—	2	1	no	—
63	C₄⋊C₈×C₆₇	—	2	1	no	—
64	C₄⋊C₈×C₆₇	—	2	1	no	—
65	C₄⋊C₈×C₆₇	—	2	1	no	—
66	C₄×C₈×C₆₇	—	2	1	no	—
67	C₄×C₈×C₆₇	—	2	1	no	—
68	C₂³×D₆₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
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	1	no	—
82	C₂×C₄×D₆₇	—	2	1	no	—
83	C₂×C₄×D₆₇	—	2	1	no	—
84	C₂×C₄×D₆₇	—	4	1	no	—
85	C₂×C₄×D₆₇	—	4	1	no	—
86	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
87	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
88	#11761611	—	2	8	no	—
89	#11761611	—	2	8	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	—	4	1	no	—
98	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
99	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
100	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
101	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
102	C₄×Dic₆₇	—	2	1	no	—
103	C₄×Dic₆₇	—	2	1	no	—
104	C₄×Dic₆₇	—	2	1	no	—
105	C₄×Dic₆₇	—	2	1	no	—
106	C₄×Dic₆₇	—	2	1	no	—
107	C₄×Dic₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₈×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	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	—	4	1	no	—
119	C₂×C₈×C₆₇	—	4	1	no	—
120	C₂×C₈×C₆₇	—	4	1	no	—
121	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
122	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
123	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₆₇	#41	1	1	yes	characteristic
126	C₄²×C₆₇	#41	1	1	yes	characteristic
127	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×D₆₇	—	2	1	no	—
129	C₂²×D₆₇	—	2	1	no	—
130	C₂²×D₆₇	—	2	1	no	—
131	C₂²×D₆₇	—	2	1	no	—
132	C₂²×D₆₇	—	2	1	no	—
133	C₂²×D₆₇	—	2	1	no	—
134	C₂²×D₆₇	C₄⋊C₈	1	2	yes	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
139	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
147	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
151	C₂×Dic₆₇	—	2	1	no	—
152	C₂×Dic₆₇	—	2	1	no	—
153	C₂×Dic₆₇	—	2	1	no	—
154	C₂×Dic₆₇	—	2	1	no	—
155	C₂×Dic₆₇	—	2	1	no	—
156	C₂×Dic₆₇	—	2	1	no	—
157	C₂×Dic₆₇	—	2	1	no	—
158	C₂×Dic₆₇	—	2	1	no	—
159	C₂×Dic₆₇	C₄⋊C₈	1	2	yes	—
160	C₄×D₆₇	—	2	2	no	—
161	C₄×D₆₇	—	2	2	no	—
162	C₄×D₆₇	—	2	2	no	—
163	C₄×D₆₇	—	2	2	no	—
164	C₄×D₆₇	—	4	2	no	—
165	C₄×D₆₇	—	4	2	no	—
166	C₄×D₆₇	—	4	1	no	—
167	C₄×D₆₇	—	4	1	no	—
168	C₄×D₆₇	—	4	1	no	—
169	C₄×D₆₇	—	4	1	no	—
170	C₈×C₆₇	—	2	2	no	—
171	C₈×C₆₇	—	2	2	no	—
172	C₈×C₆₇	—	8	1	no	—
173	C₈×C₆₇	—	8	1	no	—
174	C₆₇⋊C₈	—	2	2	no	—
175	C₆₇⋊C₈	—	8	1	no	—
176	C₆₇⋊C₈	—	8	1	no	—
177	C₆₇⋊C₈	C₂²⋊C₈	1	4	yes	complemented
178	C₂²×C₆₇	#380	1	1	yes	characteristic
179	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
180	C₂²×C₆₇	#423	1	1	yes	characteristic
181	C₂²×C₆₇	#428	1	1	yes	characteristic
182	C₂²×C₆₇	#428	1	1	yes	characteristic
183	C₂²×C₆₇	#430	1	1	yes	characteristic
184	C₂²×C₆₇	#430	1	1	yes	characteristic
185	C₂×D₆₇	—	2	2	no	—
186	C₂×D₆₇	—	2	2	no	—
187	C₂×D₆₇	—	2	2	no	—
188	C₂×D₆₇	—	4	1	no	—
189	C₂×D₆₇	—	4	1	no	—
190	C₂×D₆₇	—	4	1	no	—
191	C₂×D₆₇	—	4	1	no	—
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	2	1	no	—
194	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
195	C₄×C₆₇	#407	1	1	yes	characteristic
196	C₄×C₆₇	#431	1	1	yes	characteristic
197	C₄×C₆₇	#432	1	1	yes	characteristic
198	C₄×C₆₇	#468	1	1	yes	characteristic
199	C₄×C₆₇	#468	1	1	yes	characteristic
200	C₄×C₆₇	#469	1	1	yes	characteristic
201	C₄×C₆₇	#470	1	1	yes	characteristic
202	Dic₆₇	—	2	2	no	—
203	Dic₆₇	—	4	1	no	—
