Subgroups — C₆₇×#19955 #30699206

Order 17152.

636 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3393	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3353	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3283	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1693	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1670	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1669	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1650	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1647	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	—	2	1	no	—
13	#556×C₆₇	C₂²	1	2	yes	—
14	#547×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	C₂²	1	2	yes	—
16	#500×C₆₇	C₂²	1	2	yes	—
17	#495×C₆₇	—	2	1	no	—
18	#494×C₆₇	C₂²	1	1	yes	characteristic
19	#494×C₆₇	C₂²	1	1	yes	characteristic
20	#484×C₆₇	—	2	2	no	—
21	#482×C₆₇	—	2	2	no	—
22	#475×C₆₇	C₂²	1	2	yes	—
23	#474×C₆₇	C₂²	1	2	yes	—
24	#468×C₆₇	—	2	2	no	—
25	#467×C₆₇	—	2	1	no	—
26	#464×C₆₇	—	2	2	no	—
27	#461×C₆₇	C₂²	1	2	yes	—
28	#461×C₆₇	C₂²	1	2	yes	—
29	#452×C₆₇	—	2	2	no	—
30	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₄×Q₁₆×C₆₇	—	2	1	no	—
32	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
34	#436×C₆₇	C₂²	1	2	yes	—
35	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	#428×C₆₇	—	2	2	no	—
37	#335×C₆₇	C₂²	1	1	yes	characteristic
38	#334×C₆₇	C₂²	1	1	yes	characteristic
39	#332×C₆₇	C₂²	1	2	yes	—
40	#332×C₆₇	C₂²	1	2	yes	—
41	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#41×C₆₇	—	2	2	no	—
46	#135×C₆₇	—	4	4	no	—
47	C₂×Q₁₆×C₆₇	—	2	1	no	—
48	C₂×Q₁₆×C₆₇	—	2	1	no	—
49	C₂×Q₁₆×C₆₇	—	2	1	no	—
50	C₂×Q₁₆×C₆₇	—	2	1	no	—
51	C₂×SD₁₆×C₆₇	—	2	2	no	—
52	C₂×SD₁₆×C₆₇	—	2	2	no	—
53	C₂×SD₁₆×C₆₇	D₄	1	1	yes	characteristic
54	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
58	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
59	#126×C₆₇	—	4	2	no	—
60	#124×C₆₇	—	2	2	no	—
61	#123×C₆₇	—	4	2	no	—
62	C₂²⋊Q₈×C₆₇	—	2	2	no	—
63	C₂²⋊Q₈×C₆₇	—	2	2	no	—
64	#121×C₆₇	—	2	2	no	—
65	C₄×Q₈×C₆₇	—	2	2	no	—
66	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₆₇	C₂³	1	2	yes	—
68	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
69	#117×C₆₇	—	2	2	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
71	#107×C₆₇	—	2	2	no	—
72	#107×C₆₇	—	2	2	no	—
73	#107×C₆₇	—	2	2	no	—
74	#107×C₆₇	—	4	2	no	—
75	#107×C₆₇	D₄	1	1	yes	characteristic
76	#106×C₆₇	—	2	2	no	—
77	#106×C₆₇	D₄	1	1	yes	characteristic
78	C₄⋊C₈×C₆₇	—	2	1	no	—
79	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	#103×C₆₇	—	2	2	no	—
85	#103×C₆₇	D₄	1	1	yes	characteristic
86	#102×C₆₇	—	2	2	no	—
87	#102×C₆₇	D₄	1	2	yes	—
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₂²⋊C₈×C₆₇	—	2	2	no	—
91	C₄×C₈×C₆₇	—	2	1	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	#41×C₆₇	—	4	4	no	—
96	#41×C₆₇	—	8	2	no	—
97	C₂×Q₈×C₆₇	—	2	2	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	4	2	no	—
106	Q₁₆×C₆₇	—	4	2	no	—
107	Q₁₆×C₆₇	—	4	2	no	—
108	Q₁₆×C₆₇	—	8	1	no	—
109	Q₁₆×C₆₇	—	8	1	no	—
110	SD₁₆×C₆₇	—	2	2	no	—
111	SD₁₆×C₆₇	—	4	4	no	—
112	SD₁₆×C₆₇	—	8	2	no	—
113	D₈×C₆₇	—	2	2	no	—
114	M₁₆×C₆₇	—	4	4	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	4	2	no	—
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	4	2	no	—
130	C₄⋊C₄×C₆₇	—	4	2	no	—
131	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²⋊C₄×C₆₇	—	2	1	no	—
135	C₂²⋊C₄×C₆₇	—	2	2	no	—
136	C₂²⋊C₄×C₆₇	—	4	2	no	—
137	C₂²⋊C₄×C₆₇	—	4	2	no	—
138	C₂²⋊C₄×C₆₇	—	4	2	no	—
139	C₄²×C₆₇	—	2	1	no	—
140	C₄²×C₆₇	—	4	2	no	—
141	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	—	2	1	no	—
144	Q₈×C₆₇	—	2	1	no	—
145	Q₈×C₆₇	—	2	1	no	—
146	Q₈×C₆₇	—	4	2	no	—
147	Q₈×C₆₇	—	4	2	no	—
148	Q₈×C₆₇	—	4	2	no	—
149	Q₈×C₆₇	—	4	2	no	—
150	D₄×C₆₇	—	2	1	no	—
151	D₄×C₆₇	—	2	2	no	—
152	D₄×C₆₇	—	2	2	no	—
153	D₄×C₆₇	—	2	1	no	—
154	D₄×C₆₇	—	8	4	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	1	no	—
161	C₂×C₄×C₆₇	—	4	1	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	8	4	no	—