204	Dic₆₇	—	4	1	no	—
205	#8597	—	67	1	no	maximal, Sylow
206	C₂×C₆₇	#1565	1	1	yes	characteristic
207	C₂×C₆₇	#1565	1	1	yes	characteristic
208	C₂×C₆₇	#1569	1	1	yes	characteristic
209	C₂×C₆₇	#1569	1	1	yes	characteristic
210	C₂×C₆₇	#1797	1	1	yes	characteristic
211	C₂×C₆₇	#1809	1	1	yes	characteristic
212	C₂×C₆₇	#1811	1	1	yes	characteristic
213	D₆₇	—	4	2	no	—
214	C₂×C₄×M₁₆	—	67	1	no	—
215	#1797	—	67	1	no	—
216	#1797	—	67	1	no	—
217	C₂×#335	—	67	1	no	—
218	C₂×#334	—	67	1	no	—
219	#1277	—	67	1	no	—
220	#1277	D₆₇	1	1	yes	characteristic, complemented
221	C₆₇	#8597	1	1	yes	characteristic, complemented, Sylow
222	C₂²×M₁₆	—	67	1	no	—
223	C₂²×M₁₆	—	67	1	no	—
224	C₂²×C₄²	—	67	1	no	—
225	C₂×C₄⋊C₈	—	67	1	no	—
226	C₂×C₄⋊C₈	—	67	1	no	—
227	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
228	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
229	C₂×C₂²⋊C₈	—	134	1	no	—
230	C₂×C₂²⋊C₈	—	134	1	no	—
231	C₄×M₁₆	—	134	2	no	—
232	C₄×M₁₆	—	134	2	no	—
233	C₂×#97	—	67	1	no	—
234	C₂×#97	—	67	1	no	—
235	C₂×C₄×C₈	—	67	1	no	—
236	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
237	#336	—	134	1	no	—
238	#336	—	134	1	no	—
239	#335	—	134	2	no	—
240	#334	—	134	2	no	—
241	C₂³×C₄	—	67	1	no	—
242	C₂³×C₄	—	67	1	no	—
243	C₂³×C₄	—	67	1	no	—
244	C₂×M₁₆	—	67	4	no	—
245	C₂×M₁₆	—	134	2	no	—
246	C₂×M₁₆	—	134	2	no	—
247	C₂×M₁₆	—	134	2	no	—
248	C₂×M₁₆	—	134	2	no	—
249	C₂×M₁₆	—	134	2	no	—
250	C₂²×C₈	—	2	1	no	—
251	C₂²×C₈	—	2	1	no	—
252	C₂²×C₈	—	67	1	no	—
253	C₂²×C₈	—	67	1	no	—
254	C₂²×C₈	—	134	1	no	—
255	C₂²×C₈	—	134	1	no	—
256	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
257	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
258	C₂×C₄²	—	67	1	no	—
259	C₂×C₄²	—	67	1	no	—
260	C₂×C₄²	—	67	1	no	—
261	C₂×C₄²	—	134	1	no	—
262	C₂×C₄²	—	134	1	no	—
263	C₂×C₄²	—	134	1	no	—
264	C₂×C₄²	—	134	1	no	—
265	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	134	1	no	—
271	C₄⋊C₈	—	134	1	no	—
272	C₄⋊C₈	—	134	1	no	—
273	C₄⋊C₈	—	134	1	no	—
274	C₂²⋊C₈	—	268	2	no	—
275	C₂²⋊C₈	—	268	2	no	—
276	#97	—	134	2	no	—
277	#97	—	134	2	no	—
278	C₄×C₈	—	2	1	no	—
279	C₄×C₈	—	2	1	no	—
280	C₄×C₈	—	67	2	no	—
281	C₄×C₈	—	67	2	no	—
282	C₂⁴	—	67	1	no	—
283	C₂²×C₄	—	268	1	no	—
284	C₂²×C₄	—	268	1	no	—
285	C₂²×C₄	—	67	1	no	—
286	C₂²×C₄	—	67	2	no	—
287	C₂²×C₄	—	67	2	no	—
288	C₂²×C₄	—	67	1	no	—
289	C₂²×C₄	—	67	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	—	134	1	no	—
296	C₂²×C₄	—	134	1	no	—
297	C₂²×C₄	—	134	1	no	—
298	C₂²×C₄	—	134	1	no	—
299	C₂²×C₄	—	134	1	no	—
300	C₂²×C₄	—	134	1	no	—
301	C₂²×C₄	—	134	1	no	—
302	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
303	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
304	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
305	M₁₆	—	134	8	no	—
306	M₁₆	—	134	8	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	4	1	no	—
312	C₂×C₈	—	4	1	no	—
313	C₂×C₈	—	4	1	no	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	4	1	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	268	1	no	—
318	C₂×C₈	—	268	1	no	—
319	C₂×C₈	—	268	1	no	—
320	C₂×C₈	—	268	1	no	—
321	C₂×C₈	—	268	1	no	—
322	C₂×C₈	—	268	1	no	—
323	C₂×C₈	—	67	2	no	—
324	C₂×C₈	—	67	2	no	—
325	C₂×C₈	—	67	2	no	—
326	C₂×C₈	—	67	2	no	—
327	C₂×C₈	—	134	1	no	—
328	C₂×C₈	—	134	1	no	—
329	C₂×C₈	C₈×D₆₇	1	2	yes	—
330	C₂×C₈	#11761611	1	2	yes	—
331	C₄²	—	134	1	no	—
332	C₄²	—	134	1	no	—
333	C₄²	—	134	1	no	—