166	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
169	C₈×C₆₇	—	2	2	no	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	—	2	1	no	—
172	C₈×C₆₇	—	4	2	no	—
173	C₈×C₆₇	—	4	1	no	—
174	C₈×C₆₇	—	4	1	no	—
175	C₈×C₆₇	#135	1	2	yes	complemented
176	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
177	C₂²×C₆₇	—	4	1	no	—
178	C₂²×C₆₇	—	4	2	no	—
179	C₂²×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	4	1	no	—
181	C₂²×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	—	4	1	no	—
183	C₂²×C₆₇	D₄²	1	1	yes	characteristic
184	C₄×C₆₇	—	2	1	no	—
185	C₄×C₆₇	—	4	1	no	—
186	C₄×C₆₇	—	4	2	no	—
187	C₄×C₆₇	—	4	1	no	—
188	C₄×C₆₇	—	4	1	no	—
189	C₄×C₆₇	—	8	2	no	—
190	C₄×C₆₇	—	8	2	no	—
191	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
192	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₄×C₆₇	#576	1	1	yes	characteristic
194	C₄×C₆₇	#578	1	1	yes	characteristic
195	#19955	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂×C₆₇	—	8	1	no	—
197	C₂×C₆₇	—	8	2	no	—
198	C₂×C₆₇	#3274	1	1	yes	characteristic
199	C₂×C₆₇	#3283	1	1	yes	characteristic
200	C₂×C₆₇	#3293	1	1	yes	characteristic
201	#3393	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#3353	C₂×C₆₇	1	2	yes	complemented
203	#3283	C₂×C₆₇	1	1	yes	characteristic
204	#1713	C₂×C₆₇	1	1	yes	characteristic, complemented
205	#1693	C₂×C₆₇	1	2	yes	complemented
206	#1670	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#1669	C₂×C₆₇	1	1	yes	characteristic, complemented
208	#1650	C₂×C₆₇	1	2	yes	complemented
209	#1647	C₂×C₆₇	1	2	yes	complemented
210	#1589	C₂×C₆₇	1	1	yes	characteristic, complemented
211	C₈×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
212	C₆₇	#19955	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×#135	—	2	1	no	—
214	#556	C₂²×C₆₇	1	2	yes	—
215	#547	C₂²×C₆₇	1	2	yes	—
216	#500	C₂²×C₆₇	1	2	yes	—
217	#500	C₂²×C₆₇	1	2	yes	—
218	#495	—	2	1	no	—
219	#494	C₂²×C₆₇	1	1	yes	characteristic
220	#494	C₂²×C₆₇	1	1	yes	characteristic
221	#484	—	2	2	no	—
222	#482	—	2	2	no	—
223	#475	C₂²×C₆₇	1	2	yes	—
224	#474	C₂²×C₆₇	1	2	yes	—
225	#468	—	2	2	no	—
226	#467	—	2	1	no	—
227	#464	—	2	2	no	—
228	#461	C₂²×C₆₇	1	2	yes	—
229	#461	C₂²×C₆₇	1	2	yes	—
230	#452	—	2	2	no	—
231	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
232	C₄×Q₁₆	—	2	1	no	—
233	C₄×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
234	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
235	#436	C₂²×C₆₇	1	2	yes	—
236	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
237	#428	—	2	2	no	—
238	#335	C₂²×C₆₇	1	1	yes	characteristic
239	#334	C₂²×C₆₇	1	1	yes	characteristic
240	#332	C₂²×C₆₇	1	2	yes	—
241	#332	C₂²×C₆₇	1	2	yes	—
242	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
243	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
244	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
245	C₈²	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×#41	—	2	2	no	—
247	#135	—	4	4	no	—
248	C₂×Q₁₆	—	2	1	no	—
249	C₂×Q₁₆	—	2	1	no	—
250	C₂×Q₁₆	—	2	1	no	—
251	C₂×Q₁₆	—	2	1	no	—
252	C₂×SD₁₆	—	2	2	no	—
253	C₂×SD₁₆	—	2	2	no	—
254	C₂×SD₁₆	D₄×C₆₇	1	1	yes	characteristic
255	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
256	C₂×M₁₆	—	2	2	no	—
257	C₂²×C₈	—	2	1	no	—
258	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
259	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
260	#126	—	4	2	no	—
261	#124	—	2	2	no	—
262	#123	—	4	2	no	—
263	C₂²⋊Q₈	—	2	2	no	—
264	C₂²⋊Q₈	—	2	2	no	—
265	#121	—	2	2	no	—
266	C₄×Q₈	—	2	2	no	—
267	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
268	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
269	C₄×D₄	C₂³×C₆₇	1	2	yes	—
270	#117	—	2	2	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	#107	—	2	2	no	—
273	#107	—	2	2	no	—
274	#107	—	2	2	no	—
275	#107	—	4	2	no	—
276	#107	D₄×C₆₇	1	1	yes	characteristic
277	#106	—	2	2	no	—
278	#106	D₄×C₆₇	1	1	yes	characteristic
279	C₄⋊C₈	—	2	1	no	—
280	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
281	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
282	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
283	#103	—	2	2	no	—
284	#103	—	2	2	no	—
285	#103	—	2	2	no	—
286	#103	D₄×C₆₇	1	1	yes	characteristic
287	#102	—	2	2	no	—
288	#102	D₄×C₆₇	1	2	yes	—
289	#102	D₄×C₆₇	1	1	yes	characteristic
290	C₂²⋊C₈	—	2	1	no	—
291	C₂²⋊C₈	—	2	2	no	—
292	C₄×C₈	—	2	1	no	—
293	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
294	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
295	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
296	#41	—	4	4	no	—
297	#41	—	8	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
301	C₂×D₄	—	4	2	no	—
302	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
303	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	—	4	2	no	—
307	Q₁₆	—	4	2	no	—
308	Q₁₆	—	4	2	no	—
309	Q₁₆	—	8	1	no	—
310	Q₁₆	—	8	1	no	—
311	SD₁₆	—	2	2	no	—
312	SD₁₆	—	4	4	no	—
313	SD₁₆	—	8	2	no	—
314	D₈	—	2	2	no	—
315	M₁₆	—	4	4	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	4	2	no	—
321	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
322	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
323	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
324	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	4	2	no	—
331	C₄⋊C₄	—	4	2	no	—
332	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
333	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂²⋊C₄	—	2	1	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₂²⋊C₄	—	4	2	no	—
338	C₂²⋊C₄	—	4	2	no	—
339	C₂²⋊C₄	—	4	2	no	—
340	C₄²	—	2	1	no	—
341	C₄²	—	4	2	no	—
342	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
343	C₂³	—	2	2	no	—
344	C₂³	—	2	1	no	—
345	Q₈	—	2	1	no	—
346	Q₈	—	2	1	no	—
347	Q₈	—	4	2	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	2	no	—
351	D₄	—	2	1	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	2	no	—
354	D₄	—	2	1	no	—
355	D₄	—	8	4	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	4	no	—
367	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
369	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	4	2	no	—
374	C₈	—	4	1	no	—
375	C₈	—	4	1	no	—
376	C₈	#135×C₆₇	1	2	yes	complemented
377	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
378	C₂²	—	4	1	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	4	1	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	4	1	no	—
384	C₂²	D₄²×C₆₇	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	2	no	—
388	C₄	—	4	1	no	—
389	C₄	—	4	1	no	—
390	C₄	—	8	2	no	—
391	C₄	—	8	2	no	—
392	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
393	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
394	C₄	#576×C₆₇	1	1	yes	characteristic
395	C₄	#578×C₆₇	1	1	yes	characteristic
396	C₂	—	8	1	no	—
397	C₂	—	8	2	no	—
398	C₂	C₆₇×#3274	1	1	yes	characteristic
399	C₂	C₆₇×#3283	1	1	yes	characteristic
400	C₂	C₆₇×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3283	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3393	C₂	semidirect
2. C₆₇×#3353	C₂	semidirect
4. C₆₇×#1713	C₂	semidirect
5. C₆₇×#1693	C₂	semidirect
6. C₆₇×#1670	C₂	semidirect
7. C₆₇×#1669	C₂	semidirect
8. C₆₇×#1650	C₂	semidirect
9. C₆₇×#1647	C₂	semidirect
10. C₆₇×#1589	C₂	semidirect
11. C₈×SD₁₆×C₆₇	C₂	semidirect
175. C₈×C₆₇	#135	semidirect
195. #19955	C₆₇	direct (N × H)
201. #3393	C₂×C₆₇	semidirect
202. #3353	C₂×C₆₇	semidirect
204. #1713	C₂×C₆₇	semidirect
205. #1693	C₂×C₆₇	semidirect
206. #1670	C₂×C₆₇	semidirect
207. #1669	C₂×C₆₇	semidirect
208. #1650	C₂×C₆₇	semidirect
209. #1647	C₂×C₆₇	semidirect
210. #1589	C₂×C₆₇	semidirect
211. C₈×SD₁₆	C₂×C₆₇	semidirect
212. C₆₇	#19955	direct (N × H)
376. C₈	#135×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3393	2
C₆₇×#3353	2
C₆₇×#3283	2
C₆₇×#1713	2
C₆₇×#1693	2
C₆₇×#1670	2
C₆₇×#1669	2
C₆₇×#1650	2
C₆₇×#1647	2
C₆₇×#1589	2
C₈×SD₁₆×C₆₇	2
#19955	67

Subgroups — C67×#19955 #30699206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19955 #30699206