334	C₄²	—	134	1	no	—
335	C₄²	—	134	1	no	—
336	C₄²	—	134	1	no	—
337	C₄²	D₄×D₆₇	1	1	yes	characteristic
338	C₄²	D₄×D₆₇	1	1	yes	characteristic
339	C₄²	#11761628	1	1	yes	characteristic
340	C₄²	#11761628	1	1	yes	characteristic
341	C₂³	—	67	2	no	—
342	C₂³	—	134	1	no	—
343	C₂³	—	134	1	no	—
344	C₂³	—	134	1	no	—
345	C₂³	—	134	1	no	—
346	C₂³	—	134	1	no	—
347	C₂³	—	134	1	no	—
348	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	268	2	no	—
352	C₂×C₄	—	268	2	no	—
353	C₂×C₄	—	268	1	no	—
354	C₂×C₄	—	268	1	no	—
355	C₂×C₄	—	268	1	no	—
356	C₂×C₄	—	268	1	no	—
357	C₂×C₄	—	67	2	no	—
358	C₂×C₄	—	134	1	no	—
359	C₂×C₄	—	134	1	no	—
360	C₂×C₄	—	134	1	no	—
361	C₂×C₄	—	134	1	no	—
362	C₂×C₄	—	134	1	no	—
363	C₂×C₄	—	134	1	no	—
364	C₂×C₄	—	134	1	no	—
365	C₂×C₄	—	134	1	no	—
366	C₂×C₄	—	134	2	no	—
367	C₂×C₄	—	134	2	no	—
368	C₂×C₄	—	134	2	no	—
369	C₂×C₄	—	134	2	no	—
370	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
371	C₂×C₄	#14524000	1	1	yes	characteristic
372	C₂×C₄	#14524016	1	1	yes	characteristic
373	C₂×C₄	#14524016	1	1	yes	characteristic
374	C₂×C₄	#14524021	1	1	yes	characteristic
375	C₂×C₄	#14524021	1	1	yes	characteristic
376	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
377	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
378	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
379	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
380	C₂×C₄	#14524067	1	1	yes	characteristic
381	C₂×C₄	#14524070	1	1	yes	characteristic
382	C₂×C₄	#14524072	1	1	yes	characteristic
383	C₂×C₄	#14524075	1	1	yes	characteristic
384	C₈	—	2	2	no	—
385	C₈	—	2	2	no	—
386	C₈	—	8	1	no	—
387	C₈	—	8	1	no	—
388	C₈	—	536	1	no	—
389	C₈	—	536	1	no	—
390	C₈	—	67	4	no	—
391	C₈	—	134	2	no	—
392	C₂²	—	268	1	no	—
393	C₂²	—	268	1	no	—
394	C₂²	—	268	1	no	—
395	C₂²	—	268	1	no	—
396	C₂²	—	134	2	no	—
397	C₂²	—	134	2	no	—
398	C₂²	—	134	2	no	—
399	C₂²	#16863255	1	1	yes	characteristic
400	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
401	C₂²	#16863418	1	1	yes	characteristic
402	C₂²	#16863435	1	1	yes	characteristic
403	C₂²	#16863437	1	1	yes	characteristic
404	C₂²	#16863443	1	1	yes	characteristic
405	C₂²	#16863443	1	1	yes	characteristic
406	C₄	—	2	1	no	—
407	C₄	—	2	1	no	—
408	C₄	—	268	1	no	—
409	C₄	—	268	1	no	—
410	C₄	—	134	2	no	—
411	C₄	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
412	C₄	#16863358	1	1	yes	characteristic
413	C₄	#16863447	1	1	yes	characteristic
414	C₄	#16863451	1	1	yes	characteristic
415	C₄	#16863641	1	1	yes	characteristic
416	C₄	#16863645	1	1	yes	characteristic
417	C₄	#16863648	1	1	yes	characteristic
418	C₄	#16863653	1	1	yes	characteristic
419	C₂	—	268	2	no	—
420	C₂	#22558958	1	1	yes	characteristic
421	C₂	#22558960	1	1	yes	characteristic
422	C₂	#22558973	1	1	yes	characteristic
423	C₂	#22558975	1	1	yes	characteristic
424	C₂	#22560188	1	1	yes	characteristic
425	C₂	#22560241	1	1	yes	characteristic
426	C₂	#22560250	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	#8597
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×#16863148	C₂	semidirect
5. C₂×#16863146	C₂	semidirect
6. #22558044	C₂	semidirect
7. C₆₇×#1277	C₂	semidirect
35. C₂×#11761611	C₈	semidirect
177. C₆₇⋊C₈	C₂²⋊C₈	semidirect
220. #1277	D₆₇	semidirect
221. C₆₇	#8597	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11761611	2
#22560186	2
C₂×#16863148	2
C₂×#16863146	2
#22558044	2
C₆₇×#1277	2
#8597	67